Unit 5.3 Rotational Dynamics A skater on ice begins to spin in the camel position, as shown in the figure to the left below. As she continues to spin, she straightens up, pulls her arms to her chest, and crosses her legs to spin in the corkscrew position, as shown in the figure to the right below. Air resistance and friction are negligible as the skater spins. Camel Corkscrew Position Position 19. Which of the following statements about the skater's motion is true as she moves from the camel position to the corkscrew position? (A) Her rotational kinetic energy increases because she does work on her arms and legs to bring them inward. (B) Her rotational kinetic energy increases because her potential energy decreases. (C) Her rotational kinetic energy decreases because she does work on her arms and legs to bring them inward. (D) Her rotational kinetic energy decreases because her potential energy increases. (E) Her rotational kinetic energy remains the same because there is negligible frictional torque. Page 12 of 15 AP Physics C: Mechanics AP Scoring Guide Unit 5.3 Rotational Dynamics h A sphere of mass M, radius r, and rotational inertia / is released from rest at the top of an inclined plane of height h as shown above. 20. If the plane has friction so that the sphere rolls without slipping, what is the speed Vem of the center of mass at the bottom of the incline? (A) v2gh (B) 2Mgh (C) 2Mghr (D) 2Mohr 2M ghar (E) I+ MPAP Collegeblamed Scoring Guide Unit 5.3 Rotational Dynamics 12. Ring Disk - (behind ring) The ring and the disk shown above have identical masses, radii, and velocities, and are not attached to each other. If the ring and the disk each roll without slipping up an inclined plane, how will the distances that they move up the plane before coming to rest compare? (A) The ring will move farther than will the disk. (B) The disk will move farther than will the ring. (C) The ring and the disk will move equal distances. (D) The relative distances depend on the angle of elevation of the plane. (E) The relative distances depend on the length of the plane. 13. y 4m X N/ - m U 2R R Two uniform spheres A and Y roll without slipping along a horizontal surface, as shown above. Sphere Y has 2 times the radius, 4 times the mass, and - the translational speed of sphere X. The rotational inertia of a sphere is MR'. If Ky is the kinetic energy of sphere Y" , and Ky in the kinetic energy of X , what is the ratio ? (A) 32 (B) 16 (C) 8 (D) 4 (E) 1 AP Physics C: Mechanics Page 9 of 15Scoring Guide Unit 5.3 Rotational Dynamics 21. C d -a P A spool is made of two circular disks connected to the ends of a cylinder so that the centers of the disks and the cylinder are aligned. In the figure above, the cylinder is shown as the dashed circle, and the disk on one end of the cylinder is shown as the larger solid circle. The spool is at rest on a level horizontal surface. A string attached to the cylinder is pulled gently so that the spool can rotate without slipping on the surface at point P. A force F is pulled in the direction of arrow d, and the spool spins but does not roll across the table. If the string is pulled with a force of 2F, which of the directions indicated, if any, indicates a pull on the string that will cause the spool to spin but not roll across the table? (A) a (B) b (C) c D) d (E) The spool will roll on the table for all four directions shown. All of the following are vector quantities EXCEPT (A) rotational kinetic energy (B) torque (C) angular momentum (D) angular velocity 6 .5 (E) centripetal acceleration 2 71 R I 23. A wheel of radius 15 cm has a rotational inertia of 2.3 kg . my. The wheel is spinning at a rate of 6.5 revolutions per second. A frictional force is applied tangentially to the wheel to bring it to a stop. The work done by the torque to stop the wheel is most nearly = ? IW 2 Page 14 of 15 AP Physics C: Mechanics = 32. 3 ( 15 ) Scoring Guide Unit 5.3 Rotational Dynamics (A) zero (B) -50 J -100 J (D) -1920 J -3840 J 24. A wheel with rotational inertia 0.04 kgem" and radius 0.02 m is turning at the rate of 10 revolutions per second when a frictional torque is applied to stop it. How much work is done by the torque in stopping the wheel? (A) -0.0008 J (B) -0.4x J (C) -2J (D) -2x2J (E) -82J10. Fmotor (N) 4 t (s) 2 6 8 10 -10 A motor drives a shaft of radius R/8 that is attached to the center of a wheel of radius R. The motor is turned off, and the force that the motor exerts on the shaft, Fmotor, varies with time, as shown. There is also a constant friction force of 0.4 N applied to the rim of the wheel in the opposite direction of the motion. During which time interval does the rotational kinetic energy increase and then decrease? (A) From t = 0 stot = 1s (B) From t = 1 stot = 3 s (C) From t = 3 stot = 5s (D) From t = 5 stot = 7 s (E) From t = 7 stot = 8 s 11. In an experiment, students roll several hoops down the same incline plane. Each hoop has the same mass but a different radius. Each hoop rolls down the incline without slipping. Which of the following graphs best shows the linear speed ' of the hoops at the bottom of the incline as a function of the radius r of the hoop? AP Physics C: Mechanics Page 7 of 15 Scoring Guide Unit 5.3 Rotational Dynamics (A V (B) O (C) (D) O (E)Unit 5.3 Rotational Dynamics The following questions-are related to this scenario: TR = ' mg = = Ma 2nny = a a = 29 A solid cylinder of mass m and radius R has a string wound around it. A person holding the string pulls it vertically upward, as shown above, such that the cylinder is suspended in midair for a brief time interval At and its center of mass does not move. The tension in the string is 7, and the rotational inertia of the cylinder about its axis is - me2 . 14. The linear acceleration of the person's hand during the time interval At is (A) T-mg m (B) 2g (C) (D) I (E) zero 15. A wheel of mass and radius rolls on a level surface without slipping. If the angular velocity of the wheel is w, what is its linear momentum? (A) MWR (B) MWR P = AMV (C) MWR2 (D) MaPR (E) Zero P = AM WL Page 10 of 15 AP Physics C: Mechanics Scoring Guide Unit 5.3 Rotational Dynamics 16. 0 - Four round objects of equal mass and radius roll without slipping along a horizontal surface that then bends upward and backward into an arc of half a circle. The objects all have the same linear speed initially. The objects are a hollow cylinder, a solid cylinder, a solid sphere, and a hollow sphere. The objects to go up the are and exit the are going in the opposite direction they entered without falling off the arc. Now, several trials are run for each object. For each trial, the initial speed of the object is reduced until the object does not make it through the full arc. The speed needed for each object to just make it through the are is recorded. Which of the following correctly lists the objects in order from fastest to slowest speed needed to make it through the are? (A) Hollow cylinder, hollow sphere, solid cylinder, solid sphere B) Hollow cylinder, solid sphere, solid cylinder, hollow sphere (C) Solid sphere, solid cylinder, hollow sphere, hollow cylinder (D) Solid sphere, hollow sphere, solid cylinder, hollow cylinder (E) Hollow sphere, solid cylinder, hollow cylinder, solid sphere 17. A solid sphere, a hollow sphere, and a ring all have the same mass and radius. Each is released from rest at the top of the same incline and rolls without slipping down the incline. Which of the following indicates the object that will reach the bottom of the incline first and provides correct reasoning? (A) The ring, because it has the greatest rotational inertia. B) The ring, because it has the smallest rotational inertia. C) The hollow sphere, because it has the greatest rotational inertia. D) The solid sphere, because it has the greatest rotational inertia. (E) The solid sphere, because it has the smallest rotational inertia. 18. A hoop of mass m and radius / rolls with constant speed on a horizontal surface without slipping. What is the hoop's translational kinetic energy divided by its rotational kinetic energy' (A) 4 (B) 2 (C) 1 (D) 1/2 (E) 1/4