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PHYSICS 105 Assignment #11 Due by 10 pm November 24, 2009 NAME: __________________________________________ DISCUSSION SECTION: [ ] D7 – W 9 am [ ] D9 – W 11 am [ ] F 1 – W 1 pm [ ] F4 – W 4 pm [ ] D3 – F 11 am [ ] D8 – W 10 am [ ] F2 – W 2 pm [ ] F5 – W 7 pm [ ] HS – W 10 am [ ] F3 – W 3 pm [ ] D1 – F 9 am [ ] D4 – F 12 pm [ ] D5 – F 1 pm (Asma) [ ] D2 – F 10 am [ ] D6 – F 1 pm (John) PLEASE CHECK OFF YOUR DISCUSSION SECTION ABOVE! INSTRUCTIONS: 1. Please include appropriate units with all numerical answers. 2. Please show all steps in your solutions! If you need more space for calculations, use the back of the page preceding the question. For example, calculations for problem 3 should be done on the back of the page containing question 2. You must show correct work to receive full credit. Support your answers with brief written explanations and/or arguments based on equations. 3. Indicate clearly which part of your solution is the final answer. 4. Try answering these problems, as much as possible, without a calculator and using only the equation sheet to help you. This will help you prepare for the test. 5. Grading scheme for each problem: each part of each problem is worth 2 points. You get 0 if your answer is wrong or mostly wrong, 2 if your answer is correct, and 1 if your answer is mostly correct. Please do your draft work of the assignment elsewhere, and copy your work over neatly when you hand in the assignment. The graders will deduct points for work that is difficult to follow. Angle (θ) sin(θ) cos(θ) 30˚ 1 2 3 2 2 2 1 2 45˚ 60˚ 2 2 3 2 Problem 1: ____ tan sin cos Problem 2: ____ Problem 3: ____ Problem 4: ____ Problem 5: ____ TOTAL: _____ PROBLEM 1 – 10 points A uniform solid sphere with a mass of M = 1.6 kg and radius R = 20 cm is rolling without slipping on a horizontal surface at a constant speed of 3.0 m/s. It then encounters a ramp inclined at an angle of 10˚ with the horizontal, and proceeds to roll without slipping up the ramp. The goal of this problem is to determine the distance the sphere rolls up the ramp (measured along the ramp) before it turns around, and to use conservation of energy to do so. [2 points] (a) Sketch this situation, showing the sphere in two positions, one at the bottom of the ramp and the other when the sphere reaches its highest point. [2 points] (b) Start with the usual conservation of energy equation: K i U i Wnc K f U f . Identify all the terms that are zero in this equation, and explain why they are zero. [2 points] (c) Write out expressions for the remaining terms. Remember to account for both translational kinetic energy and rotational kinetic energy, if appropriate. Keep everything in terms of variables. [2 points] (d) How far does the sphere roll up the ramp (measured along the ramp)? First find an expression for this distance in terms of variables, simplified as much as possible, and then plug in the appropriate values. [2 points] (e) How far would the sphere have traveled up the ramp if the ramp were frictionless? PROBLEM 2 – 10 points Case 1 and case 2 show two situations of a block hanging from a string wrapped around the outside of a pulley. The blocks are identical, and the pulleys are uniform solid disks of the same radius, but the pulley in case 2 has twice the mass (and therefore twice the moment of inertia) of the pulley in case 1. The systems are released from rest at the same time, and in both cases the block accelerates down. Friction can be neglected. [2 points] (a) In which case is the acceleration of the block larger? [ ] case 1 [ ] case 2 [ ] equal in both cases [2 points] (b) In which case is the tension in the string, while the block is falling, larger? [ ] case 1 [ ] case 2 [ ] equal in both cases [2 points] (c) In which case is the net torque on the pulley, while the block is falling, larger? [ ] case 1 [ ] case 2 [ ] equal in both cases [2 points] (d) In which case does the block have a higher speed as it approaches the ground? [ ] case 1 [ ] case 2 [ ] equal in both cases [2 points] (e) In which case does the pulley have a larger rotational kinetic energy, measuring the kinetic energy just before the block reaches the ground in each case? [ ] case 1 [ ] case 2 [ ] equal in both cases PROBLEM 3 – 8 points Consider the three cases shown. Each case involves a ball of mass 2m interacting with either one or two other balls. In each case, account only for the gravitational force exerted by the balls on one another. Each ball is a distance d from the origin, except for the ball of mass m in case B, which is at the origin. [2 points] (a) Rank the cases based on the magnitude of the net force exerted on the ball of mass 2m. [ ]A>B>C [ ]C>A>B [ ]A=B>C [ ]C>B>A [ ]B>A>C [ ]B>C>A [ ]C>A=B [2 points] (b) Calculate the magnitude of the net force exerted on the ball of mass 2m in case A. [2 points] (c) Comparing case B and case C, in which case is the gravitational potential energy of the system more negative (i.e., has the smallest actual value)? [ ] case B [ ] case C [ ] neither, they’re equal [2 points] (d) Calculate the gravitational potential energy of the system in case A. PROBLEM 4 – 10 points You have two identical springs and two identical blocks. You attach each block to a spring so you have two spring-block systems, and you set the blocks up to oscillate simultaneously on a frictionless horizontal surface. You pull the blocks so they stretch their respective springs, releasing them both from rest simultaneously. However, when you release the blocks one of them (we’ll call this system 1) is displaced a distance A from equilibrium and the other (we’ll call this system 2) is displaced 2A from equilibrium. [2 points] (a) If the block in system 1 reaches a maximum speed v in its oscillations, what is the maximum speed reached by the block in system 2? [ ] v 2 [ ]v [ ] [ ] 2v 2v [ ] 4v [2 points] (b) If the block in system 1 experiences oscillations with a period T , what is the period of the oscillations experienced by the block in system 2? [ ] T 2 [ ]T [ ] [ ] 2T 2T [ ] 4T [2 points] (c) If the maximum force experienced by the block in system 1 is Fmax , what is the maximum force experienced by the block in system 2? [ ] Fmax 2 [ ] Fmax [ ] 2 Fmax [ ] 2 Fmax [ ] 4 Fmax [2 points] (d) If the potential energy stored in the spring in system 1 is U i when the block is first released from rest, what is the potential energy initially stored in the spring in system 2? [ ] Ui 2 [ ] Ui [ ] 2 Ui [ ] 2U i [ ] 4U i [2 points] (e) At a particular instant, some time after being released, the block in system 1 is 20 cm from its equilibrium position. How far from equilibrium is the block in system 2 at that same instant? [ ] 10 cm [ ] 20 cm [ ] 40 cm [ ] there is not enough information to answer this question PROBLEM 5 – 10 points A block on a horizontal frictionless surface is attached to a spring. The spring is at its natural length when the block is at x = 0. At time t = 0, the block is released from rest at the point x = A. Graph 1 represents the position of the block as a function of time for one complete oscillation. Considering one complete oscillation … (a) Which graph shows the block’s velocity as a function of time? _________ (b) Which graph shows the block’s acceleration as a function of time? ________ (c) Which graph shows the block’s kinetic energy as a function of time? ________ (d) Which graph shows the block’s potential energy as a function of position? _______ (e) Which graph shows the magnitude of the net force acting on the block as a function of position? _______ In a second experiment, instead of being released from rest, the block is given a push, so at t = 0 it starts moving from the origin to the right with speed v. Graph 3 represents the position of the block as a function of time for one complete oscillation. Considering one complete oscillation … (f) Which graph shows the block’s velocity as a function of time? _________ (g) Which graph shows the block’s acceleration as a function of time? ________ (h) Which graph shows the block’s kinetic energy as a function of time? ________ (i) Which graph shows the block’s potential energy as a function of position? _______ (j) Which graph shows the magnitude of the net force acting on the block as a function of position? (Neglect the initial push.) _______