## Chapter 6 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, March 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy PSS 6.1 Equilibrium Problems Learning Goal: To practice Problem-Solving Strategy 6.1 for equilibrium problems. A pair of students are lifting a heavy trunk on move-in day. Using two ropes tied to a small ring at the center of the top of the trunk, they pull the trunk straight up at a constant velocity . Each rope makes an angle with respect to the vertical. The gravitational force acting on the trunk has magnitude . Find the tension in each rope. PROBLEM-SOLVING STRATEGY 6.1 Equilibrium problems MODEL: Make simplifying assumptions. VISUALIZE: Establish a coordinate system, define symbols, and identify what the problem is asking you to find. This is the process of translating words into symbols. Identify all forces acting on the object, and show them on a free-body diagram. These elements form the pictorial representation of the problem. SOLVE: The mathematical representation is based on Newton’s first law: . The vector sum of the forces is found directly from the free-body diagram. v FG T F = = net i F i 0 ASSESS: Check if your result has the correct units, is reasonable, and answers the question. Model The trunk is moving at a constant velocity. This means that you can model it as a particle in dynamic equilibrium and apply the strategy above. Furthermore, you can ignore the masses of the ropes and the ring because it is reasonable to assume that their combined weight is much less than the weight of the trunk. Visualize Part A The most convenient coordinate system for this problem is one in which the y axis is vertical and the ropes both lie in the xy plane, as shown below. Identify the forces acting on the trunk, and then draw a free-body diagram of the trunk in the diagram below. The black dot represents the trunk as it is lifted by the students. Draw the vectors starting at the black dot. The location and orientation of the vectors will be graded. The length of the vectors will not be graded. ANSWER: Part B This question will be shown after you complete previous question(s). Solve Part C This question will be shown after you complete previous question(s). Assess Part D This question will be shown after you complete previous question(s). A Gymnast on a Rope A gymnast of mass 70.0 hangs from a vertical rope attached to the ceiling. You can ignore the weight of the rope and assume that the rope does not stretch. Use the value for the acceleration of gravity. Part A Calculate the tension in the rope if the gymnast hangs motionless on the rope. Express your answer in newtons. You did not open hints for this part. ANSWER: Part B Calculate the tension in the rope if the gymnast climbs the rope at a constant rate. Express your answer in newtons. You did not open hints for this part. kg 9.81m/s2 T T = N T ANSWER: Part C Calculate the tension in the rope if the gymnast climbs up the rope with an upward acceleration of magnitude 1.10 . Express your answer in newtons. You did not open hints for this part. ANSWER: Part D Calculate the tension in the rope if the gymnast slides down the rope with a downward acceleration of magnitude 1.10 . Express your answer in newtons. You did not open hints for this part. ANSWER: T = N T m/s2 T = N T m/s2 T = N Applying Newton’s 2nd Law Learning Goal: To learn a systematic approach to solving Newton’s 2nd law problems using a simple example. Once you have decided to solve a problem using Newton’s 2nd law, there are steps that will lead you to a solution. One such prescription is the following: Visualize the problem and identify special cases. Isolate each body and draw the forces acting on it. Choose a coordinate system for each body. Apply Newton’s 2nd law to each body. Write equations for the constraints and other given information. Solve the resulting equations symbolically. Check that your answer has the correct dimensions and satisfies special cases. If numbers are given in the problem, plug them in and check that the answer makes sense. Think about generalizations or simplfications of the problem. As an example, we will apply this procedure to find the acceleration of a block of mass that is pulled up a frictionless plane inclined at angle with respect to the horizontal by a perfect string that passes over a perfect pulley to a block of mass that is hanging vertically. Visualize the problem and identify special cases First examine the problem by drawing a picture and visualizing the motion. Apply Newton’s 2nd law, , to each body in your mind. Don’t worry about which quantities are given. Think about the forces on each body: How are these consistent with the direction of the acceleration for that body? Can you think of any special cases that you can solve quickly now and use to test your understanding later? m2 m1 F = ma One special case in this problem is if , in which case block 1 would simply fall freely under the acceleration of gravity: . Part A Consider another special case in which the inclined plane is vertical ( ). In this case, for what value of would the acceleration of the two blocks be equal to zero? Express your answer in terms of some or all of the variables and . ANSWER: Isolate each body and draw the forces acting on it A force diagram should include only real forces that act on the body and satisfy Newton’s 3rd law. One way to check if the forces are real is to detrmine whether they are part of a Newton’s 3rd law pair, that is, whether they result from a physical interaction that also causes an opposite force on some other body, which may not be part of the problem. Do not decompose the forces into components, and do not include resultant forces that are combinations of other real forces like centripetal force or fictitious forces like the “centrifugal” force. Assign each force a symbol, but don’t start to solve the problem at this point. Part B Which of the four drawings is a correct force diagram for this problem? = 0 m2 = −g a 1 j ^ = /2 m1 m2 g m1 = ANSWER: Choose a coordinate system for each body Newton’s 2nd law, , is a vector equation. To add or subtract vectors it is often easiest to decompose each vector into components. Whereas a particular set of vector components is only valid in a particular coordinate system, the vector equality holds in any coordinate system, giving you freedom to pick a coordinate system that most simplifies the equations that result from the component equations. It’s generally best to pick a coordinate system where the acceleration of the system lies directly on one of the coordinate axes. If there is no acceleration, then pick a coordinate system with as many unknowns as possible along the coordinate axes. Vectors that lie along the axes appear in only one of the equations for each component, rather than in two equations with trigonometric prefactors. Note that it is sometimes advantageous to use different coordinate systems for each body in the problem. In this problem, you should use Cartesian coordinates and your axes should be stationary with respect to the inclined plane. Part C Given the criteria just described, what orientation of the coordinate axes would be best to use in this problem? In the answer options, “tilted” means with the x axis oriented parallel to the plane (i.e., at angle to the horizontal), and “level” means with the x axis horizontal. ANSWER: Apply Newton’s 2nd law to each body a b c d F = ma tilted for both block 1 and block 2 tilted for block 1 and level for block 2 level for block 1 and tilted for block 2 level for both block 1 and block 2 Part D What is , the sum of the x components of the forces acting on block 2? Take forces acting up the incline to be positive. Express your answer in terms of some or all of the variables tension , , the magnitude of the acceleration of gravity , and . You did not open hints for this part. 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). Part G This question will be shown after you complete previous question(s). Lifting a Bucket A 6- bucket of water is being pulled straight up by a string at a constant speed. F2x T m2 g m2a2x =F2x = kg Part A What is the tension in the rope? ANSWER: Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Friction Force on a Dancer on a Drawbridge A dancer is standing on one leg on a drawbridge that is about to open. The coefficients of static and kinetic friction between the drawbridge and the dancer’s foot are and , respectively. represents the normal force exerted on the dancer by the bridge, and represents the gravitational force exerted on the dancer, as shown in the drawing . For all the questions, we can assume that the bridge is a perfectly flat surface and lacks the curvature characteristic of most bridges. about 42 about 60 about 78 0 because the bucket has no acceleration. N N N N μs μk n F g Part A Before the drawbridge starts to open, it is perfectly level with the ground. The dancer is standing still on one leg. What is the x component of the friction force, ? Express your answer in terms of some or all of the variables , , and/or . You did not open hints for this part. ANSWER: Part B The drawbridge then starts to rise and the dancer continues to stand on one leg. The drawbridge stops just at the point where the dancer is on the verge of slipping. What is the magnitude of the frictional force now? Express your answer in terms of some or all of the variables , , and/or . The angle should not appear in your answer. F f n μs μk Ff = Ff n μs μk You did not open hints for this part. ANSWER: Part C Then, because the bridge is old and poorly designed, it falls a little bit and then jerks. This causes the person to start to slide down the bridge at a constant speed. What is the magnitude of the frictional force now? Express your answer in terms of some or all of the variables , , and/or . The angle should not appear in your answer. ANSWER: Part D The bridge starts to come back down again. The dancer stops sliding. However, again because of the age and design of the bridge it never makes it all the way down; rather it stops half a meter short. This half a meter corresponds to an angle degree (see the diagram, which has the angle exaggerated). What is the force of friction now? Express your answer in terms of some or all of the variables , , and . Ff = Ff n μs μk Ff = 1 Ff n Fg You did not open hints for this part. ANSWER: Kinetic Friction Ranking Task Below are eight crates of different mass. The crates are attached to massless ropes, as indicated in the picture, where the ropes are marked by letters. Each crate is being pulled to the right at the same constant speed. The coefficient of kinetic friction between each crate and the surface on which it slides is the same for all eight crates. Ff = Part A Rank the ropes on the basis of the force each exerts on the crate immediately to its left. Rank from largest to smallest. To rank items as equivalent, overlap them. You did not open hints for this part. ANSWER: Pushing a Block Learning Goal: To understand kinetic and static friction. A block of mass lies on a horizontal table. The coefficient of static friction between the block and the table is . The coefficient of kinetic friction is , with . Part A m μs μk μk < μs If the block is at rest (and the only forces acting on the block are the force due to gravity and the normal force from the table), what is the magnitude of the force due to friction? You did not open hints for this part. ANSWER: Part B Suppose you want to move the block, but you want to push it with the least force possible to get it moving. With what force must you be pushing the block just before the block begins to move? Express the magnitude of in terms of some or all the variables , , and , as well as the acceleration due to gravity . You did not open hints for this part. ANSWER: Part C Suppose you push horizontally with half the force needed to just make the block move. What is the magnitude of the friction force? Express your answer in terms of some or all of the variables , , and , as well as the acceleration due to gravity . You did not open hints for this part. Ffriction = F F μs μk m g F = μs μk m g ANSWER: Part D Suppose you push horizontally with precisely enough force to make the block start to move, and you continue to apply the same amount of force even after it starts moving. Find the acceleration of the block after it begins to move. Express your answer in terms of some or all of the variables , , and , as well as the acceleration due to gravity . 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. Ffriction = a μs μk m g a =

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