## Chapter 10 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 A One-Dimensional Inelastic Collision Block 1, of mass = 3.70 , moves along a frictionless air track with speed = 15.0 . It collides with block 2, of mass = 19.0 , 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. You did not open hints for this part. ANSWER: m1 kg v1 m/s m2 kg pi 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: Part C What is the change in the two-block system’s kinetic energy due to the collision? Express your answer numerically in joules. You did not open hints for this part. ANSWER: pi = kg m/s vf vf = m/s K = Kfinal − Kinitial K = J Conservation of Energy Ranking Task Six pendulums of various masses are released from various heights above a tabletop, as shown in the figures below. All the pendulums have the same length and are mounted such that at the vertical position their lowest points are the height of the tabletop and just do not strike the tabletop when released. Assume that the size of each bob is negligible. Part A Rank each pendulum on the basis of its initial gravitational potential energy (before being released) relative to the tabletop. Rank from largest to smallest To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: m h 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). Momentum and Kinetic Energy Consider two objects (Object 1 and Object 2) moving in the same direction on a frictionless surface. Object 1 moves with speed and has mass . Object 2 moves with speed and has mass . Part A Which object has the larger magnitude of its momentum? You did not open hints for this part. ANSWER: Part B Which object has the larger kinetic energy? You did not open hints for this part. ANSWER: v1 = v m1 = 2m v2 = 2v m2 = m Object 1 has the greater magnitude of its momentum. Object 2 has the greater magnitude of its momentum. Both objects have the same magnitude of their momenta. Object 1 has the greater kinetic energy. Object 2 has the greater kinetic energy. The objects have the same kinetic energy. Projectile Motion and Conservation of Energy Ranking Task Part A Six baseball throws are shown below. In each case the baseball is thrown at the same initial speed and from the same height above the ground. Assume that the effects of air resistance are negligible. Rank these throws according to the speed of the baseball the instant before it hits the ground. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: H PSS 10.1 Conservation of Mechanical Energy Learning Goal: To practice Problem-Solving Strategy 10.1 for conservation of mechanical energy problems. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 that makes an angle of 45 with the vertical, steps off his tree limb, and swings down and then up to Jane’s open arms. When he arrives, his vine makes an angle of 30 with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan’s speed just before he reaches Jane. You can ignore air resistance and the mass of the vine. PROBLEM-SOLVING STRATEGY 10.1 Conservation of mechanical energy MODEL: Choose a system without friction or other losses of mechanical energy. m VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you’re trying to find. SOLVE: The mathematical representation is based on the law of conservation of mechanical energy: . ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model The problem does not involve friction, nor are there losses of mechanical energy, so conservation of mechanical energy applies. Model Tarzan and the vine as a pendulum. Visualize Part A Which of the following sketches can be used in drawing a before-and-after pictorial representation? ANSWER: Kf + Uf = Ki + Ui Solve Part B What is Tarzan’s speed just before he reaches Jane? Express your answer in meters per second to two significant figures. You did not open hints for this part. ANSWER: Assess Part C This question will be shown after you complete previous question(s). Bungee Jumping Diagram A Diagram B Diagram C Diagram D vf vf = m/s Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass , and the surface of the bridge is a height above the water. The bungee cord, which has length when unstretched, will first straighten and then stretch as Kate falls. Assume the following: The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant . Kate doesn’t actually jump but simply steps off the edge of the bridge and falls straight downward. Kate’s height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle. Use for the magnitude of the acceleration due to gravity. Part A How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn’t touch the water. Express the distance in terms of quantities given in the problem introduction. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Spinning Mass on a Spring An object of mass is attached to a spring with spring constant whose unstretched length is , and whose far end is fixed to a shaft that is rotating with angular speed . Neglect gravity and assume that the mass rotates with angular speed as shown. When solving this problem use an inertial coordinate system, as drawn here. m h L k g d = M k L Part A Given the angular speed , find the radius at which the mass rotates without moving toward or away from the origin. Express the radius in terms of , , , and . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C R( ) k L M R( ) = This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). ± Baby Bounce with a Hooke One of the pioneers of modern science, Sir Robert Hooke (1635-1703), studied the elastic properties of springs and formulated the law that bears his name. Hooke found the relationship among the force a spring exerts, , the distance from equilibrium the end of the spring is displaced, , and a number called the spring constant (or, sometimes, the force constant of the spring). According to Hooke, the force of the spring is directly proportional to its displacement from equilibrium, or . In its scalar form, this equation is simply . The negative sign indicates that the force that the spring exerts and its displacement have opposite directions. The value of depends on the geometry and the material of the spring; it can be easily determined experimentally using this scalar equation. Toy makers have always been interested in springs for the entertainment value of the motion they produce. One well-known application is a baby bouncer,which consists of a harness seat for a toddler, attached to a spring. The entire contraption hooks onto the top of a doorway. The idea is for the baby to hang in the seat with his or her feet just touching the ground so that a good push up will get the baby bouncing, providing potentially hours of entertainment. F x k F = −kx F = −kx k Part A The following chart and accompanying graph depict an experiment to determine the spring constant for a baby bouncer. Displacement from equilibrium, ( ) Force exerted on the spring, ( ) 0 0 0.005 2.5 0.010 5.0 0.015 7.5 0.020 10 What is the spring constant of the spring being tested for the baby bouncer? Express your answer to two significant figures in newtons per meter. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Shooting a ball into a box Two children are trying to shoot a marble of mass into a small box using a spring-loaded gun that is fixed on a table and shoots horizontally from the edge of the table. The edge of the table is a height above the top of the box (the height of which is negligibly small), and the center of the box is a distance from the edge of the table. x m F N k k = N/m m H d The spring has a spring constant . The first child compresses the spring a distance and finds that the marble falls short of its target by a horizontal distance . Part A By what distance, , should the second child compress the spring so that the marble lands in the middle of the box? (Assume that height of the box is negligible, so that there is no chance that the marble will hit the side of the box before it lands in the bottom.) Express the distance in terms of , , , , and . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). k x1 d12 x2 m k g H d x2 = Elastic Collision in One Dimension Block 1, of mass , moves across a frictionless surface with speed . It collides elastically with block 2, of mass , which is at rest ( ). After the collision, block 1 moves with speed , while block 2 moves with speed . Assume that , so that after the collision, the two objects move off in the direction of the first object before the collision. Part A This collision is elastic. What quantities, if any, are conserved in this collision? You did not open hints for this part. ANSWER: Part B What is the final speed of block 1? m1 ui m2 vi = 0 uf vf m1 > m2 kinetic energy only momentum only kinetic energy and momentum uf Express in terms of , , and . You did not open hints for this part. ANSWER: Part C What is the final speed of block 2? Express in terms of , , and . You did not open hints for this part. ANSWER: Ballistic Pendulum In a ballistic pendulum an object of mass is fired with an initial speed at a pendulum bob. The bob has a mass , which is suspended by a rod of length and negligible mass. After the collision, the pendulum and object stick together and swing to a maximum angular displacement as shown . uf m1 m2 ui uf = vf vf m1 m2 ui vf = m v0 M L Part A Find an expression for , the initial speed of the fired object. Express your answer in terms of some or all of the variables , , , and and the acceleration due to gravity, . You did not open hints for this part. ANSWER: Part B An experiment is done to compare the initial speed of bullets fired from different handguns: a 9.0 and a .44 caliber. The guns are fired into a 10- pendulum bob of length . Assume that the 9.0- bullet has a mass of 6.0 and the .44-caliber bullet has a mass of 12 . If the 9.0- bullet causes the pendulum to swing to a maximum angular displacement of 4.3 and the .44-caliber bullet causes a displacement of 10.1 , find the ratio of the initial speed of the 9.0- bullet to the speed of the .44-caliber bullet, . Express your answer numerically. You did not open hints for this part. ANSWER: v0 m M L g v0 = mm kg L mm g g mm mm (v /( 0 )9.0 v0)44 Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 0 points. (v0 )9.0/(v0 )44 =

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## Questions from Ishmael, Part 1 Type your answers to the following questions: 1. How do animals typically react to life in a zoo, according to Ishmael? And how, as a young gorilla, did zoo life affect him personally? 2. What does Ishmael mean by “Takers” and “Leavers?” Do these terms have any connotations to you (and if so, describe them), or do you see them as neutral? 3. Describe our culture’s origin myth, according to Ishmael. 4. After the events described in our culture’s origin myth, what was the next major event in human history that is essential in describing “how things came to be this way.” In the book it’s called “the middle of the story.” Describe this event, detailing what it meant for humankind. 5. According to Ishmael, how do Takers envision the role of humans “in the divine scheme?” What is humankind’s purpose? How has this perspective affected human/nature relationships? 6. According to Ishmael, how do most Takers envision the future of humanity? As part of your answer, describe how most Takers say we should deal with problems such as energy shortages and pollution. 7. What is Daniel Quinn trying to get you to perceive about the state of our culture’s relationship with nature? How does his story-telling approach affect your perception about environmental issues? To what extent is his approach successful, and why?

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## Chapter 4 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, February 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Advice for the Quarterback A quarterback is set up to throw the football to a receiver who is running with a constant velocity directly away from the quarterback and is now a distance away from the quarterback. The quarterback figures that the ball must be thrown at an angle to the horizontal and he estimates that the receiver must catch the ball a time interval after it is thrown to avoid having opposition players prevent the receiver from making the catch. In the following you may assume that the ball is thrown and caught at the same height above the level playing field. Assume that the y coordinate of the ball at the instant it is thrown or caught is and that the horizontal position of the quaterback is . Use for the magnitude of the acceleration due to gravity, and use the pictured inertial coordinate system when solving the problem. Part A Find , the vertical component of the velocity of the ball when the quarterback releases it. Express in terms of and . Hint 1. Equation of motion in y direction What is the expression for , the height of the ball as a function of time? Answer in terms of , , and . v r D tc y = 0 x = 0 g v0y v0y tc g y(t) t g v0y ANSWER: Incorrect; Try Again Hint 2. Height at which the ball is caught, Remember that after time the ball was caught at the same height as it had been released. That is, . ANSWER: Answer Requested Part B Find , the initial horizontal component of velocity of the ball. Express your answer for in terms of , , and . Hint 1. Receiver’s position Find , the receiver’s position before he catches the ball. Answer in terms of , , and . ANSWER: Football’s position y(t) = v0yt− g 1 2 t2 y(tc) tc y(tc) = y0 = 0 v0y = gtc 2 v0x v0x D tc vr xr D vr tc xr = D + vrtc Typesetting math: 100% Find , the horizontal distance that the ball travels before reaching the receiver. Answer in terms of and . ANSWER: ANSWER: Answer Requested Part C Find the speed with which the quarterback must throw the ball. Answer in terms of , , , and . Hint 1. How to approach the problem Remember that velocity is a vector; from solving Parts A and B you have the two components, from which you can find the magnitude of this vector. ANSWER: Answer Requested Part D xc v0x tc xc = v0xtc v0x = + D tc vr v0 D tc vr g v0 = ( + ) + D tc vr 2 ( ) gtc 2 2 −−−−−−−−−−−−−−−−−−− Typesetting math: 100% Assuming that the quarterback throws the ball with speed , find the angle above the horizontal at which he should throw it. Your solution should contain an inverse trig function (entered as asin, acos, or atan). Give your answer in terms of already known quantities, , , and . Hint 1. Find angle from and Think of velocity as a vector with Cartesian coordinates and . Find the angle that this vector would make with the x axis using the results of Parts A and B. ANSWER: Answer Requested Direction of Velocity at Various Times in Flight for Projectile Motion Conceptual Question For each of the motions described below, determine the algebraic sign (positive, negative, or zero) of the x component and y component of velocity of the object at the time specified. For all of the motions, the positive x axis points to the right and the positive y axis points upward. Alex, a mountaineer, must leap across a wide crevasse. The other side of the crevasse is below the point from which he leaps, as shown in the figure. Alex leaps horizontally and successfully makes the jump. v0 v0x v0y v0 v0x v0y v0xx^ v0yy^ = atan( ) v0y v0x Typesetting math: 100% Part A Determine the algebraic sign of Alex’s x velocity and y velocity at the instant he leaves the ground at the beginning of the jump. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Typesetting math: 100% Hint 1. Algebraic sign of velocity The algebraic sign of the velocity is determined solely by comparing the direction in which the object is moving with the direction that is defined to be positive. In this example, to the right is defined to be the positive x direction and upward the positive y direction. Therefore, any object moving to the right, whether speeding up, slowing down, or even simultaneously moving upward or downward, has a positive x velocity. Similarly, if the object is moving downward, regardless of any other aspect of its motion, its y velocity is negative. Hint 2. Sketch Alex’s initial velocity On the diagram below, sketch the vector representing Alex’s velocity the instant after he leaves the ground at the beginning of the jump. ANSWER: ANSWER: Typesetting math: 100% Answer Requested Part B Determine the algebraic signs of Alex’s x velocity and y velocity the instant before he lands at the end of the jump. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Typesetting math: 100% Hint 1. Sketch Alex’s final velocity On the diagram below, sketch the vector representing Alex’s velocity the instant before he safely lands on the other side of the crevasse. ANSWER: Answer Requested ANSWER: Answer Requested Typesetting math: 100% At the buzzer, a basketball player shoots a desperation shot. The ball goes in! Part C Determine the algebraic signs of the ball’s x velocity and y velocity the instant after it leaves the player’s hands. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Hint 1. Sketch the basketball’s initial velocity On the diagram below, sketch the vector representing the velocity of the basketball the instant after it leaves the player’s hands. ANSWER: Typesetting math: 100% ANSWER: Correct Part D Determine the algebraic signs of the ball’s x velocity and y velocity at the ball’s maximum height. Type the algebraic signs of the x velocity and the y velocity separated by a comma (examples: +,- and 0,+). Hint 1. Sketch the basketball’s velocity at maximum height Typesetting math: 100% On the diagram below, sketch the vector representing the velocity of the basketball the instant it reaches its maximum height. ANSWER: ANSWER: Answer Requested PSS 4.1 Projectile Motion Problems Learning Goal: Typesetting math: 100% To practice Problem-Solving Strategy 4.1 for projectile motion problems. A rock thrown with speed 9.00 and launch angle 30.0 (above the horizontal) travels a horizontal distance of = 17.0 before hitting the ground. From what height was the rock thrown? Use the value = 9.810 for the free-fall acceleration. PROBLEM-SOLVING STRATEGY 4.1 Projectile motion problems MODEL: Make simplifying assumptions, such as treating the object as a particle. Is it reasonable to ignore air resistance? VISUALIZE: Use a pictorial representation. Establish a coordinate system with the x axis horizontal and the y axis vertical. Show important points in the motion on a sketch. Define symbols, and identify what you are trying to find. SOLVE: The acceleration is known: and . Thus, the problem becomes one of two-dimensional kinematics. The kinematic equations are , . is the same for the horizontal and vertical components of the motion. Find from one component, and then use that value for the other component. ASSESS: Check that your result has the correct units, is reasonable, and answers the question. Model Start by making simplifying assumptions: Model the rock as a particle in free fall. You can ignore air resistance because the rock is a relatively heavy object moving relatively slowly. Visualize Part A Which diagram represents an accurate sketch of the rock’s trajectory? Hint 1. The launch angle In a projectile’s motion, the angle of the initial velocity above the horizontal is called the launch angle. ANSWER: m/s d m g m/s2 ax = 0 ay = −g xf = xi +vixt, yf = yi +viyt− g(t 1 2 )2 vfx = vix = constant, and vfy = viy − gt t t v i Typesetting math: 100% Typesetting math: 100% Correct Part B As stated in the strategy, choose a coordinate system where the x axis is horizontal and the y axis is vertical. Note that in the strategy, the y component of the projectile’s acceleration, , is taken to be negative. This implies that the positive y axis is upward. Use the same convention for your y axis, and take the positive x axis to be to the right. Where you choose your origin doesn’t change the answer to the question, but choosing an origin can make a problem easier to solve (even if only a bit). Usually it is nice if the majority of the quantities you are given and the quantity you are trying to solve for take positive values relative to your chosen origin. Given this goal, what location for the origin of the coordinate system would make this problem easiest? ANSWER: ay At ground level below the point where the rock is launched At the point where the rock strikes the ground At the peak of the trajectory At the point where the rock is released At ground level below the peak of the trajectory Typesetting math: 100% Correct It’s best to place the origin of the coordinate system at ground level below the launching point because in this way all the points of interest (the launching point and the landing point) will have positive coordinates. (Based on your experience, you know that it’s generally easier to work with positive coordinates.) Keep in mind, however, that this is an arbitrary choice. The correct solution of the problem will not depend on the location of the origin of your coordinate system. Now, define symbols representing initial and final position, velocity, and time. Your target variable is , the initial y coordinate of the rock. Your pictorial representation should be complete now, and similar to the picture below: Solve Part C Find the height from which the rock was launched. Express your answer in meters to three significant figures. yi yi Typesetting math: 100% Hint 1. How to approach the problem The time needed to move horizontally to the final position = 17.0 is the same time needed for the rock to rise from the initial position to the peak of its trajectory and then fall to the ground. Use the information you have about motion in the horizontal direction to solve for . Knowing this time will allow you to use the equations of motion for the vertical direction to solve for . Hint 2. Find the time spent in the air How long ( ) is the rock in the air? Express your answer in seconds to three significant figures. Hint 1. Determine which equation to use Which of the equations given in the strategy and shown below is the most appropriate to calculate the time the rock spent in the air? ANSWER: Hint 2. Find the x component of the initial velocity What is the x component of the rock’s initial velocity? Express your answer in meters per second to three significant figures. ANSWER: ANSWER: t xf = d m yi t yi t t xf = xi + vixt yf = yi + viyt− g(t 1 2 )2 vfy = viy − gt vix = 7.79 m/s Typesetting math: 100% Hint 3. Find the y component of the initial velocity What is the y component of the rock’s initial velocity? Express your answer in meters per second to three significant figures. ANSWER: ANSWER: Answer Requested Assess Part D A second rock is thrown straight upward with a speed 4.500 . If this rock takes 2.181 to fall to the ground, from what height was it released? Express your answer in meters to three significant figures. Hint 1. Identify the known variables What are the values of , , , and for the second rock? Take the positive y axis to be upward and the origin to be located on the ground where the rock lands. Express your answers to four significant figures in the units shown to the right, separated by commas. ANSWER: t = 2.18 s viy = 4.50 m/s yi = 13.5 m m/s s H yf viy t a Typesetting math: 100% Answer Requested Hint 2. Determine which equation to use to find the height Which equation should you use to find ? Keep in mind that if the positive y axis is upward and the origin is located on the ground, . ANSWER: ANSWER: Answer Requested Projectile motion is made up of two independent motions: uniform motion at constant velocity in the horizontal direction and free-fall motion in the vertical direction. Because both rocks were thrown with the same initial vertical velocity, 4.500 , and fell the same vertical distance of 13.5 , they were in the air for the same amount of time. This result was expected and helps to confirm that you did the calculation in Part C correctly. ± Arrow Hits Apple An arrow is shot at an angle of above the horizontal. The arrow hits a tree a horizontal distance away, at the same height above the ground as it was shot. Use for the magnitude of the acceleration due to gravity. Part A , , , = 0,4.500,2.181,-yf viy t a 9.810 m, m/s, s, m/s2 H yi = H yf = yi + viyt− g(t 1 2 )2 vfy = viy − gt = − 2g( − ) v2f y v2i y yf yi H = 13.5 m viy = m/s m = 45 D = 220 m g = 9.8 m/s2 Typesetting math: 100% Find , the time that the arrow spends in the air. Answer numerically in seconds, to two significant figures. Hint 1. Find the initial upward component of velocity in terms of D. Introduce the (unknown) variables and for the initial components of velocity. Then use kinematics to relate them and solve for . What is the vertical component of the initial velocity? Express your answer symbolically in terms of and . Hint 1. Find Find the horizontal component of the initial velocity. Express your answer symbolically in terms of and given symbolic quantities. ANSWER: Hint 2. Find What is the vertical component of the initial velocity? Express your answer symbolically in terms of . ANSWER: ANSWER: ta vy0 vx0 ta vy0 ta D vx0 vx0 ta vx0 = D ta vy0 vy0 vx0 vy0 = vx0 vy0 = D ta Typesetting math: 100% Hint 2. Find the time of flight in terms of the initial vertical component of velocity. From the change in the vertical component of velocity, you should be able to find in terms of and . Give your answer in terms of and . Hint 1. Find When applied to the y-component of velocity, in this problem the formula for with constant acceleration is What is , the vertical component of velocity when the arrow hits the tree? Answer symbolically in terms of only. ANSWER: ANSWER: Hint 3. Put the algebra together to find symbolically. If you have an expression for the initial vertical velocity component in terms in terms of and , and another in terms of and , you should be able to eliminate this initial component to find an expression for Express your answer symbolically in terms of given variables. ANSWER: ta vy0 g vy0 g vy(ta) v(t) −g vy(t) = vy0 − g t vy(ta ) vy0 vy(ta) = −vy0 ta = 2vy0 g ta D ta g ta ta2 t2 = a 2D g Typesetting math: 100% ANSWER: Answer Requested Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree. Part B How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as the arrow hits the tree? Express your answer numerically in seconds, to two significant figures. Hint 1. When should the apple be dropped The apple should be dropped at the time equal to the total time it takes the arrow to reach the tree minus the time it takes the apple to fall 6.0 meters. Hint 2. Find the time it takes for the apple to fall 6.0 meters How long does it take an apple to fall 6.0 meters? Express your answer numerically in seconds, to two significant figures. ANSWER: Answer Requested ANSWER: ta = 6.7 s tf = 1.1 s td = 5.6 s Typesetting math: 100% Answer Requested Video Tutor: Ball Fired Upward from Accelerating Cart First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point. Part A Consider the video you just watched. Suppose we replace the original launcher with one that fires the ball upward at twice the speed. We make no other changes. How far behind the cart will the ball land, compared to the distance in the original experiment? Hint 1. Determine how long the ball is in the air How will doubling the initial upward speed of the ball change the time the ball spends in the air? A kinematic equation may be helpful here. The time in the air will ANSWER: be cut in half. stay the same. double. quadruple. Typesetting math: 100% Hint 2. Determine the appropriate kinematic expression Which of the following kinematic equations correctly describes the horizontal distance between the ball and the cart at the moment the ball lands? The cart’s initial horizontal velocity is , its horizontal acceleration is , and is the time elapsed between launch and impact. ANSWER: ANSWER: Correct The ball will spend twice as much time in the air ( , where is the ball’s initial upward velocity), so it will land four times farther behind the cart: (where is the cart’s horizontal acceleration). Video Tutor: Ball Fired Upward from Moving Cart First, launch the video below. You will be asked to use your knowledge of physics to predict the outcome of an experiment. Then, close the video window and answer the questions at right. You can watch the video again at any point. d v0x ax t d = v0x t d = 1 2 axv0x t2 d = v0x t+ 1 2 axt2 d = 1 2 axt2 the same distance twice as far half as far four times as far by a factor not listed above t = 2v0y/g v0y d = 1 2 axt2 ax Typesetting math: 100% Part A The crew of a cargo plane wishes to drop a crate of supplies on a target below. To hit the target, when should the crew drop the crate? Ignore air resistance. Hint 1. How to approach the problem While the crate is on the plane, it shares the plane’s velocity. What is the crate’s velocity immediately after it is released? Hint 2. What affects the motion of the crate? Gravity will accelerate the crate downward. What, if anything, affects the crate’s horizontal motion? (Keep in mind that we are told to ignore air resistance, even though that’s not very realistic in this situation.) ANSWER: Correct At the moment it is released, the crate shares the plane’s horizontal velocity. In the absence of air resistance, the crate would remain directly below the plane as it fell. Score Summary: Your score on this assignment is 0%. Before the plane is directly over the target After the plane has flown over the target When the plane is directly over the target Typesetting math: 100% You received 0 out of a possible total of 0 points. Typesetting math: 100%

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## Two force vectors F1 and F2 are applied at the origin of the system of coordinates, point O of coordinates (0,0,0). The two force are expressed in Newton units (N). The vector expressed of the first fore is F1=(-120i+60j+40k) N. The second force F2 has a magnitude of 85 N and its direction is defined by the line between point O and B (4, -3, 5), where these coordinates are in meters (m). (a) Find unit vector ef1alone F1. (b) Find unit vector ef2alone F2. (c) find the angle O between F1 and F2 using the dot product operation. (d) Find the force vector resultant R=F1+F2. (e) find the direction cosine, cosOxof vector R.

## Chapter 12 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 A Spinning Grinding Wheel At time a grinding wheel has an angular velocity of 26.0 . It has a constant angular acceleration of 33.0 until a circuit breaker trips at time = 1.80 . From then on, the wheel turns through an angle of 432 as it coasts to a stop at constant angular deceleration. Part A Through what total angle did the wheel turn between and the time it stopped? Express your answer in radians. You did not open hints for this part. ANSWER: Part B At what time does the wheel stop? Express your answer in seconds. You did not open hints for this part. ANSWER: t = 0 rad/s rad/s2 t s rad t = 0 rad Part C What was the wheel’s angular acceleration as it slowed down? Express your answer in radians per second per second. You did not open hints for this part. ANSWER: An Exhausted Bicyclist An exhausted bicyclist pedals somewhat erratically when exercising on a static bicycle. The angular velocity of the wheels follows the equation , where represents time (measured in seconds), = 0.500 , = 0.250 and = 2.00 . Part A There is a spot of paint on the front wheel of the bicycle. Take the position of the spot at time to be at angle radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the wheel’s rotation. What angular displacement has the spot of paint undergone between time 0 and 2 seconds? Express your answer in radians using three significant figures. s rad/s2 (t) = at − bsin(ct) for t 0 t a rad/s2 b rad/s c rad/s t = 0 = 0 Typesetting math: 29% You did not open hints for this part. ANSWER: Part B Express the angular displacement undergone by the spot of paint at seconds in degrees. Remember to use the unrounded value from Part A, should you need it. Express your answer in degrees using three significant figures. You did not open hints for this part. ANSWER: Part C What distance has the spot of paint moved in 2 seconds if the radius of the wheel is 50 centimeters? Express your answer in centimeters using three significant figures. You did not open hints for this part. ANSWER: = rad t = 2 = d Typesetting math: 29% Part D Which one of the following statements describes the motion of the spot of paint at seconds? You did not open hints for this part. ANSWER: Flywheel Kinematics A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration . The flywheel is assumed to be at rest at time in Parts A and B of this problem. Part A Find the time it takes to accelerate the flywheel to if the angular acceleration is . Express your answer in terms of and . d = cm t = 2.0 The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is constant and the magnitude of the angular speed is increasing. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is increasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is decreasing. The angular acceleration of the spot of paint is negative and the magnitude of the angular speed is increasing. t = 0 t1 1 1 Typesetting math: 29% You did not open hints for this part. ANSWER: Part B Find the angle through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocity . Express your answer in terms of some or all of the following: , , and . You did not open hints for this part. ANSWER: Part C Assume that the motor has accelerated the wheel up to an angular velocity with angular acceleration in time . At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of . Find , the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of some or all of the following: , \texttip{\alpha }{alpha}, and \texttip{t_{\rm 1}}{t_1}. You did not open hints for this part. t1 = 1 1 1 t1 1 = 1 t1 −5 t2 1 Typesetting math: 29% ANSWER: Surprising Exploding Firework A mortar fires a shell of mass \texttip{m}{m} at speed \texttip{v_{\rm 0}}{v_0}. The shell explodes at the top of its trajectory (shown by a star in the figure) as designed. However, rather than creating a shower of colored flares, it breaks into just two pieces, a smaller piece of mass \large{\frac15m} and a larger piece of mass \large{\frac45m}. Both pieces land at exactly the same time. The smaller piece lands perilously close to the mortar (at a distance of zero from the mortar). The larger piece lands a distance \texttip{d}{d} from the mortar. If there had been no explosion, the shell would have landed a distance \texttip{r}{r} from the mortar. Assume that air resistance and the mass of the shell’s explosive charge are negligible. Part A Find the distance \texttip{d}{d} from the mortar at which the larger piece of the shell lands. Express \texttip{d}{d} in terms of \texttip{r}{r}. You did not open hints for this part. \texttip{t_{\rm 2}}{t_2} = s Typesetting math: 29% ANSWER: Kinetic Energy of a Dumbbell This problem illustrates the two contributions to the kinetic energy of an extended object: rotational kinetic energy and translational kinetic energy. You are to find the total kinetic energy \texttip{K_{\rm total}}{K_total} of a dumbbell of mass \texttip{m}{m} when it is rotating with angular speed \texttip{\omega }{omega} and its center of mass is moving translationally with speed \texttip{v}{v}. Denote the dumbbell’s moment of inertia about its center of mass by \texttip{I_{\rm cm}}{I_cm}. Note that if you approximate the spheres as point masses of mass m/2 each located a distance \texttip{r}{r} from the center and ignore the moment of inertia of the connecting rod, then the moment of inertia of the dumbbell is given by I_{\rm cm} = mr^2, but this fact will not be necessary for this problem. Part A Find the total kinetic energy \texttip{K_{\rm tot}}{K_tot} of the dumbbell. Express your answer in terms of \texttip{m}{m}, \texttip{v}{v}, \texttip{I_{\rm cm}}{I_cm}, and \texttip{\omega }{omega}. You did not open hints for this part. \texttip{d}{d} = Typesetting math: 29% ANSWER: Part B This question will be shown after you complete previous question(s). Unwinding Cylinder A cylinder with moment of inertia \texttip{I}{I} about its center of mass, mass \texttip{m}{m}, and radius \texttip{r}{r} has a string wrapped around it which is tied to the ceiling . The cylinder’s vertical position as a function of time is y(t). At time t = 0 the cylinder is released from rest at a height \texttip{h}{h} above the ground. Part A The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate \texttip{\omega }{omega} and \texttip{v}{v}, the velocity of the center of mass of the cylinder? \texttip{K_{\rm tot}}{K_tot} = Typesetting math: 29% Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity. Express \texttip{\omega }{omega} in terms of \texttip{v}{v} and other given quantities. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C Suppose that at a certain instant the velocity of the cylinder is \texttip{v}{v}. What is its total kinetic energy, \texttip{K_{\rm total}}{K_total}, at that instant? Express \texttip{K_{\rm total}}{K_total} in terms of \texttip{m}{m}, \texttip{r}{r}, \texttip{I}{I}, and \texttip{v}{v}. You did not open hints for this part. ANSWER: Part D \texttip{\omega }{omega} = \texttip{K_{\rm total}}{K_total} = Typesetting math: 29% Find \texttip{v_{\rm f \hspace{1 pt}}}{v_f}, the cylinder’s vertical velocity when it hits the ground. Express \texttip{v_{\rm f \hspace{1 pt}}}{v_f}, in terms of \texttip{g}{g}, \texttip{h}{h}, \texttip{I}{I}, \texttip{m}{m}, and \texttip{r}{r}. You did not open hints for this part. ANSWER: Kinetic Energy of a Rotating Wheel A simple wheel has the form of a solid cylinder of radius \texttip{r}{r} with a mass \texttip{m}{m} uniformly distributed throughout its volume. The wheel is pivoted on a stationary axle through the axis of the cylinder and rotates about the axle at a constant angular speed. The wheel rotates \texttip{n}{n} full revolutions in a time interval \texttip{t}{t}. Part A What is the kinetic energy \texttip{K}{K} of the rotating wheel? Express your answer in terms of \texttip{m}{m}, \texttip{r}{r}, \texttip{n}{n}, \texttip{t}{t} and, \texttip{\pi }{pi}. You did not open hints for this part. ANSWER: Finding Torque \texttip{v_{\rm f \hspace{1 pt}}}{v_f} = \texttip{K}{K} = Typesetting math: 29% A force \texttip{\vec{F}}{F_vec} of magnitude \texttip{F}{F} making an angle \texttip{\theta }{theta} with the x axis is applied to a particle located along axis of rotation A, at Cartesian coordinates (0, 0) in the figure. The vector \texttip{\vec{F}}{F_vec} lies in the xy plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the xy plane. A particle is located at a vector position \texttip{\vec{r}}{r_vec} with respect to an axis of rotation (thus \texttip{\vec{r}}{r_vec} points from the axis to the point at which the particle is located). The magnitude of the torque \texttip{\tau }{tau} about this axis due to a force \texttip{\vec{F}}{F_vec} acting on the particle is given by \tau = r F \sin(\alpha) = ({\rm moment \; arm}) \cdot F = rF_{\perp}, where \texttip{\alpha }{alpha} is the angle between \texttip{\vec{r}}{r_vec} and \texttip{\vec{F}}{F_vec}, \texttip{r}{r} is the magnitude of \texttip{\vec{r}}{r_vec}, \texttip{F}{F} is the magnitude of \texttip{\vec{F}}{F_vec}, the component of \texttip{\vec{r}}{r_vec} that is perpendicualr to \texttip{\vec{F}}{F_vec} is the moment arm, and \texttip{F_{\rm \perp}}{F_\perp} is the component of the force that is perpendicular to \texttip{\vec{r}}{r_vec}. Sign convention: You will need to determine the sign by analyzing the direction of the rotation that the torque would tend to produce. Recall that negative torque about an axis corresponds to clockwise rotation. In this problem, you must express the angle \texttip{\alpha }{alpha} in the above equation in terms of \texttip{\theta }{theta}, \texttip{\phi }{phi}, and/or \texttip{\pi }{pi} when entering your answers. Keep in mind that \pi = 180\;\rm degrees and (\pi/2) = 90\;\rm degrees . Part A What is the torque \texttip{\tau_{\rm A}}{tau_A} about axis A due to the force \texttip{\vec{F}}{F_vec}? Express the torque about axis A at Cartesian coordinates (0, 0). You did not open hints for this part. Typesetting math: 29% ANSWER: Part B What is the torque \texttip{\tau_{\rm B}}{tau_B} about axis B due to the force \texttip{\vec{F}}{F_vec}? (B is the point at Cartesian coordinates (0, b), located a distance \texttip{b}{b} from the origin along the y axis.) Express the torque about axis B in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. You did not open hints for this part. ANSWER: Part C What is the torque \texttip{\tau_{\rm C}}{tau_C} about axis C due to \texttip{\vec{F}}{F_vec}? (C is the point at Cartesian coordinates (c, 0), a distance \texttip{c}{c} along the x axis.) Express the torque about axis C in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. You did not open hints for this part. ANSWER: \texttip{\tau_{\rm A}}{tau_A} = \texttip{\tau_{\rm B}}{tau_B} = Typesetting math: 29% Part D What is the torque \texttip{\tau_{\rm D}}{tau_D} about axis D due to \texttip{\vec{F}}{F_vec}? (D is the point located at a distance \texttip{d}{d} from the origin and making an angle \texttip{\phi }{phi} with the x axis.) Express the torque about axis D in terms of \texttip{F}{F}, \texttip{\theta }{theta}, \texttip{\phi }{phi}, \texttip{\pi }{pi}, and/or other given coordinate data. ANSWER: Torque Magnitude Ranking Task The wrench in the figure has six forces of equal magnitude acting on it. \texttip{\tau_{\rm C}}{tau_C} = \texttip{\tau_{\rm D}}{tau_D} = Typesetting math: 29% Part A Rank these forces (A through F) on the basis of the magnitude of the torque they apply to the wrench, measured about an axis centered on the bolt. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: The Parallel-Axis Theorem Typesetting math: 29% Learning Goal: To understand the parallel-axis theorem and its applications To solve many problems about rotational motion, it is important to know the moment of inertia of each object involved. Calculating the moments of inertia of various objects, even highly symmetrical ones, may be a lengthy and tedious process. While it is important to be able to calculate moments of inertia from the definition (I=\sum m_ir_i^2), in most cases it is useful simply to recall the moment of inertia of a particular type of object. The moments of inertia of frequently occurring shapes (such as a uniform rod, a uniform or a hollow cylinder, a uniform or a hollow sphere) are well known and readily available from any mechanics text, including your textbook. However, one must take into account that an object has not one but an infinite number of moments of inertia. One of the distinctions between the moment of inertia and mass (the latter being the measure of tranlsational inertia) is that the moment of inertia of a body depends on the axis of rotation. The moments of inertia that you can find in the textbooks are usually calculated with respect to an axis passing through the center of mass of the object. However, in many problems the axis of rotation does not pass through the center of mass. Does that mean that one has to go through the lengthy process of finding the moment of inertia from scratch? It turns out that in many cases, calculating the moment of inertia can be done rather easily if one uses the parallel-axis theorem. Mathematically, it can be expressed as I=I_{\rm cm}+md^2, where \texttip{I_{\rm cm}}{I_cm} is the moment of inertia about an axis passing through the center of mass, \texttip{m}{m} is the total mass of the object, and \texttip{I}{I} is the moment of inertia about another axis, parallel to the one for which \texttip{I_{\rm cm}}{I_cm} is calculated and located a distance \texttip{d}{d} from the center of mass. In this problem you will show that the theorem does indeed work for at least one object: a dumbbell of length \texttip{2r}{2r} made of two small spheres of mass \texttip{m}{m} each connected by a light rod (see the figure). NOTE: Unless otherwise noted, all axes considered are perpendicular to the plane of the page. Part A Using the definition of moment of inertia, calculate I_{\rm cm}, the moment of inertia about the center of mass, for this object. Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B Using the definition of moment of inertia, calculate I_{\rm B}, the moment of inertia about an axis through point B, for this object. Point B coincides with (the center of) one of the spheres (see the figure). Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. ANSWER: Part C Now calculate I_{\rm B} for this object using the parallel-axis theorem. Express your answer in terms of \texttip{I_{\rm cm}}{I_cm}, \texttip{m}{m}, and \texttip{r}{r}. ANSWER: I_{\rm cm} = I_{\rm B} = I_{\rm B} = Typesetting math: 29% Part D Using the definition of moment of inertia, calculate I_{\rm C}, the moment of inertia about an axis through point C, for this object. Point C is located a distance \texttip{r}{r} from the center of mass (see the figure). Express your answer in terms of \texttip{m}{m} and \texttip{r}{r}. You did not open hints for this part. ANSWER: Part E Now calculate I_{\rm C} for this object using the parallel-axis theorem. Express your answer in terms of \texttip{I_{\rm cm}}{I_cm}, \texttip{m}{m}, and \texttip{r}{r}. ANSWER: Consider an irregular object of mass \texttip{m}{m}. Its moment of inertia measured with respect to axis A (parallel to the plane of the page), which passes through the center of mass (see the second diagram), is given by I_{\rm A}=0.64mr^2. Axes B, C, D, and E are parallel to axis A; their separations from axis A are shown in the diagram. In the subsequent questions, the subscript indicates the axis with respect to which the moment of inertia is measured: for instance, I_{\rm C} is the moment of inertia about axis C. I_{\rm C} = I_{\rm C} = Typesetting math: 29% Part F Which moment of inertia is the smallest? ANSWER: Part G Which moment of inertia is the largest? ANSWER: I_{\rm A} I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} I_{\rm A} I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} Typesetting math: 29% Part H Which moments of inertia are equal? ANSWER: Part I Which moment of inertia equals 4.64mr^2? ANSWER: Part J Axis X, not shown in the diagram, is parallel to the axes shown. It is known that I_{\rm X}=6mr^2. Which of the following is a possible location for axis X? ANSWER: I_{\rm A} and I_{\rm D} I_{\rm B} and I_{\rm C} I_{\rm C} and I_{\rm E} No two moments of inertia are equal. I_{\rm B} I_{\rm C} I_{\rm D} I_{\rm E} between axes A and C between axes C and D between axes D and E to the right of axis E Typesetting math: 29% Torque and Angular Acceleration Learning Goal: To understand and apply the formula \tau=I\alpha to rigid objects rotating about a fixed axis. To find the acceleration \texttip{a}{a} of a particle of mass \texttip{m}{m}, we use Newton’s second law: \vec {F}_{\rm net}=m\vec{a}, where \texttip{\vec{F}_{\rm net}}{F_vec_net} is the net force acting on the particle. To find the angular acceleration \texttip{\alpha }{alpha} of a rigid object rotating about a fixed axis, we can use a similar formula: \tau_{\rm net}=I\alpha, where \tau_{\rm net}=\sum \tau is the net torque acting on the object and \texttip{I}{I} is its moment of inertia. In this problem, you will practice applying this formula to several situations involving angular acceleration. In all of these situations, two objects of masses \texttip{m_{\rm 1}}{m_1} and \texttip{m_{\rm 2}}{m_2} are attached to a seesaw. The seesaw is made of a bar that has length \texttip{l}{l} and is pivoted so that it is free to rotate in the vertical plane without friction. You are to find the angular acceleration of the seesaw when it is set in motion from the horizontal position. In all cases, assume that m_1>m_2, and that counterclockwise is considered the positive rotational direction. Part A The seesaw is pivoted in the middle, and the mass of the swing bar is negligible. Find the angular acceleration \texttip{\alpha }{alpha} of the seesaw. Express your answer in terms of some or all of the quantities \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{l}{l}, as well as the acceleration due to gravity \texttip{g}{g}. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B In what direction will the seesaw rotate, and what will the sign of the angular acceleration be? ANSWER: Part C This question will be shown after you complete previous question(s). \texttip{\alpha }{alpha} = The rotation is in the clockwise direction and the angular acceleration is positive. The rotation is in the clockwise direction and the angular acceleration is negative. The rotation is in the counterclockwise direction and the angular acceleration is positive. The rotation is in the counterclockwise direction and the angular acceleration is negative. Typesetting math: 29% Part D In what direction will the seesaw rotate and what will the sign of the angular acceleration be? ANSWER: 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). Pivoted Rod with Unequal Masses The figure shows a simple model of a seesaw. These consist of a plank/rod of mass \texttip{m_{\rm r}}{m_r} and length 2x allowed to pivot freely about its center (or central axis), as shown in the diagram. A small sphere of mass \texttip{m_{\rm 1}}{m_1} is attached to the left end of the rod, and a small sphere of mass \texttip{m_{\rm 2}}{m_2} is attached to the right end. The spheres are small enough that they can be considered point particles. The gravitational force acts downward. The magnitude of the acceleration due to gravity is equal to \texttip{g}{g}. The rotation is in the clockwise direction and the angular acceleration is positive. The rotation is in the clockwise direction and the angular acceleration is negative. The rotation is in the counterclockwise direction and the angular acceleration is positive. The rotation is in the counterclockwise direction and the angular acceleration is negative. Typesetting math: 29% Part A What is the moment of inertia \texttip{I}{I} of this assembly about the axis through which it is pivoted? Express the moment of inertia in terms of \texttip{m_{\rm r}}{m_r}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{x}{x}. Keep in mind that the length of the rod is 2x, not \texttip{x}{x}. You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Weight and Wheel Consider a bicycle wheel that initially is not rotating. A block of mass \texttip{m}{m} is attached to the wheel and is allowed to fall a distance \texttip{h}{h}. Assume that the wheel has a moment of inertia \texttip{I}{I} about its rotation axis. Part A Consider the case that the string tied to the block is attached to the outside of the wheel, at a radius \texttip{r_{\mit A}}{r_A} \texttip{I}{I} = Typesetting math: 29% . Find \texttip{\omega _{\mit A}}{omega_A}, the angular speed of the wheel after the block has fallen a distance \texttip{h}{h}, for this case. Express \texttip{\omega _{\mit A}}{omega_A} in terms of \texttip{m}{m}, \texttip{g}{g}, \texttip{h}{h}, \texttip{r_{\mit A}}{r_A}, and \texttip{I}{I}. You did not open hints for this part. ANSWER: Part B Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius \texttip{r_{\mit B}}{r_B} . Find \texttip{\omega _{\mit B}}{omega_B}, the angular speed of the wheel after the block has fallen a distance \texttip{h}{h}, for this case. Express \texttip{\omega _{\mit B}}{omega_B} in terms of \texttip{m}{m}, \texttip{g}{g}, \texttip{h}{h}, \texttip{r_{\mit B}}{r_B}, and \texttip{I}{I}. \texttip{\omega _{\mit A}}{omega_A} = Typesetting math: 29% You did not open hints for this part. ANSWER: Part C Which of the following describes the relationship between \texttip{\omega _{\mit A}}{omega_A} and \texttip{\omega _{\mit B}}{omega_B}? You did not open hints for this part. ANSWER: A Bar Suspended by Two Vertical Strings A rigid, uniform, horizontal bar of mass \texttip{m_{\rm 1}}{m_1} and length \texttip{L}{L} is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance d < L/2 from the left end of the bar and is connected to the ceiling; string B is attached to \texttip{\omega _{\mit B}}{omega_B} = \omega_A > \omega_B \omega_B > \omega_A \omega_A = \omega_B Typesetting math: 29% the left end of the bar and is connected to the floor. A small block of mass \texttip{m_{\rm 2}}{m_2} is supported against gravity by the bar at a distance \texttip{x}{x} from the left end of the bar, as shown in the figure. Throughout this problem positive torque is that which spins an object counterclockwise. Use \texttip{g}{g} for the magnitude of the acceleration due to gravity. Part A Find \texttip{T_{\mit A}}{T_A}, the tension in string A. Express the tension in string A in terms of \texttip{g}{g}, \texttip{m_{\rm 1}}{m_1}, \texttip{L}{L}, \texttip{d}{d}, \texttip{m_{\rm 2}}{m_2}, and \texttip{x}{x}. You did not open hints for this part. ANSWER: Part B Find \texttip{T_{\mit B}}{T_B}, the magnitude of the tension in string B. Express the magnitude of the tension in string B in terms of \texttip{T_{\mit A}}{T_A}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{g}{g}. \texttip{T_{\mit A}}{T_A} = Typesetting math: 29% You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). Part D If the mass of the block is too large and the block is too close to the left end of the bar (near string B) then the horizontal bar may become unstable (i.e., the bar may no longer remain horizontal). What is the smallest possible value of \texttip{x}{x} such that the bar remains stable (call it \texttip{x_{\rm critical}}{x_critical})? Express your answer for \texttip{x_{\rm critical}}{x_critical} in terms of \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{d}{d}, and \texttip{L}{L}. You did not open hints for this part. ANSWER: Part E \texttip{T_{\mit B}}{T_B} = \texttip{x_{\rm critical}}{x_critical} = Typesetting math: 29% This question will be shown after you complete previous question(s). A Tale of Two Nutcrackers This problem explores the ways that torque can be used in everyday life. Case 1 To crack a nut a force of magnitude \texttip{F_{\rm n}}{F_n} (or greater) must be applied on both sides, as shown in the figure. One can see that a nutcracker only applies this force at the point in which it contacts the nut (at a distance \texttip{d}{d} from the nutcracker pivot). In this problem the nut is placed in a nutcracker and equal forces of magnitude \texttip{F}{F} are applied to each end, directed perpendicular to the handle, at a distance \texttip{D}{D} from the pivot. The frictional forces between the nut and the nutcracker are equal and large enough that the nut doesn’t shoot out of the nutcracker. Part A Find \texttip{F}{F}, the magnitude of the force applied to each side of the nutcracker required to crack the nut. Express the force in terms of \texttip{F_{\rm n}}{F_n}, \texttip{d}{d}, and \texttip{D}{D}. You did not open hints for this part. ANSWER: Typesetting math: 29% Case 2 The nut is now placed in a nutcracker with only one lever, as shown, and again friction keeps the nut from slipping. The top “jaw” (in black) is fixed to a stationary frame so that a person just has to apply a force to the bottom lever. Assume that \texttip{F_{\rm 2}}{F_2} is directed perpendicular to the handle. Part B Find the magnitude of the force \texttip{F_{\rm 2}}{F_2} required to crack the nut. Express your answer in terms of \texttip{F_{\rm n}}{F_n}, \texttip{d}{d}, and \texttip{D}{D}. You did not open hints for this part. ANSWER: \texttip{F}{F} = \texttip{F_{\rm 2}}{F_2} = Typesetting math: 29% Part C This question will be shown after you complete previous question(s). Precarious Lunch A uniform steel beam of length \texttip{L}{L} and mass \texttip{m_{\rm 1}}{m_1} is attached via a hinge to the side of a building. The beam is supported by a steel cable attached to the end of the beam at an angle \texttip{\theta }{theta}, as shown. Through the hinge, the wall exerts an unknown force, \texttip{F}{F}, on the beam. A workman of mass \texttip{m_{\rm 2}}{m_2} sits eating lunch a distance \texttip{d}{d} from the building. Part A Find \texttip{T}{T}, the tension in the cable. Remember to account for all the forces in the problem. Express your answer in terms of \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, \texttip{L}{L}, \texttip{d}{d}, \texttip{\theta }{theta}, and \texttip{g}{g}, the magnitude of the acceleration due to gravity. You did not open hints for this part. Typesetting math: 29% ANSWER: Part B Find \texttip{F_{\mit x}}{F_x}, the \texttip{x}{x}-component of the force exerted by the wall on the beam ( \texttip{F}{F}), using the axis shown. Remember to pay attention to the direction that the wall exerts the force. Express your answer in terms of \texttip{T}{T} and other given quantities. You did not open hints for this part. ANSWER: Part C Find \texttip{F_{\mit y}}{F_y}, the y-component of force that the wall exerts on the beam ( \texttip{F}{F}), using the axis shown. Remember to pay attention to the direction that the wall exerts the force. Express your answer in terms of \texttip{T}{T}, \texttip{\theta }{theta}, \texttip{m_{\rm 1}}{m_1}, \texttip{m_{\rm 2}}{m_2}, and \texttip{g}{g}. ANSWER: \texttip{T}{T} = \texttip{F_{\mit x}}{F_x} = \texttip{F_{\mit y}}{F_y} = Typesetting math: 29% Pulling Out a Nail A nail is hammered into a board so that it would take a force \texttip{F_{\rm nail}}{F_nail}, applied straight upward on the head of the nail, to pull it out. (Take an upward force to be positive.) A carpenter uses a crowbar to try to pry it out. The length of the handle of the crowbar is \texttip{L_{\rm h}}{L_h}, and the length of the forked portion of the crowbar (which fits around the nail) is \texttip{L_{\rm n}}{L_n}. Assume that the forked portion of the crowbar is perfectly horizontal. The handle of the crowbar makes an angle \texttip{\theta }{theta} with the horizontal, and the carpenter pulls directly along the horizontal. Typesetting math: 29% Part A With what force \texttip{F_{\rm pull}}{F_pull} must the carpenter pull on the crowbar to remove the nail? Express the force in terms of \texttip{F_{\rm nail}}{F_nail}, \texttip{L_{\rm h}}{L_h}, \texttip{L_{\rm n}}{L_n}, and \texttip{\theta }{theta}. You did not open hints for this part. ANSWER: Now, imagine that \texttip{F_{\rm pull}}{F_pull} is not large enough to dislodge the nail. In other words, the nail stays in place, and, if the surface below the crowbar weren’t present, the crowbar would rotate around the point of contact with the nail. This makes it natural to take the pivot point to be the point where the crowbar is in contact with the nail. (But you are always free to choose the pivot point to be any fixed point, even one some distance from the object.) Part B What is the magnitude of the normal force that the surface exerts on the crowbar, \texttip{F_{\rm bar}}{F_bar}? Express your answer for the normal force in terms of \texttip{F_{\rm pull}}{F_pull}, \texttip{\theta }{theta}, \texttip{L_{\rm n}}{L_n}, and \texttip{L_{\rm h}}{L_h}. Take the upward direction to be positive. You did not open hints for this part. ANSWER: Three bars are shown in the figure. Both bars A and B have \texttip{F_{\rm pull}}{F_pull} acting on them in the horizontal direction. Bar C has \texttip{F_{\rm pull}}{F_pull} = \texttip{F_{\rm bar}}{F_bar} = Typesetting math: 29% \texttip{F_{\rm pull}}{F_pull} strictly perpendicular to the bar. \texttip{L_{\rm h}}{L_h}, \texttip{L_{\rm n}}{L_n}, and \texttip{\theta }{theta} are the same quantities in each case. Part C Let the magnitude of the torque about the bend in the crowbars be denoted \texttip{\tau _{\mit A}}{tau_A}, \texttip{\tau _{\mit B}}{tau_B} and \texttip{\tau _{\mit C}}{tau_C} for each of the three cases shown. Which of the following is the correct relationship between the magnitude of of the torques? You did not open hints for this part. ANSWER: Tipping Crane \tau_A > \tau_B > \tau_C \tau_A > \tau_B = \tau_C \tau_A = \tau_B = \tau_C \tau_A < \tau_B = \tau_C \tau_A < \tau_B < \tau_C \tau_A = \tau_B > \tau_C \tau_A = \tau_B < \tau_C Typesetting math: 29% Learning Goal: To step through the application of \Sigma \vec{\tau} = 0 to prevent a crane from tipping over. A crane of weight \texttip{W}{W} has a length (wheelbase) \texttip{c}{c}, and its center of mass is midway between the wheels (i.e., the mass of the lifting arm is negligible). The arm extending from the front of the crane has a length \texttip{b}{b} and makes an angle \texttip{\theta }{theta} with the horizontal. The crane contacts the ground only at its front and rear tires. Part A While watching the crane in operation, an observer mentions to you that for a given load there is a maximum angle \texttip{\theta _{\rm max}}{theta_max} between 0 \degree and 90 \degree that the crane arm can make with the horizontal without tipping the crane over. Is this correct? ANSWER: Part B Later that week, while watching the same crane in operation, a different observer mentions to you that there is a maximum load the crane can lift without tipping, and you can find that maximum load by observing the minimum angle \texttip{\theta _{\rm min}}{theta_min} that the crane arm makes with the horizontal. Is this correct? ANSWER: yes no Typesetting math: 29% 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). 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 yes no Typesetting math: 29% Notice that we have the weight of the crane exerting a torque about the front wheels of the same crane. To create a torque, a force must be present, so it would seem that somehow the weight of the crane is exerting a force upon its front wheels. However, the crane is one object, and it follows from Newton's laws that an object cannot exert a net force upon itself. This crane seems to be defying Newton's laws. What's going on here? ANSWER: Part I Assume you get a summer job as a crane operator. On the first day you are lifting a heavy piece of machinery. Even though you have the arm at 70^\circ above the horizontal, the crane begins to tip slowly forward. Consider the following possible actions: Release the brake on the lifting cable so that the load accelerates 1. downward. 2. Release the lifting arm so that \texttip{\theta }{theta} decreases rapidly and the load accelerates downward. 3. Increase \texttip{\theta }{theta} while simultaneously letting out the lifting cable so that the load accelerates downward. 4. Put the crane wheels in gear and accelerate the crane forward. None of these solutions is ideal, but which will have the short-term effect of restoring contact of the crane's rear wheels with the ground? ANSWER: Spinning Situations Suppose you are standing on the center of a merry-go-round that is at rest. You are holding a spinning bicycle wheel over your head so that its rotation axis is pointing upward. The wheel is rotating counterclockwise when observed from above. Newton's laws don't apply to torques. The rear wheels exert a downward force on the front wheels. The crane is not accelerating so forces don't matter. The earth exerts forces on the crane and the load. all but 1 all but 2 all but 3 all but 4 all of them Typesetting math: 29% For this problem, neglect any air resistance or friction between the merry-go-round and its foundation. Part A Suppose you now grab the edge of the wheel with your hand, stopping it from spinning. What happens to the merry-go-round? You did not open hints for this part. ANSWER: Twirling a Baton A majorette in a parade is performing some acrobatic twirlings of her baton. Assume that the baton is a uniform rod of mass 0.120{\rm kg} and length 80.0{\rm cm} . Part A Initially, the baton is spinning about a line through its center at angular velocity 3.00{\rm rad/s} . What is its angular momentum? Express your answer in kilogram meters squared per second. It remains at rest. It begins to rotate counterclockwise (as observed from above). It begins to rotate clockwise (as observed from above). Typesetting math: 29% You did not open hints for this part. ANSWER: Part B 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. \rm kg \cdot m^2/s Typesetting math: 29%

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## You will receive no credit for items you complete after the assignment is due. Grading Policy Exercise 2.5 Starting from the front door of your ranch house, you walk 60.0 due east to your windmill, and then you turn around and slowly walk 35.0 west to a bench where you sit and watch the sunrise. It takes you 27.0 to walk from your house to the windmill and then 49.0 to walk from the windmill to the bench. Part A For the entire trip from your front door to the bench, what is your average velocity? Express your answer with the appropriate units. ANSWER: Correct Part B For the entire trip from your front door to the bench, what is your average speed? Express your answer with the appropriate units. ANSWER: Correct Exercise 2.7 A car is stopped at a traffic light. It then travels along a straight road so that its distance from the light is given by , where = 2.40 and = 0.110 . = -0.329 average speed = 1.25 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 1 of 16 3/23/2015 11:12 AM Part A Calculate the average velocity of the car for the time interval = 0 to = 10.0 . ANSWER: Correct Part B Calculate the instantaneous velocity of the car at =0. ANSWER: Correct Part C Calculate the instantaneous velocity of the car at =5.00 . ANSWER: Correct Part D Calculate the instantaneous velocity of the car at =10.0 . ANSWER: Correct Part E How long after starting from rest is the car again at rest? ANSWER: = 13.0 = 0 = 15.8 = 15.0 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 2 of 16 3/23/2015 11:12 AM Correct Exercise 2.9 A ball moves in a straight line (the x-axis). The graph in the figure shows this ball’s velocity as a function of time. Part A What are the ball’s average velocity during the first 2.8 ? Express your answer using two significant figures. ANSWER: Answer Requested Part B What are the ball’s average speed during the first 2.8 ? Express your answer using two significant figures. ANSWER: Correct = 14.5 = 2.3 = 2.3 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 3 of 16 3/23/2015 11:12 AM Part C Suppose that the ball moved in such a way that the graph segment after 2.0 was -3.0 instead of +3.0 . Find the ball’s and average velocity during the first 2.8 in this case. Express your answer using two significant figures. ANSWER: All attempts used; correct answer displayed Part D Suppose that the ball moved in such a way that the graph segment after 2.0 was -3.0 instead of +3.0 . Find the ball’s average speed during the first 2.8 in this case. Express your answer using two significant figures. ANSWER: Correct Exercise 2.13 Part A The table shows test data for the Bugatti Veyron, the fastest car made. The car is moving in a straight line (the x-axis). Time 0 2.10 20.0 53.0 Speed 0 60.0 205 259 Calculate the car’s average acceleration (in ) between 0 and 2.1 . ANSWER: Correct Part B Calculate the car’s average acceleration (in ) between 2.1 and 20.0 . = 0.57 = 2.3 = 12.8 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 4 of 16 3/23/2015 11:12 AM ANSWER: Correct Part C Calculate the car’s average acceleration (in ) between 20.0 and 53 . ANSWER: Correct Exercise 2.19 An antelope moving with constant acceleration covers the distance 79.0 between two points in time 7.00 . Its speed as it passes the second point is 14.5 . Part A What is its speed at the first point? ANSWER: Correct Part B What is the acceleration? ANSWER: Correct Exercise 2.22 In the fastest measured tennis serve, the ball left the racquet at 73.14 . A served tennis ball is typically in contact with = 3.62 = 0.731 = 8.07 = 0.918 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 5 of 16 3/23/2015 11:12 AM the racquet for 27.0 and starts from rest. Assume constant acceleration. Part A What was the ball’s acceleration during this serve? ANSWER: Correct Part B How far did the ball travel during the serve? ANSWER: Correct Exercise 2.30 A cat walks in a straight line, which we shall call the x-axis with the positive direction to the right. As an observant physicist, you make measurements of this cat’s motion and construct a graph of the feline’s velocity as a function of time (the figure ). Part A Find the cat’s velocity at = 5.0 . Express your answer using two significant figures. ANSWER: = 2710 = 0.987 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 6 of 16 3/23/2015 11:12 AM Correct Part B Find the cat’s velocity at = 8.0 . Express your answer using two significant figures. ANSWER: Correct Part C What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: Correct Part D What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: Correct Part E What is the cat’s acceleration at ? Express your answer using two significant figures. ANSWER: = 1.3 = -2.7 = -1.3 = -1.3 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 7 of 16 3/23/2015 11:12 AM Correct Part F What distance does the cat move during the first 4.5 ? Express your answer using two significant figures. ANSWER: Correct Part G What distance does the cat move from to ? Express your answer using two significant figures. ANSWER: Correct Part H Sketch clear graph of the cat’s acceleration as function of time, assuming that the cat started at the origin. ANSWER: = -1.3 = 23 = 26 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 8 of 16 3/23/2015 11:12 AM Correct Part I Sketch clear graph of the cat’s position as function of time, assuming that the cat started at the origin. ANSWER: Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 9 of 16 3/23/2015 11:12 AM All attempts used; correct answer displayed Exercise 2.35 Part A If a flea can jump straight up to a height of 0.510 , what is its initial speed as it leaves the ground? ANSWER: Correct Part B How long is it in the air? ANSWER: Correct = 3.16 = 0.645 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 10 of 16 3/23/2015 11:12 AM Exercise 2.36 A small rock is thrown vertically upward with a speed of 18.0 from the edge of the roof of a 39.0 tall building. The rock doesn’t hit the building on its way back down and lands in the street below. Air resistance can be neglected. Part A What is the speed of the rock just before it hits the street? Express your answer with the appropriate units. ANSWER: Correct Part B How much time elapses from when the rock is thrown until it hits the street? Express your answer with the appropriate units. ANSWER: Correct Exercise 2.38 You throw a glob of putty straight up toward the ceiling, which is 3.00 above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.70 . Part A What is the speed of the putty just before it strikes the ceiling? Express your answer with the appropriate units. ANSWER: Correct Part B = 33.0 = 5.20 = 5.94 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 11 of 16 3/23/2015 11:12 AM How much time from when it leaves your hand does it take the putty to reach the ceiling? Express your answer with the appropriate units. ANSWER: Correct Exercise 3.1 A squirrel has x- and y-coordinates ( 1.2 , 3.3 ) at time and coordinates ( 5.3 , -0.80 ) at time = 2.6 . Part A For this time interval, find the x-component of the average velocity. Express your answer using two significant figures. ANSWER: Correct Part B For this time interval, find the y-component of the average velocity. Express your answer using two significant figures. ANSWER: Correct Part C Find the magnitude of the average velocity. Express your answer using two significant figures. ANSWER: = 0.384 = 1.6 = -1.6 = 2.2 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 12 of 16 3/23/2015 11:12 AM Correct Part D Find the direction of the average velocity. Express your answer using two significant figures. ANSWER: Correct Exercise 3.3 A web page designer creates an animation in which a dot on a computer screen has a position of 4.1 2.1 4.7 . Part A Find the average velocity of the dot between and . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part B Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part C = 45 below the x-axis = 4.2,4.7 = 0,4.7 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 13 of 16 3/23/2015 11:12 AM Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part D Find the instantaneous velocity at . Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Exercise 3.5 A jet plane is flying at a constant altitude. At time it has components of velocity 89 , 108 . At time 32.5 the components are 165 , 37 . Part A For this time interval calculate the average acceleration. Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3 and the y component is 4, then you should enter 3,4. Express your answer using two significant figures. ANSWER: Correct Part B Find the magnitude of the average acceleration. Express your answer using two significant figures. ANSWER: = 4.2,4.7 = 8.4,4.7 = 2.3,-2.2 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 14 of 16 3/23/2015 11:12 AM Correct Part C Find the direction of the average acceleration (let the direction be the angle that the vector makes with the +x-axis, measured counterclockwise). ANSWER: Correct Exercise 3.4 The position of a squirrel running in a park is given by . Part A What is , the -component of the velocity of the squirrel, as function of time? ANSWER: Correct Part B What is , the y-component of the velocity of the squirrel, as function of time? ANSWER: = 3.2 = -43.1 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 15 of 16 3/23/2015 11:12 AM Correct Part C At 4.51 , how far is the squirrel from its initial position? Express your answer to three significant figures and include the appropriate units. ANSWER: All attempts used; correct answer displayed Part D At 4.51 , what is the magnitude of the squirrel’s velocity? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Part E At 4.51 , what is the direction (in degrees counterclockwise from +x-axis) of the squirrel’s velocity? Express your answer to three significant figures and include the appropriate units. ANSWER: Correct Score Summary: Your score on this assignment is 90.1%. You received 14.42 out of a possible total of 16 points. = 2.65 = 1.31 = 62.5 Week 2 https://session.masteringphysics.com/myct/assignmentPrintView?assignme… 16 of 16 3/23/2015 11:12 AM

## Chapter 14 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 Harmonic Oscillator Equations Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions. A block of mass is attached to a spring whose spring constant is . The other end of the spring is fixed so that when the spring is unstretched, the mass is located at . . Assume that the +x direction is to the right. The mass is now pulled to the right a distance beyond the equilibrium position and released, at time , with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is , and its solution, which provides the equation for , is . Part A At what time does the block come back to its original equilibrium position ( ) for the first time? Express your answer in terms of some or all of the variables: , , and . You did not open hints for this part. ANSWER: m k x = 0 A t = 0 a(t) = − x(t) km x(t) x(t) = Acos( t) km −− t1 x = 0 A k m Part B Find the velocity of the block as a function of time. Express your answer in terms of some or all of the variables: , , , and . You did not open hints for this part. ANSWER: Part C This question will be shown after you complete previous question(s). t1 = v k m A t v(t) = Typesetting math: 100% Part D Find the acceleration of the block as a function of time. Express your answer in terms of some of all of the variables: , , , and . ANSWER: Part E Specify when the magnitude of the acceleration of the block reaches its maximum value. Consider the following options: only once during one a. period of motion, b. when the block’s speed is zero, c. when the block is in the equilibrium position, d. when the block’s displacement equals either or , e. when the block’s speed is at a maximum. Choose the most complete answer. You did not open hints for this part. ANSWER: a k m A t a(t) = A −A Typesetting math: 100% Part F Find the kinetic energy of the block as a function of time. Express your answer in terms of some or all of the variables: , , , and . You did not open hints for this part. ANSWER: Part G Find , the maximum kinetic energy of the block. Express your answer in terms of some or all of the variables: , , and . ANSWER: a only b only c only d only e only b and d c and e b and c a and e d and e K k m A t K(t) = Kmax k m A Typesetting math: 100% Part H The kinetic energy of the block reaches its maximum when which of the following occurs? You did not open hints for this part. ANSWER: Mass and Simple Harmonic Motion Conceptual Question The shaker cart, shown in the figure, is the latest extreme sport craze. You stand inside of a small cart attached to a heavy-duty spring, the spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you. Kmax = The displacement of the block is zero. The displacement of the block is . The acceleration of the block is at a maximum. The velocity of the block is zero. A Typesetting math: 100% At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground. Part A What effect does dropping the sandbag out of the cart at the equilibrium position have on the amplitude of your oscillation? You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Instead of dropping the sandbag as you pass through equilibrium, you decide to drop the sandbag when the cart is at its maximum distance from equilibrium. Part C This question will be shown after you complete previous question(s). Part D It increases the amplitude. It decreases the amplitude. It has no effect on the amplitude. Typesetting math: 100% This question will be shown after you complete previous question(s). Simple Harmonic Motion Conceptual Question An object of mass is attached to a vertically oriented spring. The object is pulled a short distance below its equilibrium position and released from rest. Set the origin of the coordinate system at the equilibrium position of the object and choose upward as the positive direction. Assume air resistance is so small that it can be ignored. Refer to these graphs when answering the following questions. Part A Beginning the instant the object is released, select the graph that best matches the position vs. time graph for the object. You did not open hints for this part. ANSWER: m Typesetting math: 100% Part B Beginning the instant the object is released, select the graph that best matches the velocity vs. time graph for the object. You did not open hints for this part. ANSWER: Part C Beginning the instant the object is released, select the graph that best matches the acceleration vs. time graph for the object. A B C D E F G H A B C D E F G H Typesetting math: 100% You did not open hints for this part. ANSWER: Harmonic Oscillator Acceleration Learning Goal: To understand the application of the general harmonic equation to finding the acceleration of a spring oscillator as a function of time. One end of a spring with spring constant is attached to the wall. The other end is attached to a block of mass . The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be . The length of the relaxed spring is . The block is slowly pulled from its equilibrium position to some position along the x axis. At time , the block is released with zero initial velocity. The goal of this problem is to determine the acceleration of the block as a function of time in terms of , , and . It is known that a general solution for the position of a harmonic oscillator is , where , , and are constants. Your task, therefore, is to determine the values of , , and in terms of , ,and and then use the connection between and to find the acceleration. A B C D E F G H k m x = 0 L xinit > 0 t = 0 a(t) k m xinit x(t) = C cos (t) + S sin (t) C S C S k m xinit x(t) a(t) Typesetting math: 100% Part A Combine Newton’s 2nd law and Hooke’s law for a spring to find the acceleration of the block as a function of time. Express your answer in terms of , , and the coordinate of the block . You did not open hints for this part. ANSWER: Part B This question will be shown after you complete previous question(s). Part C a(t) k m x(t) a(t) = Typesetting math: 100% This question will be shown after you complete previous question(s). ± Introduction to Simple Harmonic Motion Consider the system shown in the figure. It consists of a block of mass attached to a spring of negligible mass and force constant . The block is free to move on a frictionless horizontal surface, while the left end of the spring is held fixed. When the spring is neither compressed nor stretched, the block is in equilibrium. If the spring is stretched, the block is displaced to the right and when it is released, a force acts on it to pull it back toward equilibrium. By the time the block has returned to the equilibrium position, it has picked up some kinetic energy, so it overshoots, stopping somewhere on the other side, where it is again pulled back toward equilibrium. As a result, the block moves back and forth from one side of the equilibrium position to the other, undergoing oscillations. Since we are ignoring friction (a good approximation to many cases), the mechanical energy of the system is conserved and the oscillations repeat themselves over and over. The motion that we have just described is typical of most systems when they are displaced from equilibrium and experience a restoring force that tends to bring them back to their equilibrium position. The resulting oscillations take the name of periodic motion. An important example of periodic motion is simple harmonic motion (SHM) and we will use the mass-spring system described here to introduce some of its properties. Part A Which of the following statements best describes the characteristic of the restoring force in the spring-mass system described in the introduction? You did not open hints for this part. ANSWER: m k The restoring force is constant. The restoring force is directly proportional to the displacement of the block. The restoring force is proportional to the mass of the block. The restoring force is maximum when the block is in the equilibrium position. Typesetting math: 100% Part B As shown in the figure, a coordinate system with the origin at the equilibrium position is chosen so that the x coordinate represents the displacement from the equilibrium position. (The positive direction is to the right.) What is the initial acceleration of the block, , when the block is released at a distance from its equilibrium position? Express your answer in terms of some or all of the variables , , and . You did not open hints for this part. ANSWER: Part C What is the acceleration of the block when it passes through its equilibrium position? Express your answer in terms of some or all of the variables , , and . You did not open hints for this part. ANSWER: a0 A A m k a0 = a1 A m k Typesetting math: 100% Part D This question will be shown after you complete previous question(s). Using the information found so far, select the correct phrases to complete the following statements. Part E You did not open hints for this part. ANSWER: Part F You did not open hints for this part. ANSWER: a1 = The magnitude of the block’s acceleration reaches its maximum value when the block is in the equilibrium position. at either its rightmost or leftmost position. between its rightmost position and the equilibrium position. between its leftmost position and the equilibrium position. Typesetting math: 100% Part G You did not open hints for this part. ANSWER: Part H Because of the periodic properties of SHM, the mathematical equations that describe this motion involve sine and cosine functions. For example, if the block is released at a distance from its equilibrium position, its displacement varies with time according to the equation , where is a constant characteristic of the system. If time is measured is seconds, must be expressed in radians per second so that the quantity is expressed in radians. Use this equation and the information you now have on the acceleration and speed of the block as it moves back and forth from one side of its equilibrium position to the other to determine the correct set of equations for the block’s x components of velocity and acceleration, and , respectively. In the expressions below, and are nonzero positive constants. You did not open hints for this part. The speed of the block is zero when it is in the equilibrium position. at either its rightmost or leftmost position. between its rightmost position and the equilibrium position. between its leftmost position and the equilibrium position. The speed of the block reaches its maximum value when the block is in the equilibrium position. at either its rightmost or leftmost position. between the rightmost position and the equilibrium position. between the leftmost position and the equilibrium position. A x t x = Acost t vx ax B C Typesetting math: 100% ANSWER: Period of a Pendulum Ranking Task Part A Six pendulums of mass and length as shown are released from rest at the same angle from vertical. Rank the pendulums according to the number of complete cycles of motion each pendulum goes through per minute. Rank from most to least complete cycles of motion per minute. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: , , , , vx = −Bsint ax = C cost vx = Bcost ax = C sint vx = −Bcost ax = −C cost vx = −Bsint ax = −C cost m L Typesetting math: 100% ± Gravity on Another Planet After landing on an unfamiliar planet, a space explorer constructs a simple pendulum of length 47.0 . The explorer finds that the pendulum completes 108 full swing cycles in a time of 138 . Part A What is the magnitude of the gravitational acceleration on this planet? Express your answer in meters per second per second. You did not open hints for this part. cm s Typesetting math: 100% ANSWER: ± Tactics Box 14.1 Identifying and Analyzing Simple Harmonic Motion Learning Goal: To practice Tactics Box 14.1 Identifying and analyzing simple harmonic motion. A complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations among them. This Tactics Box summarizes the essential information needed to solve oscillation problems of this type. TACTICS BOX 14.1 Identifying and analyzing simple harmonic motion If the net force acting on a particle is a linear restoring force, the motion will be simple harmonic motion around the equilibrium 1. position. The position as a function of time is . The velocity as a function of time is . The maximum speed is . The equations are given here in terms of , but they can be written in terms of , or some other parameter if the situation calls for it. 2. 3. The amplitude and the phase constant are determined by the initial conditions through and . 4. The angular frequency (and hence the period ) depends on the physical properties of the situtaion. But does not depend on or . Mechanical energy is conserved. Thus .Energy conservation provides a relationship between position and velocity that is independent of time. 5. Part A The position of a 60 oscillating mass is given by , where is in seconds. Determine the velocity at . Express your answer in meters per second to two significant figures. You did not open hints for this part. ANSWER: gplanet = m/s2 x(t) = Acos(t + 0 ) vx(t) = −Asin(t + 0 ) vmax = A x y A 0 x0 = Acos 0 v0x = −Asin 0 T = 2/ A 0 1 m + k = k = m( 2 v2 x 1 2 x2 1 2 A2 1 2 vmax)2 g x(t) = (2.0 cm) cos(10t) t t = 0.40 s Typesetting math: 100% Part B Assume that the oscillating mass described in Part A is attached to a spring. What would the spring constant of this spring be? Express your answer in newtons per meter to two significant figures. You did not open hints for this part. ANSWER: Part C What is the total energy of the mass described in the previous parts? Express your answer in joules 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. vx = m/s k k = N/m E E = J Typesetting math: 100%

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## To identify the correct notation for a point and a vector, determine the position vector of a point relative to another point , and calculate the corresponding unit vector. although vectors are often constructed from points , points are not vectors, vectors are commonly constructed from either : (1) the origin to a point or (2) a starting point to an ending point. Part A ) as shown on the coordinate system, points A and B have the following distance from the origin : xA = 2.70 ft , zA = 2.50 ft , xB = 1.10 ft , and zB = 1.70 ft. which of the following statements correctly describes the location and position vector of point A from the origin ? Use appropriate notation for the location and position vector of a point,

## University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Exam 1 26 April 2013 Name: Problem 1 (25 points) Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Answer the following questions: a. Find cov(i; ^1). b. If 0 = 4 what is the least squares estimate of 1? c. What is the variance of the estimate of part (b)? d. Is the estimate of part (b) unbiased? Problem 2 (25 points) Answer the following questions: a. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ^0 is BLUE (it has the smallest variance among all the linear unbiased estimators of 0). b. Consider the model of part (a). Find cov(ei; ^ Yi). c. Consider the simple regression model through the origin yi = 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that Pn i=1 xiei where ei = Yi ? ^ Yi. d. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2xi, and cov(i; j) = 0. Also, x is nonrandom. Is the assumption of constant variance satised in the following model? Please explain. Yi p xi = 0 p xi + 1 xi p xi + i p xi : Problem 3 (25 points): Answer the following questions: a. Consider the model yi = 0 +1xi +i. Assume that E(i) = 0, var(i) = 2, and cov(i; j) = 0. Suppose we rescale the x values as x = x ? , and we want to t the model yi = 0 + 1xi + i. Find the least squares estimates of 0 and 1 . b. Refer to the model yi = 0 + 1xi +i of part (a). Find the SSE of this model and compare it to the SSE of the model yi = 0 + 1xi + i. What is your conclusion? c. Consider the simple regression model yi = 0 + 1xi + i, with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ESY Y = (n ? 1)2 + 2 1SXX, where SY Y = Pn i=1(yi ? y)2 and SXX = Pn i=1(xi ? x)2. d. Refer to the model of part (c). Find cov(i; ei). Problem 4 (25 points) Suppose that a simple linear regression of miles per gallon (Y ) on car weight (x) has been performed on 32 observations. The least squares estimates are ^0 = 68:17 and ^1 = ?1:112, with se = 4:281. Other useful information: x = 30:91 and P32 i=1(xi ? x)2 = 2054:8. Answer the following questions: a. Construct a 95% condence interval for 1. b. Construct a 95% condence interval for 2. c. What is the value of R2? d. Construct a condence interval for 30 ? 21 ? 50.

University of California, Los Angeles Department of Statistics Statistics 100C … Read More...