AP Physics: Rotational Dynamics 2

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1 Name: Assignment Due Date: March 30, 2012 AP Physics: Rotational Dynamics 2 Problem A solid cylinder with mass M, radius R, and rotational inertia 1 2 MR2 rolls without slipping down the inclined plane shown above. The cylinder starts from rest at a height H. The inclined plane makes an angle θ with the horizontal. Express all solutions in terms of M, R, H, θ, and g. 1. Determine the translational speed of the cylinder when it reaches the bottom of the inclined plane. 2. On the figure below, draw and label the forces acting on the cylinder as it rolls down the inclined plane. Your arrows should begin at the point of application of each force. 1

2 Name: 3. Show that the acceleration of the center of mass of the cylinder while it is rolling down the inclined plane is 2 3 g sinθ. 4. Determine the minimum coefficient of friction between the cylinder and the inclined plane that is required for the cylinder to roll without slipping. 5. The coefficient of friction µ is now made less than the value determined in question 11, so that the cylinder both rotates and slips. I. Indicate whether the translational speed of the cylinder at the bottom of the inclined plane is greater than, less than, or equal to the translational speed calculated in question 8. Justify your answer. II. Indicate whether the total kinetic energy of the cylinder at the bottom of the inclined plane is greater than, less than, or equal to the total kinetic energy for the previous case of rolling without slipping. Justify your answer. 2

3 Name: As shown above, a uniform disk is mounted to an axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached to the end of the rod. Properties of the disk, rod, and block are as follows. Disk: mass = 3m, radius = R, moment of inertia about center I D = 3 2 mr2 Rod: mass = m, length = 2R, moment of inertia about one end I R = 4 3 mr2 Block: mass = 2m The system is held in equilibrium with the rod at an angle θ 0 to the vertical, as shown above, by a horizontal string of negligible mass with one end attached to the disk and the other to a wall. Express your answers to the following in terms of m, R, θ 0, and g. 6. Determine the tension in the string. 3

4 Name: 7. The string is now cut, and the disk-rod-block system is free to rotate. Determine the following for the instant immediately after the string is cut. i. The magnitude of the angular acceleration of the disk 8. ii. The magnitude of the linear acceleration of the mass at the end of the rod As the disk rotates, the rod passes the horizontal position shown above. Determine the linear speed of the mass at the end of the rod for the instant the rod is in the horizontal position. 4

5 Name: A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r, as shown in Experiment A above. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The rotational inertia of the pole and the rod are negligible. 9. \Determine the rotational inertia of the rod-and-block apparatus attached to the top of the pole. 10. Determine the downward acceleration of the large block. 5

6 Name: 11. When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD? Check the appropriate space below. Greater than 4mgD Equal to 4mgD Less than 4mgD Justify your answer. 12. The system is now reset. The string is rewound around the pole to bring the large block back to its original location. The small blocks are detached from the rod and then suspended from each end of the rod, using strings of length P. The system is again released from rest so that as the large block descends and the apparatus rotates, the small blocks swing outward, as shown in Experiment B above. This time the downward acceleration of the block decreases with time after the system is released. When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare to that in part (c)? Check the appropriate space below. Justify your answer. Greater Equal Less 6

7 Name: A thin hoop of mass M, radius R, and rotational inertia MR 2 is released from rest from the top of the ramp of length L above. The ramp makes an angle θ with respect to a horizontal tabletop to which the ramp is fixed. The table is a height H above the floor. Assume that the hoop rolls without slipping down the ramp and across the table. Express all algebraic answers in terms of given quantities and fundamental constants. 13. Derive an expression for the acceleration of the center of mass of the hoop as it rolls down the ramp. 14. Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp. 7

8 Name: 15. Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor. 16. Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge of the table to where the disk lands on the floor compare with the distance determined in part (c) for the hoop? Briefly justify your response. Less than The same as Greater than 8

9 AP Physics: Rotational Dynamics 2 Answer Section PROBLEM 1. ANS: REF: Free Resp Ques 1997 Mech #3a 1

10 2. ANS: REF: Free Resp Ques 1997 Mech #3b 2

11 3. ANS: REF: Free Resp Ques 1997 Mech #3c 3

12 4. ANS: REF: Free Resp Ques 1997 Mech #3d 4

13 5. ANS: REF: Free Resp Ques 1997 Mech #3e 5

14 6. ANS: REF: Free Resp Ques 1999 Mech #3a 6

15 7. ANS: REF: Free Resp Ques 1999 Mech #3b 7

16 8. ANS: REF: Free Resp Ques 1999 Mech #3c 9. ANS: REF: Free Resp Ques 2001 Mech #3a 8

17 10. ANS: REF: Free Resp Ques 2001 Mech #3b 11. ANS: REF: Free Resp Ques 2001 Mech #3c 9

18 12. ANS: REF: Free Resp Ques 2001 Mech #3d 10

19 13. ANS: 11

20 REF: AP Physics Final 2006 Free Resp (Mech) #3a 14. ANS: REF: AP Physics Final 2006 Free Resp (Mech) #3b 12

21 15. ANS: REF: AP Physics Final 2006 Free Resp (Mech) #3c 16. ANS: REF: AP Physics Final 2006 Free Resp (Mech) #3d 13

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