Chapter 15 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Fluid Pressure in a U-Tube A U-tube is filled with water, and the two arms are capped. The tube is cylindrical, and the right arm has twice the radius of the left arm. The caps have negligible mass, are watertight, and can freely slide up and down the tube. Part A A one-inch depth of sand is poured onto the cap on each arm. After the caps have moved (if necessary) to reestablish equilibrium, is the right cap higher, lower, or the same height as the left cap? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Pressure in the Ocean The pressure at 10 below the surface of the ocean is about 2.00×105 . Part A higher lower the same height m Pa Which of the following statements is true? You did not open hints for this part. ANSWER: Part B Now consider the pressure 20 below the surface of the ocean. Which of the following statements is true? You did not open hints for this part. ANSWER: Relating Pressure and Height in a Container Learning Goal: To understand the derivation of the law relating height and pressure in a container. The weight of a column of seawater 1 in cross section and 10 high is about 2.00×105 . The weight of a column of seawater 1 in cross section and 10 high plus the weight of a column of air with the same cross section extending up to the top of the atmosphere is about 2.00×105 . The weight of 1 of seawater at 10 below the surface of the ocean is about 2.00×105 . The density of seawater is about 2.00×105 times the density of air at sea level. m2 m N m2 m N m3 m N m The pressure is twice that at a depth of 10 . The pressure is the same as that at a depth of 10 . The pressure is equal to that at a depth of 10 plus the weight per 1 cross sectional area of a column of seawater 10 high. The pressure is equal to the weight per 1 cross sectional area of a column of seawater 20 high. m m m m2 m m2 m In this problem, you will derive the law relating pressure to height in a container by analyzing a particular system. A container of uniform cross-sectional area is filled with liquid of uniform density . Consider a thin horizontal layer of liquid (thickness ) at a height as measured from the bottom of the container. Let the pressure exerted upward on the bottom of the layer be and the pressure exerted downward on the top be . Assume throughout the problem that the system is in equilibrium (the container has not been recently shaken or moved, etc.). Part A What is , the magnitude of the force exerted upward on the bottom of the liquid? You did not open hints for this part. ANSWER: Part B What is , the magnitude of the force exerted downward on the top of the liquid? A  dy y p p + dp Fup Fup = Fdown You did not open hints for this part. ANSWER: Part C What is the weight of the thin layer of liquid? Express your answer in terms of quantities given in the problem introduction and , the magnitude of the acceleration due to gravity. You did not open hints for this part. ANSWER: Part D Since the liquid is in equilibrium, the net force on the thin layer of liquid is zero. Complete the force equation for the sum of the vertical forces acting on the liquid layer described in the problem introduction. Express your answer in terms of quantities given in the problem introduction and taking upward forces to be positive. You did not open hints for this part. ANSWER: Fdown = wlayer g wlayer = Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). A Submerged Ball A ball of mass and volume is lowered on a string into a fluid of density . Assume that the object would sink to the bottom if it were not supported by the string. Part A  = = i Fy,i mb V f What is the tension in the string when the ball is fully submerged but not touching the bottom, as shown in the figure? Express your answer in terms of any or all of the given quantities and , the magnitude of the acceleration due to gravity. You did not open hints for this part. ANSWER: Archimedes’ Principle Learning Goal: To understand the applications of Archimedes’ principle. Archimedes’ principle is a powerful tool for solving many problems involving equilibrium in fluids. It states the following: When a body is partially or completely submerged in a fluid (either a liquid or a gas), the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body. As a result of the upward Archimedes force (often called the buoyant force), some objects may float in a fluid, and all of them appear to weigh less. This is the familiar phenomenon of buoyancy. Quantitatively, the buoyant force can be found as , where is the force, is the density of the fluid, is the magnitude of the acceleration due to gravity, and is the volume of the displaced fluid. In this problem, you will be asked several qualitative questions that should help you develop a feel for Archimedes’ principle. An object is placed in a fluid and then released. Assume that the object either floats to the surface (settling so that the object is partly above and partly below the fluid surface) or sinks to the bottom. (Note that for Parts A through D, you should assume that the object has settled in equilibrium.) Part A Consider the following statement: The magnitude of the buoyant force is equal to the weight of fluid displaced by the object. Under what circumstances is this statement true? T g T = Fbuoyant = fluidgV Fbuoyant fluid g V You did not open hints for this part. ANSWER: Part B Consider the following statement: The magnitude of the buoyant force is equal to the weight of the amount of fluid that has the same total volume as the object. Under what circumstances is this statement true? You did not open hints for this part. ANSWER: Part C Consider the following statement: The magnitude of the buoyant force equals the weight of the object. Under what circumstances is this statement true? for every object submerged partially or completely in a fluid only for an object that floats only for an object that sinks for no object submerged in a fluid for an object that is partially submerged in a fluid only for an object that floats for an object completely submerged in a fluid for no object partially or completely submerged in a fluid You did not open hints for this part. ANSWER: Part D Consider the following statement: The magnitude of the buoyant force is less than the weight of the object. Under what circumstances is this statement true? ANSWER: Now apply what you know to some more complicated situations. Part E An object is floating in equilibrium on the surface of a liquid. The object is then removed and placed in another container, filled with a denser liquid. What would you observe? You did not open hints for this part. ANSWER: for every object submerged partially or completely in a fluid for an object that floats only for an object that sinks for no object submerged in a fluid for every object submerged partially or completely in a fluid for an object that floats for an object that sinks for no object submerged in a fluid Part F An object is floating in equilibrium on the surface of a liquid. The object is then removed and placed in another container, filled with a less dense liquid. What would you observe? You did not open hints for this part. ANSWER: Part G Two objects, T and B, have identical size and shape and have uniform density. They are carefully placed in a container filled with a liquid. Both objects float in equilibrium. Less of object T is submerged than of object B, which floats, fully submerged, closer to the bottom of the container. Which of the following statements is true? ANSWER: The object would sink all the way to the bottom. The object would float submerged more deeply than in the first container. The object would float submerged less deeply than in the first container. More than one of these outcomes is possible. The object would sink all the way to the bottom. The object would float submerged more deeply than in the first container. The object would float submerged less deeply than in the first container. More than one of these outcomes is possible. Object T has a greater density than object B. Object B has a greater density than object T. Both objects have the same density. ± Buoyant Force Conceptual Question A rectangular wooden block of weight floats with exactly one-half of its volume below the waterline. Part A What is the buoyant force acting on the block? You did not open hints for this part. ANSWER: Part B W The buoyant force cannot be determined. 2W W 1 W 2 The density of water is 1.00 . What is the density of the block? You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). g/cm3 2.00 between 1.00 and 2.00 1.00 between 0.50 and 1.00 0.50 The density cannot be determined. g/cm3 g/cm3 g/cm3 g/cm3 g/cm3 Flow Velocity of Blood Conceptual Question Arteriosclerotic plaques forming on the inner walls of arteries can decrease the effective cross-sectional area of an artery. Even small changes in the effective area of an artery can lead to very large changes in the blood pressure in the artery and possibly to the collapse of the blood vessel. Imagine a healthy artery, with blood flow velocity of and mass per unit volume of . The kinetic energy per unit volume of blood is given by Imagine that plaque has narrowed an artery to one-fifth of its normal cross-sectional area (an 80% blockage). Part A Compared to normal blood flow velocity, , what is the velocity of blood as it passes through this blockage? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C v0 = 0.14 m/s  = 1050 kg/m3 K0 =  . 1 2 v20 v0 80v0 20v0 5v0 v0/5 This question will be shown after you complete previous question(s). For parts D – F imagine that plaque has grown to a 90% blockage. Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). ± Playing with a Water Hose Two children, Ferdinand and Isabella, are playing with a water hose on a sunny summer day. Isabella is holding the hose in her hand 1.0 meters above the ground and is trying to spray Ferdinand, who is standing 10.0 meters away. Part A Will Isabella be able to spray Ferdinand if the water is flowing out of the hose at a constant speed of 3.5 meters per second? Assume that the hose is pointed parallel to the ground and take the magnitude of the acceleration due to gravity to be 9.81 meters per second, per second. You did not open hints for this part. v0 g ANSWER: Part B This question will be shown after you complete previous question(s). Tactics Box 15.2 Finding Whether an Object Floats or Sinks Learning Goal: To practice Tactics Box 15.2 Finding whether an object floats or sinks. If you hold an object underwater and then release it, it can float to the surface, sink, or remain “hanging” in the water, depending on whether the fluid density is larger than, smaller than, or equal to the object’s average density . These conditions are summarized in this Tactics Box. Yes No f avg TACTICS BOX 15.2 Finding whether an object floats or sinks Object sinks Object floats Object has neutral buoyancy An object sinks if it weighs more than the fluid it displaces, that is, if its average density is greater than the density of the fluid: . An object floats on the surface if it weighs less than the fluid it displaces, that is, if its average density is less than the density of the fluid: . An object hangs motionless in the fluid if it weighs exactly the same as the fluid it displaces. It has neutral buoyancy if its average density equals the density of the fluid: . Part A Ice at 0.0 has a density of 917 . A 3.00 ice cube is gently released inside a small container filled with oil and is observed to be neutrally buoyant. What is the density of the oil, ? Express your answer in kilograms per meter cubed to three significant figures. ANSWER: Part B Once the ice cube melts, what happens to the liquid water that it produces? You did not open hints for this part. ANSWER: avg > f avg < f avg = f 'C kg/m3 cm3 oil oil = kg/m3 Part C What happens if some ethyl alcohol of density 790 is poured into the container after the ice cube has melted? ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. The liquid water sinks to the bottom of the container. The liquid water rises to the surface and floats on top of the oil. The liquid water is in static equilibrium at the location where the ice cube was originally placed. kg/m3 A layer of ethyl alcohol forms between the oil and the water. The layer of ethyl alcohol forms at the bottom of the container. The layer of ethyl alcohol forms on the surface.

Chapter 15 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Fluid Pressure in a U-Tube A U-tube is filled with water, and the two arms are capped. The tube is cylindrical, and the right arm has twice the radius of the left arm. The caps have negligible mass, are watertight, and can freely slide up and down the tube. Part A A one-inch depth of sand is poured onto the cap on each arm. After the caps have moved (if necessary) to reestablish equilibrium, is the right cap higher, lower, or the same height as the left cap? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Pressure in the Ocean The pressure at 10 below the surface of the ocean is about 2.00×105 . Part A higher lower the same height m Pa Which of the following statements is true? You did not open hints for this part. ANSWER: Part B Now consider the pressure 20 below the surface of the ocean. Which of the following statements is true? You did not open hints for this part. ANSWER: Relating Pressure and Height in a Container Learning Goal: To understand the derivation of the law relating height and pressure in a container. The weight of a column of seawater 1 in cross section and 10 high is about 2.00×105 . The weight of a column of seawater 1 in cross section and 10 high plus the weight of a column of air with the same cross section extending up to the top of the atmosphere is about 2.00×105 . The weight of 1 of seawater at 10 below the surface of the ocean is about 2.00×105 . The density of seawater is about 2.00×105 times the density of air at sea level. m2 m N m2 m N m3 m N m The pressure is twice that at a depth of 10 . The pressure is the same as that at a depth of 10 . The pressure is equal to that at a depth of 10 plus the weight per 1 cross sectional area of a column of seawater 10 high. The pressure is equal to the weight per 1 cross sectional area of a column of seawater 20 high. m m m m2 m m2 m In this problem, you will derive the law relating pressure to height in a container by analyzing a particular system. A container of uniform cross-sectional area is filled with liquid of uniform density . Consider a thin horizontal layer of liquid (thickness ) at a height as measured from the bottom of the container. Let the pressure exerted upward on the bottom of the layer be and the pressure exerted downward on the top be . Assume throughout the problem that the system is in equilibrium (the container has not been recently shaken or moved, etc.). Part A What is , the magnitude of the force exerted upward on the bottom of the liquid? You did not open hints for this part. ANSWER: Part B What is , the magnitude of the force exerted downward on the top of the liquid? A  dy y p p + dp Fup Fup = Fdown You did not open hints for this part. ANSWER: Part C What is the weight of the thin layer of liquid? Express your answer in terms of quantities given in the problem introduction and , the magnitude of the acceleration due to gravity. You did not open hints for this part. ANSWER: Part D Since the liquid is in equilibrium, the net force on the thin layer of liquid is zero. Complete the force equation for the sum of the vertical forces acting on the liquid layer described in the problem introduction. Express your answer in terms of quantities given in the problem introduction and taking upward forces to be positive. You did not open hints for this part. ANSWER: Fdown = wlayer g wlayer = Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). A Submerged Ball A ball of mass and volume is lowered on a string into a fluid of density . Assume that the object would sink to the bottom if it were not supported by the string. Part A  = = i Fy,i mb V f What is the tension in the string when the ball is fully submerged but not touching the bottom, as shown in the figure? Express your answer in terms of any or all of the given quantities and , the magnitude of the acceleration due to gravity. You did not open hints for this part. ANSWER: Archimedes’ Principle Learning Goal: To understand the applications of Archimedes’ principle. Archimedes’ principle is a powerful tool for solving many problems involving equilibrium in fluids. It states the following: When a body is partially or completely submerged in a fluid (either a liquid or a gas), the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body. As a result of the upward Archimedes force (often called the buoyant force), some objects may float in a fluid, and all of them appear to weigh less. This is the familiar phenomenon of buoyancy. Quantitatively, the buoyant force can be found as , where is the force, is the density of the fluid, is the magnitude of the acceleration due to gravity, and is the volume of the displaced fluid. In this problem, you will be asked several qualitative questions that should help you develop a feel for Archimedes’ principle. An object is placed in a fluid and then released. Assume that the object either floats to the surface (settling so that the object is partly above and partly below the fluid surface) or sinks to the bottom. (Note that for Parts A through D, you should assume that the object has settled in equilibrium.) Part A Consider the following statement: The magnitude of the buoyant force is equal to the weight of fluid displaced by the object. Under what circumstances is this statement true? T g T = Fbuoyant = fluidgV Fbuoyant fluid g V You did not open hints for this part. ANSWER: Part B Consider the following statement: The magnitude of the buoyant force is equal to the weight of the amount of fluid that has the same total volume as the object. Under what circumstances is this statement true? You did not open hints for this part. ANSWER: Part C Consider the following statement: The magnitude of the buoyant force equals the weight of the object. Under what circumstances is this statement true? for every object submerged partially or completely in a fluid only for an object that floats only for an object that sinks for no object submerged in a fluid for an object that is partially submerged in a fluid only for an object that floats for an object completely submerged in a fluid for no object partially or completely submerged in a fluid You did not open hints for this part. ANSWER: Part D Consider the following statement: The magnitude of the buoyant force is less than the weight of the object. Under what circumstances is this statement true? ANSWER: Now apply what you know to some more complicated situations. Part E An object is floating in equilibrium on the surface of a liquid. The object is then removed and placed in another container, filled with a denser liquid. What would you observe? You did not open hints for this part. ANSWER: for every object submerged partially or completely in a fluid for an object that floats only for an object that sinks for no object submerged in a fluid for every object submerged partially or completely in a fluid for an object that floats for an object that sinks for no object submerged in a fluid Part F An object is floating in equilibrium on the surface of a liquid. The object is then removed and placed in another container, filled with a less dense liquid. What would you observe? You did not open hints for this part. ANSWER: Part G Two objects, T and B, have identical size and shape and have uniform density. They are carefully placed in a container filled with a liquid. Both objects float in equilibrium. Less of object T is submerged than of object B, which floats, fully submerged, closer to the bottom of the container. Which of the following statements is true? ANSWER: The object would sink all the way to the bottom. The object would float submerged more deeply than in the first container. The object would float submerged less deeply than in the first container. More than one of these outcomes is possible. The object would sink all the way to the bottom. The object would float submerged more deeply than in the first container. The object would float submerged less deeply than in the first container. More than one of these outcomes is possible. Object T has a greater density than object B. Object B has a greater density than object T. Both objects have the same density. ± Buoyant Force Conceptual Question A rectangular wooden block of weight floats with exactly one-half of its volume below the waterline. Part A What is the buoyant force acting on the block? You did not open hints for this part. ANSWER: Part B W The buoyant force cannot be determined. 2W W 1 W 2 The density of water is 1.00 . What is the density of the block? You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). g/cm3 2.00 between 1.00 and 2.00 1.00 between 0.50 and 1.00 0.50 The density cannot be determined. g/cm3 g/cm3 g/cm3 g/cm3 g/cm3 Flow Velocity of Blood Conceptual Question Arteriosclerotic plaques forming on the inner walls of arteries can decrease the effective cross-sectional area of an artery. Even small changes in the effective area of an artery can lead to very large changes in the blood pressure in the artery and possibly to the collapse of the blood vessel. Imagine a healthy artery, with blood flow velocity of and mass per unit volume of . The kinetic energy per unit volume of blood is given by Imagine that plaque has narrowed an artery to one-fifth of its normal cross-sectional area (an 80% blockage). Part A Compared to normal blood flow velocity, , what is the velocity of blood as it passes through this blockage? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C v0 = 0.14 m/s  = 1050 kg/m3 K0 =  . 1 2 v20 v0 80v0 20v0 5v0 v0/5 This question will be shown after you complete previous question(s). For parts D – F imagine that plaque has grown to a 90% blockage. Part D This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). ± Playing with a Water Hose Two children, Ferdinand and Isabella, are playing with a water hose on a sunny summer day. Isabella is holding the hose in her hand 1.0 meters above the ground and is trying to spray Ferdinand, who is standing 10.0 meters away. Part A Will Isabella be able to spray Ferdinand if the water is flowing out of the hose at a constant speed of 3.5 meters per second? Assume that the hose is pointed parallel to the ground and take the magnitude of the acceleration due to gravity to be 9.81 meters per second, per second. You did not open hints for this part. v0 g ANSWER: Part B This question will be shown after you complete previous question(s). Tactics Box 15.2 Finding Whether an Object Floats or Sinks Learning Goal: To practice Tactics Box 15.2 Finding whether an object floats or sinks. If you hold an object underwater and then release it, it can float to the surface, sink, or remain “hanging” in the water, depending on whether the fluid density is larger than, smaller than, or equal to the object’s average density . These conditions are summarized in this Tactics Box. Yes No f avg TACTICS BOX 15.2 Finding whether an object floats or sinks Object sinks Object floats Object has neutral buoyancy An object sinks if it weighs more than the fluid it displaces, that is, if its average density is greater than the density of the fluid: . An object floats on the surface if it weighs less than the fluid it displaces, that is, if its average density is less than the density of the fluid: . An object hangs motionless in the fluid if it weighs exactly the same as the fluid it displaces. It has neutral buoyancy if its average density equals the density of the fluid: . Part A Ice at 0.0 has a density of 917 . A 3.00 ice cube is gently released inside a small container filled with oil and is observed to be neutrally buoyant. What is the density of the oil, ? Express your answer in kilograms per meter cubed to three significant figures. ANSWER: Part B Once the ice cube melts, what happens to the liquid water that it produces? You did not open hints for this part. ANSWER: avg > f avg < f avg = f 'C kg/m3 cm3 oil oil = kg/m3 Part C What happens if some ethyl alcohol of density 790 is poured into the container after the ice cube has melted? ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. The liquid water sinks to the bottom of the container. The liquid water rises to the surface and floats on top of the oil. The liquid water is in static equilibrium at the location where the ice cube was originally placed. kg/m3 A layer of ethyl alcohol forms between the oil and the water. The layer of ethyl alcohol forms at the bottom of the container. The layer of ethyl alcohol forms on the surface.

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Chapter 5 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, March 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Tactics Box 5.1 Drawing Force Vectors Learning Goal: To practice Tactics Box 5.1 Drawing Force Vectors. To visualize how forces are exerted on objects, we can use simple diagrams such as vectors. This Tactics Box illustrates the process of drawing a force vector by using the particle model, in which objects are treated as points. TACTICS BOX 5.1 Drawing force vectors Represent the object 1. as a particle. 2. Place the tail of the force vector on the particle. 3. Draw the force vector as an arrow pointing in the proper direction and with a length proportional to the size of the force. 4. Give the vector an appropriate label. The resulting diagram for a force exerted on an object is shown in the drawing. Note that the object is represented as a black dot. Part A A book lies on a table. A pushing force parallel to the table top and directed to the right is exerted on the book. Follow the steps above to draw the force vector . Use the black dot as the particle representing the book. F  F push F push Draw the vector starting at the black dot. The location and orientation of the vector will be graded. The length of the vector will not be graded. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Tactics Box 5.2 Identifying Forces Learning Goal: To practice Tactics Box 5.2 Identifying Forces. The first basic step in solving force and motion problems generally involves identifying all of the forces acting on an object. This tactics box provides a step-by-step method for identifying each force in a problem. TACTICS BOX 5.2 Identifying forces Identify the object of interest. This is the object whose motion 1. you wish to study. 2. Draw a picture of the situation. Show the object of interest and all other objects—such as ropes, springs, or surfaces—that touch it. 3. Draw a closed curve around the object. Only the object of interest is inside the curve; everything else is outside. 4. Locate every point on the boundary of this curve where other objects touch the object of interest. These are the points where contact forces are exerted on the object. Name and label each contact force acting on the object. There is at least one force at each point of contact; there may be more than one. When necessary, use subscripts to distinguish forces of the same type. 5. 6. Name and label each long-range force acting on the object. For now, the only long-range force is the gravitational force. Apply these steps to the following problem: A crate is pulled up a rough inclined wood board by a tow rope. Identify the forces on the crate. Part A Which of the following objects are of interest? Check all that apply. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Conceptual Questions on Newton’s 1st and 2nd Laws Learning Goal: To understand the meaning and the basic applications of Newton’s 1st and 2nd laws. In this problem, you are given a diagram representing the motion of an object–a motion diagram. The dots represent the object’s position at moments separated by equal intervals of time. The dots are connected by arrows representing the object’s average velocity during the corresponding time interval. Your goal is to use this motion diagram to determine the direction of the net force acting on the object. You will then determine which force diagrams and which situations may correspond to such a motion. crate earth rope wood board Part A What is the direction of the net force acting on the object at position A? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D upward downward to the left to the right The net force is zero. This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s). Part J This question will be shown after you complete previous question(s). Understanding Newton’s Laws Part A An object cannot remain at rest unless which of the following holds? You did not open hints for this part. ANSWER: Part B If a block is moving to the left at a constant velocity, what can one conclude? You did not open hints for this part. ANSWER: The net force acting on it is zero. The net force acting on it is constant and nonzero. There are no forces at all acting on it. There is only one force acting on it. Part C A block of mass is acted upon by two forces: (directed to the left) and (directed to the right). What can you say about the block’s motion? You did not open hints for this part. ANSWER: Part D A massive block is being pulled along a horizontal frictionless surface by a constant horizontal force. The block must be __________. You did not open hints for this part. ANSWER: There is exactly one force applied to the block. The net force applied to the block is directed to the left. The net force applied to the block is zero. There must be no forces at all applied to the block. 2 kg 3 N 4 N It must be moving to the left. It must be moving to the right. It must be at rest. It could be moving to the left, moving to the right, or be instantaneously at rest. Part E Two forces, of magnitude and , are applied to an object. The relative direction of the forces is unknown. The net force acting on the object __________. Check all that apply. You did not open hints for this part. ANSWER: Tactics Box 5.3 Drawing a Free-Body Diagram Learning Goal: To practice Tactics Box 5.3 Drawing a Free-Body Diagram. A free-body diagram is a diagram that represents the object as a particle and shows all of the forces acting on the object. Learning how to draw such a diagram is a very important skill in solving physics problems. This tactics box explains the essential steps to construct a correct free-body diagram. TACTICS BOX 5.3 Drawing a free-body diagram Identify all forces acting on the object. This step was described 1. in Tactics Box 5.2. continuously changing direction moving at constant velocity moving with a constant nonzero acceleration moving with continuously increasing acceleration 4 N 10 N cannot have a magnitude equal to cannot have a magnitude equal to cannot have the same direction as the force with magnitude must have a magnitude greater than 5 N 10 N 10 N 10 N Draw a coordinate system. Use the axes defined in your pictorial representation. If those axes are tilted, for motion along an incline, then the axes of the free-body diagram should be similarly tilted. 2. Represent the object as a dot at the origin of the coordinate axes. This is 3. the particle model. 4. Draw vectors representing each of the identified forces. This was described in Tactics Box 5.1. Be sure to label each force vector. Draw and label the net force vector . Draw this vector beside the diagram, not on the particle. Or, if appropriate, write . Then, check that points in the same direction as the acceleration vector on your motion diagram. 5. Apply these steps to the following problem: Your physics book is sliding on the carpet. Draw a free-body diagram. Part A Which forces are acting on the book? Check all that apply. You did not open hints for this part. ANSWER: F  net F =  net 0 F  net a Part B Draw the most appropriate set of coordinate axes for this problem. The orientation of your vectors will be graded. ANSWER: gravity normal force drag static friction tension kinetic friction spring force Part C This question will be shown after you complete previous question(s). Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points.

Chapter 5 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, March 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Tactics Box 5.1 Drawing Force Vectors Learning Goal: To practice Tactics Box 5.1 Drawing Force Vectors. To visualize how forces are exerted on objects, we can use simple diagrams such as vectors. This Tactics Box illustrates the process of drawing a force vector by using the particle model, in which objects are treated as points. TACTICS BOX 5.1 Drawing force vectors Represent the object 1. as a particle. 2. Place the tail of the force vector on the particle. 3. Draw the force vector as an arrow pointing in the proper direction and with a length proportional to the size of the force. 4. Give the vector an appropriate label. The resulting diagram for a force exerted on an object is shown in the drawing. Note that the object is represented as a black dot. Part A A book lies on a table. A pushing force parallel to the table top and directed to the right is exerted on the book. Follow the steps above to draw the force vector . Use the black dot as the particle representing the book. F  F push F push Draw the vector starting at the black dot. The location and orientation of the vector will be graded. The length of the vector will not be graded. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Tactics Box 5.2 Identifying Forces Learning Goal: To practice Tactics Box 5.2 Identifying Forces. The first basic step in solving force and motion problems generally involves identifying all of the forces acting on an object. This tactics box provides a step-by-step method for identifying each force in a problem. TACTICS BOX 5.2 Identifying forces Identify the object of interest. This is the object whose motion 1. you wish to study. 2. Draw a picture of the situation. Show the object of interest and all other objects—such as ropes, springs, or surfaces—that touch it. 3. Draw a closed curve around the object. Only the object of interest is inside the curve; everything else is outside. 4. Locate every point on the boundary of this curve where other objects touch the object of interest. These are the points where contact forces are exerted on the object. Name and label each contact force acting on the object. There is at least one force at each point of contact; there may be more than one. When necessary, use subscripts to distinguish forces of the same type. 5. 6. Name and label each long-range force acting on the object. For now, the only long-range force is the gravitational force. Apply these steps to the following problem: A crate is pulled up a rough inclined wood board by a tow rope. Identify the forces on the crate. Part A Which of the following objects are of interest? Check all that apply. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Conceptual Questions on Newton’s 1st and 2nd Laws Learning Goal: To understand the meaning and the basic applications of Newton’s 1st and 2nd laws. In this problem, you are given a diagram representing the motion of an object–a motion diagram. The dots represent the object’s position at moments separated by equal intervals of time. The dots are connected by arrows representing the object’s average velocity during the corresponding time interval. Your goal is to use this motion diagram to determine the direction of the net force acting on the object. You will then determine which force diagrams and which situations may correspond to such a motion. crate earth rope wood board Part A What is the direction of the net force acting on the object at position A? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D upward downward to the left to the right The net force is zero. This question will be shown after you complete previous question(s). Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s). Part J This question will be shown after you complete previous question(s). Understanding Newton’s Laws Part A An object cannot remain at rest unless which of the following holds? You did not open hints for this part. ANSWER: Part B If a block is moving to the left at a constant velocity, what can one conclude? You did not open hints for this part. ANSWER: The net force acting on it is zero. The net force acting on it is constant and nonzero. There are no forces at all acting on it. There is only one force acting on it. Part C A block of mass is acted upon by two forces: (directed to the left) and (directed to the right). What can you say about the block’s motion? You did not open hints for this part. ANSWER: Part D A massive block is being pulled along a horizontal frictionless surface by a constant horizontal force. The block must be __________. You did not open hints for this part. ANSWER: There is exactly one force applied to the block. The net force applied to the block is directed to the left. The net force applied to the block is zero. There must be no forces at all applied to the block. 2 kg 3 N 4 N It must be moving to the left. It must be moving to the right. It must be at rest. It could be moving to the left, moving to the right, or be instantaneously at rest. Part E Two forces, of magnitude and , are applied to an object. The relative direction of the forces is unknown. The net force acting on the object __________. Check all that apply. You did not open hints for this part. ANSWER: Tactics Box 5.3 Drawing a Free-Body Diagram Learning Goal: To practice Tactics Box 5.3 Drawing a Free-Body Diagram. A free-body diagram is a diagram that represents the object as a particle and shows all of the forces acting on the object. Learning how to draw such a diagram is a very important skill in solving physics problems. This tactics box explains the essential steps to construct a correct free-body diagram. TACTICS BOX 5.3 Drawing a free-body diagram Identify all forces acting on the object. This step was described 1. in Tactics Box 5.2. continuously changing direction moving at constant velocity moving with a constant nonzero acceleration moving with continuously increasing acceleration 4 N 10 N cannot have a magnitude equal to cannot have a magnitude equal to cannot have the same direction as the force with magnitude must have a magnitude greater than 5 N 10 N 10 N 10 N Draw a coordinate system. Use the axes defined in your pictorial representation. If those axes are tilted, for motion along an incline, then the axes of the free-body diagram should be similarly tilted. 2. Represent the object as a dot at the origin of the coordinate axes. This is 3. the particle model. 4. Draw vectors representing each of the identified forces. This was described in Tactics Box 5.1. Be sure to label each force vector. Draw and label the net force vector . Draw this vector beside the diagram, not on the particle. Or, if appropriate, write . Then, check that points in the same direction as the acceleration vector on your motion diagram. 5. Apply these steps to the following problem: Your physics book is sliding on the carpet. Draw a free-body diagram. Part A Which forces are acting on the book? Check all that apply. You did not open hints for this part. ANSWER: F  net F =  net 0 F  net a Part B Draw the most appropriate set of coordinate axes for this problem. The orientation of your vectors will be graded. ANSWER: gravity normal force drag static friction tension kinetic friction spring force Part C This question will be shown after you complete previous question(s). Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points.

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Chapter 8 Practice Problems (Practice – no credit) Due: 12:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Circular Launch A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 . Part A How far from the bottom of the chute does the ball land? Your answer for the distance the ball travels from the end of the chute should contain . You did not open hints for this part. ANSWER: g R Normal Force and Centripetal Force Ranking Task A roller-coaster track has six semicircular “dips” with different radii of curvature. The same roller-coaster cart rides through each dip at a different speed. Part A For the different values given for the radius of curvature and speed , rank the magnitude of the force of the roller-coaster track on the cart at the bottom of each dip. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: D = R v Two Cars on a Curving Road Part A A small car of mass and a large car of mass drive along a highway. They approach a curve of radius . Both cars maintain the same acceleration as they travel around the curve. How does the speed of the small car compare to the speed of the large car as they round the curve? You did not open hints for this part. m 4m R a vS vL ANSWER: Part B Now assume that two identical cars of mass drive along a highway. One car approaches a curve of radius at speed . The second car approaches a curve of radius at a speed of . How does the magnitude of the net force exerted on the first car compare to the magnitude of the net force exerted on the second car? You did not open hints for this part. ANSWER: ± A Ride on the Ferris Wheel A woman rides on a Ferris wheel of radius 16 that maintains the same speed throughout its motion. To better understand physics, she takes along a digital bathroom scale (with memory) and sits on it. When she gets off the ride, she uploads the scale readings to a computer and creates a graph of scale reading versus time. Note that the graph has a minimum value of 510 and a maximum value of 666 . vS = 1 4 vL vS = 1 2 vL vS = vL vS = 2vL vS = 4vL m 2R v 6R 3v F1 F2 F1 = 1 3 F2 F1 = 3 4 F2 F1 = F2 F1 = 3F2 F1 = 27F2 m N N Part A What is the woman’s mass? Express your answer in kilograms. You did not open hints for this part. ANSWER: ± Mass on Turntable A small metal cylinder rests on a circular turntable that is rotating at a constant speed as illustrated in the diagram . The small metal cylinder has a mass of 0.20 , the coefficient of static friction between the cylinder and the turntable is 0.080, and the cylinder is located 0.15 from the center of the turntable. Take the magnitude of the acceleration due to gravity to be 9.81 . m = kg kg m m/s2 Part A What is the maximum speed that the cylinder can move along its circular path without slipping off the turntable? Express your answer numerically in meters per second to two significant figures. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. vmax vmax = m/s

Chapter 8 Practice Problems (Practice – no credit) Due: 12:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Circular Launch A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 . Part A How far from the bottom of the chute does the ball land? Your answer for the distance the ball travels from the end of the chute should contain . You did not open hints for this part. ANSWER: g R Normal Force and Centripetal Force Ranking Task A roller-coaster track has six semicircular “dips” with different radii of curvature. The same roller-coaster cart rides through each dip at a different speed. Part A For the different values given for the radius of curvature and speed , rank the magnitude of the force of the roller-coaster track on the cart at the bottom of each dip. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: D = R v Two Cars on a Curving Road Part A A small car of mass and a large car of mass drive along a highway. They approach a curve of radius . Both cars maintain the same acceleration as they travel around the curve. How does the speed of the small car compare to the speed of the large car as they round the curve? You did not open hints for this part. m 4m R a vS vL ANSWER: Part B Now assume that two identical cars of mass drive along a highway. One car approaches a curve of radius at speed . The second car approaches a curve of radius at a speed of . How does the magnitude of the net force exerted on the first car compare to the magnitude of the net force exerted on the second car? You did not open hints for this part. ANSWER: ± A Ride on the Ferris Wheel A woman rides on a Ferris wheel of radius 16 that maintains the same speed throughout its motion. To better understand physics, she takes along a digital bathroom scale (with memory) and sits on it. When she gets off the ride, she uploads the scale readings to a computer and creates a graph of scale reading versus time. Note that the graph has a minimum value of 510 and a maximum value of 666 . vS = 1 4 vL vS = 1 2 vL vS = vL vS = 2vL vS = 4vL m 2R v 6R 3v F1 F2 F1 = 1 3 F2 F1 = 3 4 F2 F1 = F2 F1 = 3F2 F1 = 27F2 m N N Part A What is the woman’s mass? Express your answer in kilograms. You did not open hints for this part. ANSWER: ± Mass on Turntable A small metal cylinder rests on a circular turntable that is rotating at a constant speed as illustrated in the diagram . The small metal cylinder has a mass of 0.20 , the coefficient of static friction between the cylinder and the turntable is 0.080, and the cylinder is located 0.15 from the center of the turntable. Take the magnitude of the acceleration due to gravity to be 9.81 . m = kg kg m m/s2 Part A What is the maximum speed that the cylinder can move along its circular path without slipping off the turntable? Express your answer numerically in meters per second to two significant figures. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. vmax vmax = m/s

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Assignment 11 Due: 11:59pm on Wednesday, April 30, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 13.2 The gravitational force of a star on orbiting planet 1 is . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force . Part A What is the ratio ? ANSWER: Correct Conceptual Question 13.3 A 1500 satellite and a 2200 satellite follow exactly the same orbit around the earth. Part A What is the ratio of the force on the first satellite to that on the second satellite? ANSWER: Correct F1 F2 F1 F2 = 2 F1 F2 kg kg F1 F2 = 0.682 F1 F2 Part B What is the ratio of the acceleration of the first satellite to that of the second satellite? ANSWER: Correct Problem 13.2 The centers of a 15.0 lead ball and a 90.0 lead ball are separated by 9.00 . Part A What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER: Correct Part B What is the ratio of this gravitational force to the weight of the 90.0 ball? ANSWER: a1 a2 = 1 a1 a2 kg g cm 1.11×10−8 N g 1.26×10−8 Correct Problem 13.6 The space shuttle orbits 310 above the surface of the earth. Part A What is the gravitational force on a 7.5 sphere inside the space shuttle? Express your answer with the appropriate units. ANSWER: Correct ± A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 2310 and is in a circular orbit at a height of 650 above the surface of the earth. Part A What is the gravitational force on the satellite? Take the gravitational constant to be = 6.67×10−11 , the mass of the earth to be = 5.97×1024 , and the radius of the Earth to be = 6.38×106 . Express your answer in newtons. Hint 1. How to approach the problem Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations. km kg Fe on s = 67.0 N kg km Fgrav G N m2/kg2 me kg re m Hint 2. Law of gravitation According to Newton’s law of gravitation, , where is the gravitational constant, and are the masses of the two objects, and is the distance between the centers of mass of the two objects. Hint 3. Calculate the distance between the centers of mass What is the distance from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters. ANSWER: ANSWER: Correct Part B What fraction is this of the satellite’s weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be = 9.80 . Hint 1. How to approach the problem All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth. ANSWER: F = Gm1m2/r2 G m1 m2 r r = 7.03×10r 6 m = 1.86×10Fgrav 4 N g m/s2 0.824 Correct Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation: . Dividing the gravitational force on the satellite by , we find that the ratio of the forces due to the earth’s gravity is simply the square of the ratio of the earth’s radius to the sum of the earth’s radius and the height of the orbit of the satellite above the earth, . This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut’s weight is never zero. When people speak of “weightlessness” in space, what they really mean is “free fall.” Problem 13.8 Part A What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units. ANSWER: Correct Part B What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units. ANSWER: Correct w = G m/ me r2e Fgrav = Gmem/(re + h)2 w [re/(re + h)]2 gmoon = 1.62 m s2 gJupiter = 25.9 m s2 Enhanced EOC: Problem 13.14 A rocket is launched straight up from the earth’s surface at a speed of 1.90×104 . You may want to review ( pages 362 – 365) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A What is its speed when it is very far away from the earth? Express your answer with the appropriate units. Hint 1. How to approach the problem What is conserved in this problem? What is the rocket’s initial kinetic energy in terms of its unknown mass, ? What is the rocket’s initial gravitational potential energy in terms of its unknown mass, ? When the rocket is very far away from the Earth, what is its gravitational potential energy? Using conservation of energy, what is the rocket’s kinetic energy when it is very far away from the Earth? Therefore, what is the rocket’s velocity when it is very far away from the Earth? ANSWER: Correct Problem 13.13 Part A m/s m m 1.54×104 ms What is the escape speed from Venus? Express your answer with the appropriate units. ANSWER: Correct Problem 13.17 The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years. Part A What is the asteroid’s orbital radius? Express your answer with the appropriate units. ANSWER: Correct Part B What is the asteroid’s orbital speed? Express your answer with the appropriate units. ANSWER: vescape = 10.4 km s = 3.89×1011 R m = 1.85×104 v ms Correct Problem 13.32 Part A At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units. ANSWER: Correct Part B What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER: Correct Problem 13.36 Two meteoroids are heading for earth. Their speeds as they cross the moon’s orbit are 2 . 1.01×107 m 4920 ms km/s Part A The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units. ANSWER: Correct Part B The second misses the earth by 5500 . What is its speed at its closest point? Express your answer with the appropriate units. ANSWER: Incorrect; Try Again Problem 14.2 An air-track glider attached to a spring oscillates between the 11.0 mark and the 67.0 mark on the track. The glider completes 11.0 oscillations in 32.0 . Part A What is the period of the oscillations? Express your answer with the appropriate units. v1 = 11.3 km s km v2 = cm cm s ANSWER: Correct Part B What is the frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part C What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part D What is the amplitude? Express your answer with the appropriate units. 2.91 s 0.344 Hz 2.16 rad s ANSWER: Correct Part E What is the maximum speed of the glider? Express your answer with the appropriate units. ANSWER: Correct Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object’s displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the 28.0 cm 60.5 cms F  F = −kx x k m k x = 0 block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from , it will ANSWER: Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at . After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we’ve assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds. The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ). A A x = A remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there. move to the left until it reaches and then begin to move to the right. x = −A x = −A x = −A x = −A x = A T f f = 1/T f Hz Part B If the period is doubled, the frequency is ANSWER: Correct Part C An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER: Correct unchanged. doubled. halved. s s f f = 10 Hz Part D If the frequency is 40 , what is the period ? Express your answer in seconds. ANSWER: Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance from the equilibrium position? ANSWER: Hz T T = 0.025 s A Correct Part F Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER: Correct Now assume for the remaining Parts G – J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 . Part G What is the period ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the R only Q only both R and Q T T K and L K and M K and P L and N M and P m s T t = 0 a aT = 0.005 s a period . ANSWER: Correct Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: Correct Part I What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER: Correct Part J What distance does the object cover between the moments labeled K and N on the graph? T T = 0.02 s t t = 0.01 s d d = 0.48 m d Express your answer in meters. ANSWER: Correct Problem 14.4 Part A What is the amplitude of the oscillation shown in the figure? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct d = 0.36 m A = 20.0 cm Part B What is the frequency of this oscillation? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the phase constant? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.10 An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 4.60 left of the equilibrium position and moving to the right at 33.4 . Part A What is the phase constant? Express your answer to three significant figures and include the appropriate units. ANSWER: f = 0.25 Hz 0 = -60 % s t = 0 s cm cm/s 0 = -2.09 rad Correct Part B What is the phase at ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C What is the phase at ? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part D What is the phase at ? Express your answer to three significant figures and include the appropriate units. ANSWER: t = 0.5 s  = 0 rad t = 1.0 s  = 2.09 rad t = 1.5 s  = 4.19 rad Correct Problem 14.12 A 140 air-track glider is attached to a spring. The glider is pushed in 12.2 and released. A student with a stopwatch finds that 14.0 oscillations take 19.0 . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: Correct Problem 14.14 The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine: Part A The amplitude. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct g cm s 3.00 Nm x(t) = (2.0 cm)cos(10t − /4) t 2.00 cm Part B The period. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C The spring constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part D The phase constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct 0.628 s 5.00 Nm -0.785 rad Part E The initial coordinate of the mass. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part F The initial velocity. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part G The maximum speed. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct 1.41 cm 14.1 cms 20.0 cms Part H The total energy. Express your answer to one decimal place and include the appropriate units. ANSWER: Correct Part I The velocity at . Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 14.17 A spring with spring constant 16 hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0 and released. The ball makes 35 oscillations in 18 seconds. You may want to review ( pages 389 – 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions Part A What is its the mass of the ball? 1.0 mJ t = 0.40 s 1.46 cms N/m cm s Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass? ANSWER: Correct Part B What is its maximum speed? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the amplitude of the oscillations? How is the maximum speed related to the amplitude of the oscillations and the angular frequency? ANSWER: Correct Changing the Period of a Pendulum m = 110 g vmax = 49 cms A simple pendulum consisting of a bob of mass attached to a string of length swings with a period . Part A If the bob’s mass is doubled, approximately what will the pendulum’s new period be? Hint 1. Period of a simple pendulum The period of a simple pendulum of length is given by , where is the acceleration due to gravity. ANSWER: Correct Part B If the pendulum is brought on the moon where the gravitational acceleration is about , approximately what will its period now be? Hint 1. How to approach the problem Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. m L T T L T = 2 Lg −−  g T/2 T ‘2T 2T g/6 ANSWER: Correct Part C If the pendulum is taken into the orbiting space station what will happen to the bob? Hint 1. How to approach the problem Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth’s gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall. ANSWER: Correct In the space station, where all objects undergo the same acceleration due to the earth’s gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached. T/6 T/’6 ‘6T 6T It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero. Problem 14.20 A 175 ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13 . Part A How long is the string? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.22 Part A What is the length of a pendulum whose period on the moon matches the period of a 2.1- -long pendulum on the earth? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.42 An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( = 0.17 ) driven back and forth in SHM at by an electromagnetic coil. g % s L = 19 cm m lmoon = 0.35 m m g 1.0 MHz Part A The maximum restoring force that can be applied to the disk without breaking it is 4.4×104 . What is the maximum oscillation amplitude that won’t rupture the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the disk’s maximum speed at this amplitude? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 94.2%. You received 135.71 out of a possible total of 144 points. N amax = 6.6 μm vmax = 41 ms

Assignment 11 Due: 11:59pm on Wednesday, April 30, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 13.2 The gravitational force of a star on orbiting planet 1 is . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force . Part A What is the ratio ? ANSWER: Correct Conceptual Question 13.3 A 1500 satellite and a 2200 satellite follow exactly the same orbit around the earth. Part A What is the ratio of the force on the first satellite to that on the second satellite? ANSWER: Correct F1 F2 F1 F2 = 2 F1 F2 kg kg F1 F2 = 0.682 F1 F2 Part B What is the ratio of the acceleration of the first satellite to that of the second satellite? ANSWER: Correct Problem 13.2 The centers of a 15.0 lead ball and a 90.0 lead ball are separated by 9.00 . Part A What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER: Correct Part B What is the ratio of this gravitational force to the weight of the 90.0 ball? ANSWER: a1 a2 = 1 a1 a2 kg g cm 1.11×10−8 N g 1.26×10−8 Correct Problem 13.6 The space shuttle orbits 310 above the surface of the earth. Part A What is the gravitational force on a 7.5 sphere inside the space shuttle? Express your answer with the appropriate units. ANSWER: Correct ± A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 2310 and is in a circular orbit at a height of 650 above the surface of the earth. Part A What is the gravitational force on the satellite? Take the gravitational constant to be = 6.67×10−11 , the mass of the earth to be = 5.97×1024 , and the radius of the Earth to be = 6.38×106 . Express your answer in newtons. Hint 1. How to approach the problem Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations. km kg Fe on s = 67.0 N kg km Fgrav G N m2/kg2 me kg re m Hint 2. Law of gravitation According to Newton’s law of gravitation, , where is the gravitational constant, and are the masses of the two objects, and is the distance between the centers of mass of the two objects. Hint 3. Calculate the distance between the centers of mass What is the distance from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters. ANSWER: ANSWER: Correct Part B What fraction is this of the satellite’s weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be = 9.80 . Hint 1. How to approach the problem All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth. ANSWER: F = Gm1m2/r2 G m1 m2 r r = 7.03×10r 6 m = 1.86×10Fgrav 4 N g m/s2 0.824 Correct Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation: . Dividing the gravitational force on the satellite by , we find that the ratio of the forces due to the earth’s gravity is simply the square of the ratio of the earth’s radius to the sum of the earth’s radius and the height of the orbit of the satellite above the earth, . This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut’s weight is never zero. When people speak of “weightlessness” in space, what they really mean is “free fall.” Problem 13.8 Part A What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units. ANSWER: Correct Part B What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units. ANSWER: Correct w = G m/ me r2e Fgrav = Gmem/(re + h)2 w [re/(re + h)]2 gmoon = 1.62 m s2 gJupiter = 25.9 m s2 Enhanced EOC: Problem 13.14 A rocket is launched straight up from the earth’s surface at a speed of 1.90×104 . You may want to review ( pages 362 – 365) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A What is its speed when it is very far away from the earth? Express your answer with the appropriate units. Hint 1. How to approach the problem What is conserved in this problem? What is the rocket’s initial kinetic energy in terms of its unknown mass, ? What is the rocket’s initial gravitational potential energy in terms of its unknown mass, ? When the rocket is very far away from the Earth, what is its gravitational potential energy? Using conservation of energy, what is the rocket’s kinetic energy when it is very far away from the Earth? Therefore, what is the rocket’s velocity when it is very far away from the Earth? ANSWER: Correct Problem 13.13 Part A m/s m m 1.54×104 ms What is the escape speed from Venus? Express your answer with the appropriate units. ANSWER: Correct Problem 13.17 The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years. Part A What is the asteroid’s orbital radius? Express your answer with the appropriate units. ANSWER: Correct Part B What is the asteroid’s orbital speed? Express your answer with the appropriate units. ANSWER: vescape = 10.4 km s = 3.89×1011 R m = 1.85×104 v ms Correct Problem 13.32 Part A At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units. ANSWER: Correct Part B What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER: Correct Problem 13.36 Two meteoroids are heading for earth. Their speeds as they cross the moon’s orbit are 2 . 1.01×107 m 4920 ms km/s Part A The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units. ANSWER: Correct Part B The second misses the earth by 5500 . What is its speed at its closest point? Express your answer with the appropriate units. ANSWER: Incorrect; Try Again Problem 14.2 An air-track glider attached to a spring oscillates between the 11.0 mark and the 67.0 mark on the track. The glider completes 11.0 oscillations in 32.0 . Part A What is the period of the oscillations? Express your answer with the appropriate units. v1 = 11.3 km s km v2 = cm cm s ANSWER: Correct Part B What is the frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part C What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER: Correct Part D What is the amplitude? Express your answer with the appropriate units. 2.91 s 0.344 Hz 2.16 rad s ANSWER: Correct Part E What is the maximum speed of the glider? Express your answer with the appropriate units. ANSWER: Correct Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object’s displacement from its equilibrium position. Mathematically, the restoring force is given by , where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass attached to a spring with force constant , as shown in the figure. The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at . If the 28.0 cm 60.5 cms F  F = −kx x k m k x = 0 block is pulled to the right a distance and then released, will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from , it will ANSWER: Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at . After is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches , completing one cycle of motion. The motion will then repeat; if, as we’ve assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period. Usually, the period is denoted and is measured in seconds. The frequency, denoted , is the number of cycles that are completed per unit of time: . In SI units, is measured in inverse seconds, or hertz ( ). A A x = A remain at rest. move to the left until it reaches equilibrium and stop there. move to the left until it reaches and stop there. move to the left until it reaches and then begin to move to the right. x = −A x = −A x = −A x = −A x = A T f f = 1/T f Hz Part B If the period is doubled, the frequency is ANSWER: Correct Part C An oscillating object takes 0.10 to complete one cycle; that is, its period is 0.10 . What is its frequency ? Express your answer in hertz. ANSWER: Correct unchanged. doubled. halved. s s f f = 10 Hz Part D If the frequency is 40 , what is the period ? Express your answer in seconds. ANSWER: Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance from the equilibrium position? ANSWER: Hz T T = 0.025 s A Correct Part F Suppose that the period is . Which of the following points on the t axis are separated by the time interval ? ANSWER: Correct Now assume for the remaining Parts G – J, that the x coordinate of point R is 0.12 and the t coordinate of point K is 0.0050 . Part G What is the period ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point to the point K, what fraction of a full wavelength is covered? Call that fraction . Then you can set . Dividing by the fraction will give the R only Q only both R and Q T T K and L K and M K and P L and N M and P m s T t = 0 a aT = 0.005 s a period . ANSWER: Correct Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: Correct Part I What distance does the object cover during one period of oscillation? Express your answer in meters. ANSWER: Correct Part J What distance does the object cover between the moments labeled K and N on the graph? T T = 0.02 s t t = 0.01 s d d = 0.48 m d Express your answer in meters. ANSWER: Correct Problem 14.4 Part A What is the amplitude of the oscillation shown in the figure? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct d = 0.36 m A = 20.0 cm Part B What is the frequency of this oscillation? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the phase constant? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.10 An air-track glider attached to a spring oscillates with a period of 1.50 . At the glider is 4.60 left of the equilibrium position and moving to the right at 33.4 . Part A What is the phase constant? Express your answer to three significant figures and include the appropriate units. ANSWER: f = 0.25 Hz 0 = -60 % s t = 0 s cm cm/s 0 = -2.09 rad Correct Part B What is the phase at ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C What is the phase at ? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part D What is the phase at ? Express your answer to three significant figures and include the appropriate units. ANSWER: t = 0.5 s  = 0 rad t = 1.0 s  = 2.09 rad t = 1.5 s  = 4.19 rad Correct Problem 14.12 A 140 air-track glider is attached to a spring. The glider is pushed in 12.2 and released. A student with a stopwatch finds that 14.0 oscillations take 19.0 . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: Correct Problem 14.14 The position of a 50 g oscillating mass is given by , where is in s. If necessary, round your answers to three significant figures. Determine: Part A The amplitude. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct g cm s 3.00 Nm x(t) = (2.0 cm)cos(10t − /4) t 2.00 cm Part B The period. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part C The spring constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part D The phase constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct 0.628 s 5.00 Nm -0.785 rad Part E The initial coordinate of the mass. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part F The initial velocity. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part G The maximum speed. Express your answer to three significant figures and include the appropriate units. ANSWER: Correct 1.41 cm 14.1 cms 20.0 cms Part H The total energy. Express your answer to one decimal place and include the appropriate units. ANSWER: Correct Part I The velocity at . Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Enhanced EOC: Problem 14.17 A spring with spring constant 16 hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0 and released. The ball makes 35 oscillations in 18 seconds. You may want to review ( pages 389 – 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions Part A What is its the mass of the ball? 1.0 mJ t = 0.40 s 1.46 cms N/m cm s Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass? ANSWER: Correct Part B What is its maximum speed? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the amplitude of the oscillations? How is the maximum speed related to the amplitude of the oscillations and the angular frequency? ANSWER: Correct Changing the Period of a Pendulum m = 110 g vmax = 49 cms A simple pendulum consisting of a bob of mass attached to a string of length swings with a period . Part A If the bob’s mass is doubled, approximately what will the pendulum’s new period be? Hint 1. Period of a simple pendulum The period of a simple pendulum of length is given by , where is the acceleration due to gravity. ANSWER: Correct Part B If the pendulum is brought on the moon where the gravitational acceleration is about , approximately what will its period now be? Hint 1. How to approach the problem Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. m L T T L T = 2 Lg −−  g T/2 T ‘2T 2T g/6 ANSWER: Correct Part C If the pendulum is taken into the orbiting space station what will happen to the bob? Hint 1. How to approach the problem Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth’s gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall. ANSWER: Correct In the space station, where all objects undergo the same acceleration due to the earth’s gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached. T/6 T/’6 ‘6T 6T It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero. Problem 14.20 A 175 ball is tied to a string. It is pulled to an angle of 8.0 and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13 . Part A How long is the string? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.22 Part A What is the length of a pendulum whose period on the moon matches the period of a 2.1- -long pendulum on the earth? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 14.42 An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( = 0.17 ) driven back and forth in SHM at by an electromagnetic coil. g % s L = 19 cm m lmoon = 0.35 m m g 1.0 MHz Part A The maximum restoring force that can be applied to the disk without breaking it is 4.4×104 . What is the maximum oscillation amplitude that won’t rupture the disk? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the disk’s maximum speed at this amplitude? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 94.2%. You received 135.71 out of a possible total of 144 points. N amax = 6.6 μm vmax = 41 ms

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Chapter 9 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: The blocks can move in only one dimension, namely, 1. along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. m1 m2 t v1(t) a1 (t) v2 (t) a2 (t)

Chapter 9 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: The blocks can move in only one dimension, namely, 1. along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. m1 m2 t v1(t) a1 (t) v2 (t) a2 (t)

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Chapter 04 Reading Questions Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Chapter 4 Reading Quiz Question 1 Part A The cells of all plants have _____. ANSWER: Chapter 4 Reading Quiz Question 2 Part A Which of the following is a difference between cellular respiration and anaerobic respiration? ANSWER: Chapter 4 Reading Quiz Question 16 Part A Which of the following are found in the cells of a dog but not in the bacteria that are found on a dog’s fur? ANSWER: chloroplasts but not mitochondria and use carbohydrates to power their functions chloroplasts and mitochondria and use carbohydrates to power their functions mitochondria but not chloroplasts and use proteins to power their functions chloroplasts but not mitochondria and use proteins to power their functions Only anaerobic respiration produces carbon dioxide. Only anaerobic respiration produces water. Only cellular respiration breaks down carbohydrates. Only cellular respiration uses oxygen to break down carbohydrates. membrane-enclosed nucleus, chloroplasts, and cytoplasm mitochondria, nuclei, and cytoplasm membrane-enclosed nucleus and mitochondria mitochondria and chloroplasts Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 7 5/21/2014 7:58 PM Chapter 4 Reading Quiz Question 18 Part A The products of photosynthesis are the materials that react in the process of _____. ANSWER: Chapter 4 Reading Quiz Question 4 Part A Tissues within an organism’s body are different from each other because the cells in each tissue _____. ANSWER: Chapter 4 Reading Quiz Question 5 Part A An egg and a sperm are _____. ANSWER: Chapter 4 Reading Quiz Question 6 chemosynthesis cellular respiration osmosis anaerobic respiration are produced by different combinations of eggs and sperm activate different portions of the identical DNA remove the DNA that they do not need have different types of DNA gametes that combine in asexual reproduction to produce a zygote zygotes that combine in asexual reproduction to produce a gamete zygotes that combine in sexual reproduction to produce a gamete gametes that combine in sexual reproduction to produce a zygote Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 7 5/21/2014 7:58 PM Part A The growth of a population of sparrows accelerates with each generation. The chances of dying are about the same for these sparrows, regardless of age. This population of sparrows therefore demonstrates _____. ANSWER: Chapter 4 Reading Quiz Question 7 Part A A population will grow the fastest when total fertility rates are _____. ANSWER: Chapter 4 Reading Quiz Question 20 Part A A population will not change in size if the _____. ANSWER: Chapter 4 Reading Quiz Question 8 Part A exponential growth and a type III survivorship curve exponential growth and a type II survivorship curve arithmetic growth and a type II survivorship curve arithmetic growth and a type I survivorship curve low and generation times are long high and generation times are short high and generation times are long low and generation times are short birth rate equals the emigration rate birth rate equals the immigration rate and the death rate equals the emigration rate death rate equals the emigration rate birth rate equals the death rate and the immigration rate equals the emigration rate Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 7 5/21/2014 7:58 PM As a population increases to near its carrying capacity, _____. ANSWER: Chapter 4 Reading Quiz Question 10 Part A The various activities that define an organism’s role in an ecosystem are that organism’s _____. ANSWER: Chapter 4 Reading Quiz Question 22 Part A Largemouth bass and rainbow trout are stocked in a small, deep 10-acre pond. The bass are most active in 20-30 °C water, and the rainbow trout prefer water at about 6-22 °C. We expect to find few places in this pond where the two species interact because they have different _____. ANSWER: Chapter 4 Reading Quiz Question 11 Part A The evolutionary concept of fitness is most closely associated with _____. birth rates decline, death rates increase, and the overall rate of population growth declines birth rates decline, death rates decline, and the overall rate of population growth increases birth rates decline, death rates decline, and the overall rate of population growth declines birth rates increase, death rates increase, and the overall rate of population growth declines range of tolerance habitat ecological niche fitness generation times ranges of tolerance growth rates carrying capacities Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 7 5/21/2014 7:58 PM ANSWER: Chapter 4 Reading Quiz Question 12 Part A The ultimate source of new inherited traits in a population is _____. ANSWER: Chapter 4 Reading Quiz Question 23 Part A Studies of human birth weight and infant health reveal that babies who are heavier than 10 pounds or lighter than 6 pounds have decreased survival rates. This pattern of survival is an example of _____. ANSWER: Chapter 4 Reading Quiz Question 14 Part A From an evolutionary perspective, the most important property that defines a species is _____. ANSWER: feeding reproduction carrying capacity mutations adaptation survivorship natural selection mutation disruptive selection directional selection bimodal selection stabilizing selection Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 7 5/21/2014 7:58 PM Chapter 4 Reading Quiz Question 24 Part A In Missouri, the marbled salamander breeds in the late fall (October-December). In the same region, the closely related spotted salamander breeds in early spring (February-March). Thus, these species are kept separate because of _____. ANSWER: Chapter 4 Reading Quiz Question 15 Part A Which one of the following correctly lists the levels of classification from specific to general? ANSWER: Chapter 4 Reading Quiz Question 25 Part A In a phylogenetic tree, all of the species in one family will _____. ANSWER: the ability of a species to extend the its range its type of natural selection its dietary habits reproductive isolation behavioral isolation structural isolation temporal isolation geographic isolation genus, species, order, family, class, phylum, kingdom species, genus, class, family, order, phylum, kingdom species, genus, family, order, class, phylum, kingdom kingdom, phylum, class, order, family, genus, species Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 7 5/21/2014 7:58 PM Score Summary: Your score on this assignment is 0.0%. You received 0 out of a possible total of 19 points. have the same scientific name have identical ecological niches be scattered throughout the tree be clustered together in the tree Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 7 5/21/2014 7:58 PM

Chapter 04 Reading Questions Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Chapter 4 Reading Quiz Question 1 Part A The cells of all plants have _____. ANSWER: Chapter 4 Reading Quiz Question 2 Part A Which of the following is a difference between cellular respiration and anaerobic respiration? ANSWER: Chapter 4 Reading Quiz Question 16 Part A Which of the following are found in the cells of a dog but not in the bacteria that are found on a dog’s fur? ANSWER: chloroplasts but not mitochondria and use carbohydrates to power their functions chloroplasts and mitochondria and use carbohydrates to power their functions mitochondria but not chloroplasts and use proteins to power their functions chloroplasts but not mitochondria and use proteins to power their functions Only anaerobic respiration produces carbon dioxide. Only anaerobic respiration produces water. Only cellular respiration breaks down carbohydrates. Only cellular respiration uses oxygen to break down carbohydrates. membrane-enclosed nucleus, chloroplasts, and cytoplasm mitochondria, nuclei, and cytoplasm membrane-enclosed nucleus and mitochondria mitochondria and chloroplasts Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 7 5/21/2014 7:58 PM Chapter 4 Reading Quiz Question 18 Part A The products of photosynthesis are the materials that react in the process of _____. ANSWER: Chapter 4 Reading Quiz Question 4 Part A Tissues within an organism’s body are different from each other because the cells in each tissue _____. ANSWER: Chapter 4 Reading Quiz Question 5 Part A An egg and a sperm are _____. ANSWER: Chapter 4 Reading Quiz Question 6 chemosynthesis cellular respiration osmosis anaerobic respiration are produced by different combinations of eggs and sperm activate different portions of the identical DNA remove the DNA that they do not need have different types of DNA gametes that combine in asexual reproduction to produce a zygote zygotes that combine in asexual reproduction to produce a gamete zygotes that combine in sexual reproduction to produce a gamete gametes that combine in sexual reproduction to produce a zygote Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 7 5/21/2014 7:58 PM Part A The growth of a population of sparrows accelerates with each generation. The chances of dying are about the same for these sparrows, regardless of age. This population of sparrows therefore demonstrates _____. ANSWER: Chapter 4 Reading Quiz Question 7 Part A A population will grow the fastest when total fertility rates are _____. ANSWER: Chapter 4 Reading Quiz Question 20 Part A A population will not change in size if the _____. ANSWER: Chapter 4 Reading Quiz Question 8 Part A exponential growth and a type III survivorship curve exponential growth and a type II survivorship curve arithmetic growth and a type II survivorship curve arithmetic growth and a type I survivorship curve low and generation times are long high and generation times are short high and generation times are long low and generation times are short birth rate equals the emigration rate birth rate equals the immigration rate and the death rate equals the emigration rate death rate equals the emigration rate birth rate equals the death rate and the immigration rate equals the emigration rate Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 7 5/21/2014 7:58 PM As a population increases to near its carrying capacity, _____. ANSWER: Chapter 4 Reading Quiz Question 10 Part A The various activities that define an organism’s role in an ecosystem are that organism’s _____. ANSWER: Chapter 4 Reading Quiz Question 22 Part A Largemouth bass and rainbow trout are stocked in a small, deep 10-acre pond. The bass are most active in 20-30 °C water, and the rainbow trout prefer water at about 6-22 °C. We expect to find few places in this pond where the two species interact because they have different _____. ANSWER: Chapter 4 Reading Quiz Question 11 Part A The evolutionary concept of fitness is most closely associated with _____. birth rates decline, death rates increase, and the overall rate of population growth declines birth rates decline, death rates decline, and the overall rate of population growth increases birth rates decline, death rates decline, and the overall rate of population growth declines birth rates increase, death rates increase, and the overall rate of population growth declines range of tolerance habitat ecological niche fitness generation times ranges of tolerance growth rates carrying capacities Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 7 5/21/2014 7:58 PM ANSWER: Chapter 4 Reading Quiz Question 12 Part A The ultimate source of new inherited traits in a population is _____. ANSWER: Chapter 4 Reading Quiz Question 23 Part A Studies of human birth weight and infant health reveal that babies who are heavier than 10 pounds or lighter than 6 pounds have decreased survival rates. This pattern of survival is an example of _____. ANSWER: Chapter 4 Reading Quiz Question 14 Part A From an evolutionary perspective, the most important property that defines a species is _____. ANSWER: feeding reproduction carrying capacity mutations adaptation survivorship natural selection mutation disruptive selection directional selection bimodal selection stabilizing selection Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 7 5/21/2014 7:58 PM Chapter 4 Reading Quiz Question 24 Part A In Missouri, the marbled salamander breeds in the late fall (October-December). In the same region, the closely related spotted salamander breeds in early spring (February-March). Thus, these species are kept separate because of _____. ANSWER: Chapter 4 Reading Quiz Question 15 Part A Which one of the following correctly lists the levels of classification from specific to general? ANSWER: Chapter 4 Reading Quiz Question 25 Part A In a phylogenetic tree, all of the species in one family will _____. ANSWER: the ability of a species to extend the its range its type of natural selection its dietary habits reproductive isolation behavioral isolation structural isolation temporal isolation geographic isolation genus, species, order, family, class, phylum, kingdom species, genus, class, family, order, phylum, kingdom species, genus, family, order, class, phylum, kingdom kingdom, phylum, class, order, family, genus, species Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 7 5/21/2014 7:58 PM Score Summary: Your score on this assignment is 0.0%. You received 0 out of a possible total of 19 points. have the same scientific name have identical ecological niches be scattered throughout the tree be clustered together in the tree Chapter 04 Reading Questions http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 7 5/21/2014 7:58 PM

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Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to ? ANSWER: Correct mg Equal to Larger than Smaller than mg mg mg Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension in the third rope? Express your answer using two significant figures. ANSWER: Correct Part B What is the direction of the tension in the third rope? Express your answer using two significant figures. T  3 T3 = 94 N T  3 Typesetting math: 100% ANSWER: Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, , to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude ? = 58   below horizontal n ws n F = 0 fpull Typesetting math: 100% ANSWER: ANSWER: Correct Part B A B C D The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase’s weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Typesetting math: 100% Now assume that the man of weight is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled is a force that has the same magnitude as the man’s weight. ANSWER: wm n wm Typesetting math: 100% ANSWER: Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880 stands on a roof that is sloped at 18 . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct A B C D The magnitude of the normal force is equal to the magnitude of the suitcase’s weight. The magnitude of the normal force is equal to the magnitude of the suitcase’s weight minus the magnitude of the man’s weight. The magnitude of the normal force is equal to the sum of the magnitude of the man’s weight and the magnitude of the suitcase’s weight. The magnitude of the normal force is less than the magnitude of the suitcase’s weight. N  n = 840 N Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B For diagram the part A, find the value of the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: kg ax x ax = -0.67 m s2 ay, y Typesetting math: 100% Correct Part C For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D For diagram the part C, find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 ax x ax = 0.67 m s2 ay y Typesetting math: 100% Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay = 0 m s2 kg ax x ax = 0.99 m s2 Typesetting math: 100% Part B Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay y ay = 0 m s2 ax x ax = -0.18 m s2 ay y ay = 0 m s2 Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0 box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B The box moves at a steady = 4.80 ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C The box = 4.80 and = 4.60 ? Express your answer to three significant figures and include the appropriate units. ANSWER: kg T = 0 N vx m/s T = 0 N vx m/s ax m/s2 Typesetting math: 100% Correct Problem 6.14 It takes the elevator in a skyscraper 4.5 to reach its cruising speed of 11 . A 60 passenger gets aboard on the ground floor. Part A What is the passenger’s weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the passenger’s weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the passenger’s weight after the elevator reaches its cruising speed? T = 244 N s m/s kg w = 590 N w = 730 N Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Block on an Incline A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: w = 590 N  F  w F n F  f Typesetting math: 100% Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: Correct only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn’t accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the y-axis is also , as shown in the figure. ANSWER: F  n Fn Fn Ff Fw  F n Fny F  n Fn  F  n  Typesetting math: 100% Hint 2. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the x-axis is also , as shown in the figure. ANSWER: ANSWER: Fny = Fncos() F f Ffy F f Ff  F  f  Ffy = Ffsin() Fy = 0 = Fncos() + Ffsin() − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the x component of Write an expression for , the x component of the force , using coordinate system b. Express your answer in terms of and . ANSWER: ANSWER: Correct Part E To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not . Hint 1. How to approach the problem From your answers to the previous two parts you should have two force equations ( and ). Combine these equations to eliminate . The key is to multiply the Fn Ff Fw  F n Fnx F  n Fn  Fnx = −Fnsin() Fx = 0 = −Fnsin() + Ffcos() Fn Fw Ff Fn Fw  Ff Typesetting math: 100% Fy = 0 Fx = 0 Ff equation for the y components by and the equation for the x components by , then add or subtract the two equations to eliminate the term . An alternative motivation for the algebra is to eliminate the trig functions in front of by using the trig identity . At the very least this would result in an equation that is simple to solve for . ANSWER: Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed (although they can be smaller than this or even zero). Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by and (or , or something similar) respectively. These are the components of the overall contact force: perpendicular to and parallel to the plane of contact. Kinetic friction when surfaces slide cos  sin  Ff cos() sin() Fn sin2() + cos2 () = 1 Fn Fn = Fwcos() Fy = Fn − FW cos() = 0 Fn F = ma μn n f Ffric n f Typesetting math: 100% When one surface is sliding past the other, experiments show three things about the friction force (denoted ): The frictional force opposes the relative motion at the 1. point of contact, 2. is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated . As long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other). Static friction when surfaces don’t slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed . The equation is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: Correct Part B fk fk μk fk = μkn μsn μs μs μk fs ! μsn μsn fs = μsn The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μkn μkn μkn Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed . If the magnitude of static friction needed to keep acceleration equal to zero exceeds , then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than , or equal to depends on the magnitude of the other forces (if any) as well as the acceleration of the object through . Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point “break loose” and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μsn μsn μsn μsn μsn fs = μsn μsn μsn F = ma Fg The force of kinetic friction is smaller than that of maximum static friction, but remains the same. Once the box is moving, is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, is larger than the force of maximum static friction. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Fg Fg Fg Fg Typesetting math: 100% Correct At the point when the box finally does “break loose,” you know that the component of the box’s weight that is parallel to the board just exceeds (i.e., this component of gravitational force on the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box’s weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction must be less than the force of static friction which implies , as expected. Part D Consider a problem in which a car of mass is on a road tilted at an angle . The normal force Select the best answer. ANSWER: Correct The key point is that contact forces must be determined from Newton’s equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems–you must determine from . Problem 6.17 Bonnie and Clyde are sliding a 323 bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375 of force while Bonnie pulls forward on a rope with 335 of force. μsn μsn μkn μsn μk < μs M  is found using n = Mg n = Mg cos() n = Mg cos() F  = Ma  = 0 n F = ma kg N N Typesetting math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: Correct Problem 6.19 A crate is placed on a horizontal conveyor belt. The materials are such that and . Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: 0.224 10 kg μs = 0.5 μk = 0.3 Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct amax = 4.9 m s2 Typesetting math: 100% Problem 6.28 A 1100 steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T1 = 7000 N Typesetting math: 100% Correct Problem 6.35 The position of a 1.4 mass is given by , where is in seconds. Part A What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.39 T2 = 4800 N kg x = (2t3 − 3t2 )m t t = 0 s F = -8.4 N t = 1 s F = 8.4 N Typesetting math: 100% A rifle with a barrel length of 61 fires a 8 bullet with a horizontal speed of 400 . The bullet strikes a block of wood and penetrates to a depth of 11 . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21 . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0 away, asks you for the box of nails. Rather than carry the 2.0 box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. cm g m/s cm fk = 5800 N = 5.5×10−4 t s  m kg Typesetting math: 100% ANSWER: Correct Problem 6.54 The 2.0 wood box in the figure slides down a vertical wood wall while you push on it at a 45 angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct v = 3.9 ms kg  F = 23 N Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%

Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to ? ANSWER: Correct mg Equal to Larger than Smaller than mg mg mg Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension in the third rope? Express your answer using two significant figures. ANSWER: Correct Part B What is the direction of the tension in the third rope? Express your answer using two significant figures. T  3 T3 = 94 N T  3 Typesetting math: 100% ANSWER: Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, , to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude ? = 58   below horizontal n ws n F = 0 fpull Typesetting math: 100% ANSWER: ANSWER: Correct Part B A B C D The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase’s weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Typesetting math: 100% Now assume that the man of weight is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled is a force that has the same magnitude as the man’s weight. ANSWER: wm n wm Typesetting math: 100% ANSWER: Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880 stands on a roof that is sloped at 18 . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct A B C D The magnitude of the normal force is equal to the magnitude of the suitcase’s weight. The magnitude of the normal force is equal to the magnitude of the suitcase’s weight minus the magnitude of the man’s weight. The magnitude of the normal force is equal to the sum of the magnitude of the man’s weight and the magnitude of the suitcase’s weight. The magnitude of the normal force is less than the magnitude of the suitcase’s weight. N  n = 840 N Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B For diagram the part A, find the value of the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: kg ax x ax = -0.67 m s2 ay, y Typesetting math: 100% Correct Part C For diagram , find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D For diagram the part C, find the value of , the -component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 ax x ax = 0.67 m s2 ay y Typesetting math: 100% Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0 object. Part A Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay = 0 m s2 kg ax x ax = 0.99 m s2 Typesetting math: 100% Part B Find the value of , the component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part D Find the value of , the component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: Correct ay y ay = 0 m s2 ax x ax = -0.18 m s2 ay y ay = 0 m s2 Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0 box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part B The box moves at a steady = 4.80 ? Express your answer as an integer and include the appropriate units. ANSWER: Correct Part C The box = 4.80 and = 4.60 ? Express your answer to three significant figures and include the appropriate units. ANSWER: kg T = 0 N vx m/s T = 0 N vx m/s ax m/s2 Typesetting math: 100% Correct Problem 6.14 It takes the elevator in a skyscraper 4.5 to reach its cruising speed of 11 . A 60 passenger gets aboard on the ground floor. Part A What is the passenger’s weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the passenger’s weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part C What is the passenger’s weight after the elevator reaches its cruising speed? T = 244 N s m/s kg w = 590 N w = 730 N Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Block on an Incline A block lies on a plane raised an angle from the horizontal. Three forces act upon the block: , the force of gravity; , the normal force; and , the force of friction. The coefficient of friction is large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: w = 590 N  F  w F n F  f Typesetting math: 100% Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: Correct only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w only only only and and and and and F  f F  n F  w F  f F  n F  f F  w F  n F w F  f F  n F w Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn’t accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, , using vertical coordinate system b. In these coordinates you will find the magnitude appearing in both the x and y equations, each multiplied by a trigonometric function. Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the y-axis is also , as shown in the figure. ANSWER: F  n Fn Fn Ff Fw  F n Fny F  n Fn  F  n  Typesetting math: 100% Hint 2. Find the y component of Write an expression for , the y component of the force , using coordinate system b. Express your answer in terms of and . Hint 1. Some geometry help – a useful angle The smaller angle between and the x-axis is also , as shown in the figure. ANSWER: ANSWER: Fny = Fncos() F f Ffy F f Ff  F  f  Ffy = Ffsin() Fy = 0 = Fncos() + Ffsin() − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables , , , and . Hint 1. Find the x component of Write an expression for , the x component of the force , using coordinate system b. Express your answer in terms of and . ANSWER: ANSWER: Correct Part E To find the magnitude of the normal force, you must express in terms of since is an unknown. Using the equations you found in the two previous parts, find an expression for involving and but not . Hint 1. How to approach the problem From your answers to the previous two parts you should have two force equations ( and ). Combine these equations to eliminate . The key is to multiply the Fn Ff Fw  F n Fnx F  n Fn  Fnx = −Fnsin() Fx = 0 = −Fnsin() + Ffcos() Fn Fw Ff Fn Fw  Ff Typesetting math: 100% Fy = 0 Fx = 0 Ff equation for the y components by and the equation for the x components by , then add or subtract the two equations to eliminate the term . An alternative motivation for the algebra is to eliminate the trig functions in front of by using the trig identity . At the very least this would result in an equation that is simple to solve for . ANSWER: Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is , which leads immediately to the result obtained here for . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed (although they can be smaller than this or even zero). Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by and (or , or something similar) respectively. These are the components of the overall contact force: perpendicular to and parallel to the plane of contact. Kinetic friction when surfaces slide cos  sin  Ff cos() sin() Fn sin2() + cos2 () = 1 Fn Fn = Fwcos() Fy = Fn − FW cos() = 0 Fn F = ma μn n f Ffric n f Typesetting math: 100% When one surface is sliding past the other, experiments show three things about the friction force (denoted ): The frictional force opposes the relative motion at the 1. point of contact, 2. is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated . As long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other). Static friction when surfaces don’t slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed . The equation is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: Correct Part B fk fk μk fk = μkn μsn μs μs μk fs ! μsn μsn fs = μsn The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μkn μkn μkn Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed . If the magnitude of static friction needed to keep acceleration equal to zero exceeds , then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than , or equal to depends on the magnitude of the other forces (if any) as well as the acceleration of the object through . Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point “break loose” and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The frictional force is always equal to . The frictional force is always less than . The frictional force is determined by other forces on the objects so it can be either equal to or less than . μsn μsn μsn μsn μsn fs = μsn μsn μsn F = ma Fg The force of kinetic friction is smaller than that of maximum static friction, but remains the same. Once the box is moving, is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, is larger than the force of maximum static friction. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Fg Fg Fg Fg Typesetting math: 100% Correct At the point when the box finally does “break loose,” you know that the component of the box’s weight that is parallel to the board just exceeds (i.e., this component of gravitational force on the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box’s weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction must be less than the force of static friction which implies , as expected. Part D Consider a problem in which a car of mass is on a road tilted at an angle . The normal force Select the best answer. ANSWER: Correct The key point is that contact forces must be determined from Newton’s equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems–you must determine from . Problem 6.17 Bonnie and Clyde are sliding a 323 bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375 of force while Bonnie pulls forward on a rope with 335 of force. μsn μsn μkn μsn μk < μs M  is found using n = Mg n = Mg cos() n = Mg cos() F  = Ma  = 0 n F = ma kg N N Typesetting math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: Correct Problem 6.19 A crate is placed on a horizontal conveyor belt. The materials are such that and . Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: 0.224 10 kg μs = 0.5 μk = 0.3 Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct amax = 4.9 m s2 Typesetting math: 100% Problem 6.28 A 1100 steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: kg T1 = 7000 N Typesetting math: 100% Correct Problem 6.35 The position of a 1.4 mass is given by , where is in seconds. Part A What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B What is the net horizontal force on the mass at ? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.39 T2 = 4800 N kg x = (2t3 − 3t2 )m t t = 0 s F = -8.4 N t = 1 s F = 8.4 N Typesetting math: 100% A rifle with a barrel length of 61 fires a 8 bullet with a horizontal speed of 400 . The bullet strikes a block of wood and penetrates to a depth of 11 . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21 . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0 away, asks you for the box of nails. Rather than carry the 2.0 box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. cm g m/s cm fk = 5800 N = 5.5×10−4 t s  m kg Typesetting math: 100% ANSWER: Correct Problem 6.54 The 2.0 wood box in the figure slides down a vertical wood wall while you push on it at a 45 angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: Correct v = 3.9 ms kg  F = 23 N Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%

Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You … Read More...
Chapter 07 Homework Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy BioFlix Quiz: The Carbon Cycle Watch the animation at left before answering the questions below. Part A An organism gets carbon by using carbon dioxide in the atmosphere to make sugar molecules. This organism is a Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: Correct During photosynthesis, producers use carbon dioxide to make sugar molecules. Part B Which organisms play a role in returning carbon to the atmosphere? Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: higher-level consumer. producer. primary consumer. decomposer. None of the above Consumers and decomposers, but not producers. Producers only. Decomposers only. Consumers only. Producers, consumers, and decomposers. Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 7 5/21/2014 8:02 PM Correct Producers, consumers, and decomposers all return carbon dioxide to the atmosphere during cellular respiration. Part C Every carbon atom in the organic molecules that make up your body MUST recently have been part of Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: Correct You are a consumer, and all your carbon comes ultimately from plants and other producers. Part D Imagine following a single carbon atom through the carbon cycle. Which of the following is a possible path for the carbon atom to take? Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: Correct Carbon moves from the atmosphere into a producer (such as a plant), up the food chain, and then back to the atmosphere during cellular respiration. Part E Which process or processes return carbon to the atmosphere? Hint 1. Review the animation. ANSWER: Correct Cellular respiration results in the release of carbon dioxide to the atmosphere. a higher-level consumer. a primary consumer. a decomposer. a producer. a sugar molecule made in one of your chloroplasts. The atmosphere; a plant; a higher-level consumer; then back to the atmosphere. The atmosphere; a plant; an herbivore; another plant; then back to the atmosphere. The atmosphere, a plant, a herbivore, a decomposer, then back to the atmosphere The atmosphere; a decomposer; a higher-level consumer; then back to the atmosphere. The atmosphere; a decomposer; then back to the atmosphere. Cellular respiration only Photosynthesis only Cellular respiration and photosynthesis Breakdown of large organic molecules into smaller organic molecules Cellular respiration and the breakdown of large organic molecules into smaller organic molecules Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 7 5/21/2014 8:02 PM Activity: The Nitrogen Cycle Click here to complete this activity. Then answer the questions. Part A Nitrifying bacteria convert _____ to _____. ANSWER: Correct Nitrifying bacteria convert ammonium to nitrites. Part B _____ removes nitrogen from the atmosphere. ANSWER: Correct Nitrogen fixation is the conversion of nitrogen gas to a form that can be used by plants (and other organisms). Part C Assimilation is indicated by the letter(s) _____. nitrogen gas … ammonium nitrogen gas … nitrates ammonium … nitrites nitrates … nitrogen gas ammonium … nitrogen gas Denitrification Nitrification Mineralization Nitrogen fixation Assimilation Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 7 5/21/2014 8:02 PM ANSWER: Correct Assimilation is the uptake of nutrients into an organism. Part D Nitrogen-fixing bacteria is(are) indicated by the letter(s) _____. ANSWER: Correct Both of these pointers are indicating nitrogen-fixing bacteria. Nitrogen fixation is the conversion of nitrogen to a form that plants can use. Part E Nitrification is indicated by the letter(s) _____. ANSWER: C B A D and E C and D B and C A and B D and E C and D A Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 7 5/21/2014 8:02 PM Correct Nitrification is the conversion of organic nitrogen-containing compounds to nitrites and nitrates. Part F Denitrifying bacteria convert _____ to _____. ANSWER: Correct Denitrifying bacteria convert nitrates to nitrogen gas. Part G Which one of these is a nitrate? ANSWER: Correct NO3 – is a nitrate. Part H Which one of these is a nitrite? ANSWER: Correct This is a nitrite. GeoScience: Earth’s Water and the Hydrologic Cycle A B B and C D and E B and E nitrogen gas … nitrites nitrogen gas … ammonium nitrates … nitrogen gas ammonium … nitrogen gas nitrogen gas … nitrates NO2 – NH4 – NH2 SH NO3 – PO4 – NH2 NH4 – NO2 – NO3 – Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 7 5/21/2014 8:02 PM When you have finished, answer the questions. Part A The largest percentage of fresh water today is located in: ANSWER: Correct Ice sheets and glaciers are the greatest single repository of fresh water: they contain 77.3% of all Earth’s fresh water and 99.357% of all Earth’s surface fresh water. Part B Earth’s oceans hold: ANSWER: Correct The oceans contain 97.22% of all water, comprising about 1.321 billion cubic kilometers of salt water. This leaves only 2.78% of all of Earth’s water as fresh water (non-oceanic). Part C Which of the following is true of the hydrologic cycle? ANSWER: Correct About 20% of the moisture evaporated from the ocean combines with 2% of land-derived moisture to produce 22% of all precipitation that falls over land. Clearly, the bulk of continental precipitation comes from the oceanic portion of the cycle. Concept Review: Eutrophication Can you sequence the steps in the eutrophication process that occurs in a body of water? Part A Drag each statement to the appropriate location in the flowchart of the eutrophication process. ANSWER: soil. ice sheets and glaciers. the rivers and lakes of the world. groundwater resources. about the same amount of water as all groundwater sources combined. most of the fresh water on Earth. the bulk of all of the water found on Earth. about the same amount of water as all Earth’s rivers and lakes combined. Atmospheric water and surface water do not mix. Most evaporation on Earth occurs over the continents. The bulk of the precipitation occurs over land. Most of the water that falls on the continents is derived from the oceans. Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 7 5/21/2014 8:02 PM Concept Review: Biogeochemical Cycles Can you sort the items by which biogeochemical cycle they apply to? Part A Drag each description to the appropriate bin. ANSWER: Score Summary: Your score on this assignment is 62.3%. You received 12.45 out of a possible total of 20 points. Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 7 5/21/2014 8:02 PM

Chapter 07 Homework Due: 11:59pm on Friday, May 23, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy BioFlix Quiz: The Carbon Cycle Watch the animation at left before answering the questions below. Part A An organism gets carbon by using carbon dioxide in the atmosphere to make sugar molecules. This organism is a Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: Correct During photosynthesis, producers use carbon dioxide to make sugar molecules. Part B Which organisms play a role in returning carbon to the atmosphere? Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: higher-level consumer. producer. primary consumer. decomposer. None of the above Consumers and decomposers, but not producers. Producers only. Decomposers only. Consumers only. Producers, consumers, and decomposers. Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 1 of 7 5/21/2014 8:02 PM Correct Producers, consumers, and decomposers all return carbon dioxide to the atmosphere during cellular respiration. Part C Every carbon atom in the organic molecules that make up your body MUST recently have been part of Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: Correct You are a consumer, and all your carbon comes ultimately from plants and other producers. Part D Imagine following a single carbon atom through the carbon cycle. Which of the following is a possible path for the carbon atom to take? Hint 1. Review the animation or your Study Sheet for The Carbon Cycle. ANSWER: Correct Carbon moves from the atmosphere into a producer (such as a plant), up the food chain, and then back to the atmosphere during cellular respiration. Part E Which process or processes return carbon to the atmosphere? Hint 1. Review the animation. ANSWER: Correct Cellular respiration results in the release of carbon dioxide to the atmosphere. a higher-level consumer. a primary consumer. a decomposer. a producer. a sugar molecule made in one of your chloroplasts. The atmosphere; a plant; a higher-level consumer; then back to the atmosphere. The atmosphere; a plant; an herbivore; another plant; then back to the atmosphere. The atmosphere, a plant, a herbivore, a decomposer, then back to the atmosphere The atmosphere; a decomposer; a higher-level consumer; then back to the atmosphere. The atmosphere; a decomposer; then back to the atmosphere. Cellular respiration only Photosynthesis only Cellular respiration and photosynthesis Breakdown of large organic molecules into smaller organic molecules Cellular respiration and the breakdown of large organic molecules into smaller organic molecules Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 2 of 7 5/21/2014 8:02 PM Activity: The Nitrogen Cycle Click here to complete this activity. Then answer the questions. Part A Nitrifying bacteria convert _____ to _____. ANSWER: Correct Nitrifying bacteria convert ammonium to nitrites. Part B _____ removes nitrogen from the atmosphere. ANSWER: Correct Nitrogen fixation is the conversion of nitrogen gas to a form that can be used by plants (and other organisms). Part C Assimilation is indicated by the letter(s) _____. nitrogen gas … ammonium nitrogen gas … nitrates ammonium … nitrites nitrates … nitrogen gas ammonium … nitrogen gas Denitrification Nitrification Mineralization Nitrogen fixation Assimilation Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 3 of 7 5/21/2014 8:02 PM ANSWER: Correct Assimilation is the uptake of nutrients into an organism. Part D Nitrogen-fixing bacteria is(are) indicated by the letter(s) _____. ANSWER: Correct Both of these pointers are indicating nitrogen-fixing bacteria. Nitrogen fixation is the conversion of nitrogen to a form that plants can use. Part E Nitrification is indicated by the letter(s) _____. ANSWER: C B A D and E C and D B and C A and B D and E C and D A Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 4 of 7 5/21/2014 8:02 PM Correct Nitrification is the conversion of organic nitrogen-containing compounds to nitrites and nitrates. Part F Denitrifying bacteria convert _____ to _____. ANSWER: Correct Denitrifying bacteria convert nitrates to nitrogen gas. Part G Which one of these is a nitrate? ANSWER: Correct NO3 – is a nitrate. Part H Which one of these is a nitrite? ANSWER: Correct This is a nitrite. GeoScience: Earth’s Water and the Hydrologic Cycle A B B and C D and E B and E nitrogen gas … nitrites nitrogen gas … ammonium nitrates … nitrogen gas ammonium … nitrogen gas nitrogen gas … nitrates NO2 – NH4 – NH2 SH NO3 – PO4 – NH2 NH4 – NO2 – NO3 – Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 5 of 7 5/21/2014 8:02 PM When you have finished, answer the questions. Part A The largest percentage of fresh water today is located in: ANSWER: Correct Ice sheets and glaciers are the greatest single repository of fresh water: they contain 77.3% of all Earth’s fresh water and 99.357% of all Earth’s surface fresh water. Part B Earth’s oceans hold: ANSWER: Correct The oceans contain 97.22% of all water, comprising about 1.321 billion cubic kilometers of salt water. This leaves only 2.78% of all of Earth’s water as fresh water (non-oceanic). Part C Which of the following is true of the hydrologic cycle? ANSWER: Correct About 20% of the moisture evaporated from the ocean combines with 2% of land-derived moisture to produce 22% of all precipitation that falls over land. Clearly, the bulk of continental precipitation comes from the oceanic portion of the cycle. Concept Review: Eutrophication Can you sequence the steps in the eutrophication process that occurs in a body of water? Part A Drag each statement to the appropriate location in the flowchart of the eutrophication process. ANSWER: soil. ice sheets and glaciers. the rivers and lakes of the world. groundwater resources. about the same amount of water as all groundwater sources combined. most of the fresh water on Earth. the bulk of all of the water found on Earth. about the same amount of water as all Earth’s rivers and lakes combined. Atmospheric water and surface water do not mix. Most evaporation on Earth occurs over the continents. The bulk of the precipitation occurs over land. Most of the water that falls on the continents is derived from the oceans. Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 6 of 7 5/21/2014 8:02 PM Concept Review: Biogeochemical Cycles Can you sort the items by which biogeochemical cycle they apply to? Part A Drag each description to the appropriate bin. ANSWER: Score Summary: Your score on this assignment is 62.3%. You received 12.45 out of a possible total of 20 points. Chapter 07 Homework http://session.masteringenvironmentalscience.com/myct/assignmentPrintV… 7 of 7 5/21/2014 8:02 PM

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Chapter 9 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: The blocks can move in only one dimension, namely, 1. along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. m1 m2 t v1(t) a1 (t) v2 (t) a2 (t) Part A Find , the x component of the total momentum of the system at time . Express your answer in terms of , , , and . ANSWER: Part B Find the time derivative of the x component of the system’s total momentum. Express your answer in terms of , , , and . You did not open hints for this part. ANSWER: Why did we bother with all this math? The expression for the derivative of momentum that we just obtained will be useful in reaching our desired conclusion, if only for this very special case. Part C The quantity (mass times acceleration) is dimensionally equivalent to which of the following? ANSWER: p(t) t m1 m2 v1 (t) v2 (t) p(t) = dp(t)/dt a1 (t) a2 (t) m1 m2 dp(t)/dt = ma Part D Acceleration is due to which of the following physical quantities? ANSWER: Part E Since we have assumed that the system composed of blocks 1 and 2 is closed, what could be the reason for the acceleration of block 1? You did not open hints for this part. ANSWER: momentum energy force acceleration inertia velocity speed energy momentum force Part F This question will be shown after you complete previous question(s). Part G Let us denote the x component of the force exerted by block 1 on block 2 by , and the x component of the force exerted by block 2 on block 1 by . Which of the following pairs equalities is a direct consequence of Newton’s second law? ANSWER: Part H Let us recall that we have denoted the force exerted by block 1 on block 2 by , and the force exerted by block 2 on block 1 by . If we suppose that is greater than , which of the following statements about forces is true? You did not open hints for this part. the large mass of block 1 air resistance Earth’s gravitational attraction a force exerted by block 2 on block 1 a force exerted by block 1 on block 2 F12 F21 and and and and F12 = m2a2 F21 = m1a1 F12 = m1a1 F21 = m2a2 F12 = m1a2 F21 = m2a1 F12 = m2a1 F21 = m1a2 F12 F21 m1 m2 ANSWER: Part I Now recall the expression for the time derivative of the x component of the system’s total momentum: . Considering the information that you now have, choose the best alternative for an equivalent expression to . You did not open hints for this part. ANSWER: Impulse and Momentum Ranking Task Six automobiles are initially traveling at the indicated velocities. The automobiles have different masses and velocities. The drivers step on the brakes and all automobiles are brought to rest. Part A Rank these automobiles based on the magnitude of their momentum before the brakes are applied, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. ANSWER: Both forces have equal magnitudes. |F12 | > |F21| |F21 | > |F12| dpx(t)/dt = Fx dpx(t)/dt 0 nonzero constant kt kt2 Part B Rank these automobiles based on the magnitude of the impulse needed to stop them, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. You did not open hints for this part. ANSWER: Part C Rank the automobiles based on the magnitude of the force needed to stop them, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. You did not open hints for this part. ANSWER: A Game of Frictionless Catch Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, , is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest. Chuck then picks up a ball of mass and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is . The speed of the thrown ball relative to the ground is . Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie’s speed relative to the ground after she catches the ball is . When answering the questions in this problem, keep the following in mind: The original mass of Chuck and his cart does not include the 1. mass of the ball. 2. The speed of an object is the magnitude of its velocity. An object’s speed will always be a nonnegative quantity. mcart mball vc vb vj mcart Part A Find the relative speed between Chuck and the ball after Chuck has thrown the ball. Express the speed in terms of and . You did not open hints for this part. ANSWER: Part B What is the speed of the ball (relative to the ground) while it is in the air? Express your answer in terms of , , and . You did not open hints for this part. ANSWER: Part C What is Chuck’s speed (relative to the ground) after he throws the ball? Express your answer in terms of , , and . u vc vb u = vb mball mcart u vb = vc mball mcart u You did not open hints for this part. ANSWER: Part D Find Jackie’s speed (relative to the ground) after she catches the ball, in terms of . Express in terms of , , and . You did not open hints for this part. ANSWER: Part E Find Jackie’s speed (relative to the ground) after she catches the ball, in terms of . Express in terms of , , and . You did not open hints for this part. ANSWER: vc = vj vb vj mball mcart vb vj = vj u vj mball mcart u Momentum in an Explosion A giant “egg” explodes as part of a fireworks display. The egg is at rest before the explosion, and after the explosion, it breaks into two pieces, with the masses indicated in the diagram, traveling in opposite directions. Part A What is the momentum of piece A before the explosion? Express your answer numerically in kilogram meters per second. You did not open hints for this part. ANSWER: vj = pA,i Part B During the explosion, is the force of piece A on piece B greater than, less than, or equal to the force of piece B on piece A? You did not open hints for this part. ANSWER: Part C The momentum of piece B is measured to be 500 after the explosion. Find the momentum of piece A after the explosion. Enter your answer numerically in kilogram meters per second. You did not open hints for this part. ANSWER: pA,i = kg  m/s greater than less than equal to cannot be determined kg  m/s pA,f pA,f = kg  m/s ± PSS 9.1 Conservation of Momentum Learning Goal: To practice Problem-Solving Strategy 9.1 for conservation of momentum problems. An 80- quarterback jumps straight up in the air right before throwing a 0.43- football horizontally at 15 . How fast will he be moving backward just after releasing the ball? PROBLEM-SOLVING STRATEGY 9.1 Conservation of momentum MODEL: Clearly define the system. If possible, choose a system that is isolated ( ) or within which the interactions are sufficiently short and intense that you can ignore external forces for the duration of the interaction (the impulse approximation). Momentum is conserved. If it is not possible to choose an isolated system, try to divide the problem into parts such that momentum is conserved during one segment of the motion. Other segments of the motion can be analyzed using Newton’s laws or, as you will learn later, conservation of energy. VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you are trying to find. SOLVE: The mathematical representation is based on the law of conservation of momentum: . In component form, this is ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model The interaction at study in this problem is the action of throwing the ball, performed by the quarterback while being off the ground. To apply conservation of momentum to this interaction, you will need to clearly define a system that is isolated or within which the impulse approximation can be applied. Part A Sort the following objects as part of the system or not. Drag the appropriate objects to their respective bins. ANSWER: kg kg m/s F = net 0 P = f P  i (pfx + ( + ( += ( + ( + ( + )1 pfx)2 pfx)3 pix)1 pix)2 pix)3 (pfy + ( + ( += ( + ( + ( + )1 pfy)2 pfy)3 piy)1 piy)2 piy)3 Part B This question will be shown after you complete previous question(s). Visualize Solve Part C This question will be shown after you complete previous question(s). Assess Part D This question will be shown after you complete previous question(s). Conservation of Momentum in Inelastic Collisions Learning Goal: To understand the vector nature of momentum in the case in which two objects collide and stick together. In this problem we will consider a collision of two moving objects such that after the collision, the objects stick together and travel off as a single unit. The collision is therefore completely inelastic. You have probably learned that “momentum is conserved” in an inelastic collision. But how does this fact help you to solve collision problems? The following questions should help you to clarify the meaning and implications of the statement “momentum is conserved.” Part A What physical quantities are conserved in this collision? ANSWER: Part B Two cars of equal mass collide inelastically and stick together after the collision. Before the collision, their speeds are and . What is the speed of the two-car system after the collision? the magnitude of the momentum only the net momentum (considered as a vector) only the momentum of each object considered individually v1 v2 You did not open hints for this part. ANSWER: Part C Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, what is the magnitude of their combined momentum? You did not open hints for this part. ANSWER: The answer depends on the directions in which the cars were moving before the collision. v1 + v2 v1 − v2 v2 − v1 v1v2 −−−− ” v1+v2 2 v1 + 2 v2 2 −−−−−−−  p1 p2 Part D Two cars collide inelastically and stick together after the collision. Before the collision, their momenta are and . After the collision, their combined momentum is . Of what can one be certain? You did not open hints for this part. ANSWER: Part E Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, the magnitude of their combined momentum is . Of what can one be certain? The answer depends on the directions in which the cars were moving before the collision. p1 + p2 p1 − p2 p2 − p1 p1p2 −−−− ” p1+p2 2 p1 + 2 p2 2 −−−−−−−  p 1 p 2 p p = p1 + # p2 # p = p1 − # p2 # p = p2 − # p1 # p1 p2 p You did not open hints for this part. ANSWER: Colliding Cars In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic. Two cars of masses and collide at an intersection. Before the collision, car 1 was traveling eastward at a speed of , and car 2 was traveling northward at a speed of . After the collision, the two cars stick together and travel off in the direction shown. Part A p1 + p2 $ p $ p1p2 −−−− ” p1 +p2 $ p $ p1+p2 2 p1 + p2 $ p $ |p1 − p2 | p1 + p2 $ p $ p1 + 2 p2 2 −−−−−−−  m1 m2 v1 v2 First, find the magnitude of , that is, the speed of the two-car unit after the collision. Express in terms of , , and the cars’ initial speeds and . You did not open hints for this part. ANSWER: Part B Find the tangent of the angle . Express your answer in terms of the momenta of the two cars, and . ANSWER: Part C Suppose that after the collision, ; in other words, is . This means that before the collision: ANSWER: v v v m1 m2 v1 v2 v = p1 p2 tan( ) = tan = 1 45′ The magnitudes of the momenta of the cars were equal. The masses of the cars were equal. The velocities of the cars were equal. ± Catching a Ball on Ice Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass 0.400 that is traveling horizontally at 11.2 . Olaf’s mass is 67.1 . Part A If Olaf catches the ball, with what speed do Olaf and the ball move afterward? Express your answer numerically in meters per second. You did not open hints for this part. ANSWER: Part B kg m/s kg vf vf = m/s If the ball hits Olaf and bounces off his chest horizontally at 8.00 in the opposite direction, what is his speed after the collision? Express your answer numerically in meters per second. You did not open hints for this part. ANSWER: A One-Dimensional Inelastic Collision Block 1, of mass = 2.90 , moves along a frictionless air track with speed = 25.0 . It collides with block 2, of mass = 17.0 , which was initially at rest. The blocks stick together after the collision. Part A Find the magnitude of the total initial momentum of the two-block system. Express your answer numerically. m/s vf vf = m/s m1 kg v1 m/s m2 kg pi You did not open hints for this part. ANSWER: Part B Find , the magnitude of the final velocity of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. pi = kg  m/s vf vf = m/s

Chapter 9 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Momentum and Internal Forces Learning Goal: To understand the concept of total momentum for a system of objects and the effect of the internal forces on the total momentum. We begin by introducing the following terms: System: Any collection of objects, either pointlike or extended. In many momentum-related problems, you have a certain freedom in choosing the objects to be considered as your system. Making a wise choice is often a crucial step in solving the problem. Internal force: Any force interaction between two objects belonging to the chosen system. Let us stress that both interacting objects must belong to the system. External force: Any force interaction between objects at least one of which does not belong to the chosen system; in other words, at least one of the objects is external to the system. Closed system: a system that is not subject to any external forces. Total momentum: The vector sum of the individual momenta of all objects constituting the system. In this problem, you will analyze a system composed of two blocks, 1 and 2, of respective masses and . To simplify the analysis, we will make several assumptions: The blocks can move in only one dimension, namely, 1. along the x axis. 2. The masses of the blocks remain constant. 3. The system is closed. At time , the x components of the velocity and the acceleration of block 1 are denoted by and . Similarly, the x components of the velocity and acceleration of block 2 are denoted by and . In this problem, you will show that the total momentum of the system is not changed by the presence of internal forces. m1 m2 t v1(t) a1 (t) v2 (t) a2 (t) Part A Find , the x component of the total momentum of the system at time . Express your answer in terms of , , , and . ANSWER: Part B Find the time derivative of the x component of the system’s total momentum. Express your answer in terms of , , , and . You did not open hints for this part. ANSWER: Why did we bother with all this math? The expression for the derivative of momentum that we just obtained will be useful in reaching our desired conclusion, if only for this very special case. Part C The quantity (mass times acceleration) is dimensionally equivalent to which of the following? ANSWER: p(t) t m1 m2 v1 (t) v2 (t) p(t) = dp(t)/dt a1 (t) a2 (t) m1 m2 dp(t)/dt = ma Part D Acceleration is due to which of the following physical quantities? ANSWER: Part E Since we have assumed that the system composed of blocks 1 and 2 is closed, what could be the reason for the acceleration of block 1? You did not open hints for this part. ANSWER: momentum energy force acceleration inertia velocity speed energy momentum force Part F This question will be shown after you complete previous question(s). Part G Let us denote the x component of the force exerted by block 1 on block 2 by , and the x component of the force exerted by block 2 on block 1 by . Which of the following pairs equalities is a direct consequence of Newton’s second law? ANSWER: Part H Let us recall that we have denoted the force exerted by block 1 on block 2 by , and the force exerted by block 2 on block 1 by . If we suppose that is greater than , which of the following statements about forces is true? You did not open hints for this part. the large mass of block 1 air resistance Earth’s gravitational attraction a force exerted by block 2 on block 1 a force exerted by block 1 on block 2 F12 F21 and and and and F12 = m2a2 F21 = m1a1 F12 = m1a1 F21 = m2a2 F12 = m1a2 F21 = m2a1 F12 = m2a1 F21 = m1a2 F12 F21 m1 m2 ANSWER: Part I Now recall the expression for the time derivative of the x component of the system’s total momentum: . Considering the information that you now have, choose the best alternative for an equivalent expression to . You did not open hints for this part. ANSWER: Impulse and Momentum Ranking Task Six automobiles are initially traveling at the indicated velocities. The automobiles have different masses and velocities. The drivers step on the brakes and all automobiles are brought to rest. Part A Rank these automobiles based on the magnitude of their momentum before the brakes are applied, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. ANSWER: Both forces have equal magnitudes. |F12 | > |F21| |F21 | > |F12| dpx(t)/dt = Fx dpx(t)/dt 0 nonzero constant kt kt2 Part B Rank these automobiles based on the magnitude of the impulse needed to stop them, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. You did not open hints for this part. ANSWER: Part C Rank the automobiles based on the magnitude of the force needed to stop them, from largest to smallest. Rank from largest to smallest. To rank items as equivalent, overlap them. If the ranking cannot be determined, check the box below. You did not open hints for this part. ANSWER: A Game of Frictionless Catch Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, , is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest. Chuck then picks up a ball of mass and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is . The speed of the thrown ball relative to the ground is . Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie’s speed relative to the ground after she catches the ball is . When answering the questions in this problem, keep the following in mind: The original mass of Chuck and his cart does not include the 1. mass of the ball. 2. The speed of an object is the magnitude of its velocity. An object’s speed will always be a nonnegative quantity. mcart mball vc vb vj mcart Part A Find the relative speed between Chuck and the ball after Chuck has thrown the ball. Express the speed in terms of and . You did not open hints for this part. ANSWER: Part B What is the speed of the ball (relative to the ground) while it is in the air? Express your answer in terms of , , and . You did not open hints for this part. ANSWER: Part C What is Chuck’s speed (relative to the ground) after he throws the ball? Express your answer in terms of , , and . u vc vb u = vb mball mcart u vb = vc mball mcart u You did not open hints for this part. ANSWER: Part D Find Jackie’s speed (relative to the ground) after she catches the ball, in terms of . Express in terms of , , and . You did not open hints for this part. ANSWER: Part E Find Jackie’s speed (relative to the ground) after she catches the ball, in terms of . Express in terms of , , and . You did not open hints for this part. ANSWER: vc = vj vb vj mball mcart vb vj = vj u vj mball mcart u Momentum in an Explosion A giant “egg” explodes as part of a fireworks display. The egg is at rest before the explosion, and after the explosion, it breaks into two pieces, with the masses indicated in the diagram, traveling in opposite directions. Part A What is the momentum of piece A before the explosion? Express your answer numerically in kilogram meters per second. You did not open hints for this part. ANSWER: vj = pA,i Part B During the explosion, is the force of piece A on piece B greater than, less than, or equal to the force of piece B on piece A? You did not open hints for this part. ANSWER: Part C The momentum of piece B is measured to be 500 after the explosion. Find the momentum of piece A after the explosion. Enter your answer numerically in kilogram meters per second. You did not open hints for this part. ANSWER: pA,i = kg  m/s greater than less than equal to cannot be determined kg  m/s pA,f pA,f = kg  m/s ± PSS 9.1 Conservation of Momentum Learning Goal: To practice Problem-Solving Strategy 9.1 for conservation of momentum problems. An 80- quarterback jumps straight up in the air right before throwing a 0.43- football horizontally at 15 . How fast will he be moving backward just after releasing the ball? PROBLEM-SOLVING STRATEGY 9.1 Conservation of momentum MODEL: Clearly define the system. If possible, choose a system that is isolated ( ) or within which the interactions are sufficiently short and intense that you can ignore external forces for the duration of the interaction (the impulse approximation). Momentum is conserved. If it is not possible to choose an isolated system, try to divide the problem into parts such that momentum is conserved during one segment of the motion. Other segments of the motion can be analyzed using Newton’s laws or, as you will learn later, conservation of energy. VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you are trying to find. SOLVE: The mathematical representation is based on the law of conservation of momentum: . In component form, this is ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model The interaction at study in this problem is the action of throwing the ball, performed by the quarterback while being off the ground. To apply conservation of momentum to this interaction, you will need to clearly define a system that is isolated or within which the impulse approximation can be applied. Part A Sort the following objects as part of the system or not. Drag the appropriate objects to their respective bins. ANSWER: kg kg m/s F = net 0 P = f P  i (pfx + ( + ( += ( + ( + ( + )1 pfx)2 pfx)3 pix)1 pix)2 pix)3 (pfy + ( + ( += ( + ( + ( + )1 pfy)2 pfy)3 piy)1 piy)2 piy)3 Part B This question will be shown after you complete previous question(s). Visualize Solve Part C This question will be shown after you complete previous question(s). Assess Part D This question will be shown after you complete previous question(s). Conservation of Momentum in Inelastic Collisions Learning Goal: To understand the vector nature of momentum in the case in which two objects collide and stick together. In this problem we will consider a collision of two moving objects such that after the collision, the objects stick together and travel off as a single unit. The collision is therefore completely inelastic. You have probably learned that “momentum is conserved” in an inelastic collision. But how does this fact help you to solve collision problems? The following questions should help you to clarify the meaning and implications of the statement “momentum is conserved.” Part A What physical quantities are conserved in this collision? ANSWER: Part B Two cars of equal mass collide inelastically and stick together after the collision. Before the collision, their speeds are and . What is the speed of the two-car system after the collision? the magnitude of the momentum only the net momentum (considered as a vector) only the momentum of each object considered individually v1 v2 You did not open hints for this part. ANSWER: Part C Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, what is the magnitude of their combined momentum? You did not open hints for this part. ANSWER: The answer depends on the directions in which the cars were moving before the collision. v1 + v2 v1 − v2 v2 − v1 v1v2 −−−− ” v1+v2 2 v1 + 2 v2 2 −−−−−−−  p1 p2 Part D Two cars collide inelastically and stick together after the collision. Before the collision, their momenta are and . After the collision, their combined momentum is . Of what can one be certain? You did not open hints for this part. ANSWER: Part E Two cars collide inelastically and stick together after the collision. Before the collision, the magnitudes of their momenta are and . After the collision, the magnitude of their combined momentum is . Of what can one be certain? The answer depends on the directions in which the cars were moving before the collision. p1 + p2 p1 − p2 p2 − p1 p1p2 −−−− ” p1+p2 2 p1 + 2 p2 2 −−−−−−−  p 1 p 2 p p = p1 + # p2 # p = p1 − # p2 # p = p2 − # p1 # p1 p2 p You did not open hints for this part. ANSWER: Colliding Cars In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic. Two cars of masses and collide at an intersection. Before the collision, car 1 was traveling eastward at a speed of , and car 2 was traveling northward at a speed of . After the collision, the two cars stick together and travel off in the direction shown. Part A p1 + p2 $ p $ p1p2 −−−− ” p1 +p2 $ p $ p1+p2 2 p1 + p2 $ p $ |p1 − p2 | p1 + p2 $ p $ p1 + 2 p2 2 −−−−−−−  m1 m2 v1 v2 First, find the magnitude of , that is, the speed of the two-car unit after the collision. Express in terms of , , and the cars’ initial speeds and . You did not open hints for this part. ANSWER: Part B Find the tangent of the angle . Express your answer in terms of the momenta of the two cars, and . ANSWER: Part C Suppose that after the collision, ; in other words, is . This means that before the collision: ANSWER: v v v m1 m2 v1 v2 v = p1 p2 tan( ) = tan = 1 45′ The magnitudes of the momenta of the cars were equal. The masses of the cars were equal. The velocities of the cars were equal. ± Catching a Ball on Ice Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass 0.400 that is traveling horizontally at 11.2 . Olaf’s mass is 67.1 . Part A If Olaf catches the ball, with what speed do Olaf and the ball move afterward? Express your answer numerically in meters per second. You did not open hints for this part. ANSWER: Part B kg m/s kg vf vf = m/s If the ball hits Olaf and bounces off his chest horizontally at 8.00 in the opposite direction, what is his speed after the collision? Express your answer numerically in meters per second. You did not open hints for this part. ANSWER: A One-Dimensional Inelastic Collision Block 1, of mass = 2.90 , moves along a frictionless air track with speed = 25.0 . It collides with block 2, of mass = 17.0 , which was initially at rest. The blocks stick together after the collision. Part A Find the magnitude of the total initial momentum of the two-block system. Express your answer numerically. m/s vf vf = m/s m1 kg v1 m/s m2 kg pi You did not open hints for this part. ANSWER: Part B Find , the magnitude of the final velocity of the two-block system. Express your answer numerically. You did not open hints for this part. ANSWER: Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. pi = kg  m/s vf vf = m/s

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