AP PHYSICS C: MECHANICS Scoring Guide Unit 5 Progress Check: MCQ 1. A sphere rotates about an axis passing through its center. The angular velocity, , of the sphere in by the following equations, where is the time in seconds. What is the angular displacement of the sphere at time (A) radian (B) radians (C) radians (D) radians is given seconds? Answer D Correct. The angular displacement is found by integrating the angular velocity. For the time interval , the displacement is simply integrate or to get , which evaluates to displacements for the two intervals gives 2. . For the time interval , . Adding the radians. A disk of radius rotates about an axis passing through its center and perpendicular to the plane of the disk. The is positive, and the disk is given a constant, negative angular acceleration . Which of initial angular speed the following expressions correctly represents the magnitude of the net linear acceleration for a point located at the edge of the disk? (A) (B) (C) (D) Answer D Correct. First, the centripetal acceleration can be found by combining the expressions to get the angular acceleration by and . Next, the tangential acceleration has a magnitude related to . Finally, the total linear acceleration is found by using the AP Physics C: Mechanics Page 1 of 16 Scoring Guide Unit 5 Progress Check: MCQ Pythagorean theorem with the centripetal and tangential acceleration components: . 3. Three disks are attached together and can rotate together about an axis through their common center, as shown in meter, meter, and meters, and each disk has a force of newtons the figure. The disks have radii of exerted at the locations and in the directions shown. What is the magnitude of the net torque about the center of the three-disk system? (A) (B) (C) (D) Answer B Correct. The inner and outer disks have counterclockwise torques exerted on them, while the middle disk has a clockwise torque that must be subtracted to get the net torque. Taking the counterclockwise direction as positive, the torques exerted on the disks are for the small disk, for the middle disk, and Page 2 of 16 AP Physics C: Mechanics for the large disk. Scoring Guide Unit 5 Progress Check: MCQ The net torque is the sum of these, or . 4. The uniform rod shown in the figure has length and mass . The rod’s rotational inertia is about its about Point , which is located a distance from Point . What is the distance ? center at Point and (A) (B) (C) (D) Answer B Correct. Using the parallel-axis theorem, . AP Physics C: Mechanics Page 3 of 16 Scoring Guide Unit 5 Progress Check: MCQ 5. A uniform rod of mass and length is supported on a vertical wall at its left end by a pivot and at its center by is placed on the rod a distance from the rod’s right a string, as shown in the figure. A small block of mass end. What is the magnitude of the force that the pivot exerts on the rod? (A) (B) (C) (D) Answer C Correct. Both the string tension and the force from the pivot are unknown. The vertical or y-component of the tension is found from applying Newton’s first law in rotational form about the pivot: . Because the string is at a 45-degree angle, the horizontal or x-component of tension is the same: . The y-component of the force from the pivot is found from applying Newton’s second law in the y-direction: . The x-component of the force from the pivot is found from applying Newton’s first law in the x- Page 4 of 16 AP Physics C: Mechanics Scoring Guide Unit 5 Progress Check: MCQ direction: . The magnitude of the force from the pivot is found by vector addition of the two components via the Pythagorean theorem: . 6. A pulley with a rotational inertia of and radius can rotate with negligible friction in its axle. A block is attached to a lightweight cord that passes over the pulley, while the other end of the cord is pulled with a constant force of magnitude , as shown in the figure. The string does not slip relative to the pulley. The block on the cord as it accelerates downward, and the pulley has a clockwise angular acceleration exerts a force of . What is ? (A) (B) (C) (D) Answer B Correct. The net torque exerted on the pulley must be . This means that the tensions in the vertical and horizontal sections of the cord differ by an amount , which is AP Physics C: Mechanics Page 5 of 16 Scoring Guide Unit 5 Progress Check: MCQ . Since the horizontal cord section has tension in the vertical section, is a tension that is less than the . 7. Points and are located on a disk that rotates about its center, as shown in the figure. The disk has an angular velocity and an angular acceleration that are both in the counterclockwise direction. How do the magnitudes of the centripetal and tangential accelerations of the two points compare to each other? Tangential acceleration Centripetal acceleration A Greater for Point Greater for Point B Greater for Point Greater for Point C Greater for Point Greater for Point D Greater for Point Greater for Point (A) A (B) B (C) C (D) D Answer D Correct. For a point located a distance from the axis of rotation, the tangential acceleration and centripetal acceleration are both proportional to and therefore larger for Point . Note that the angular velocity and angular acceleration are the same for all points on the disk. Page 6 of 16 AP Physics C: Mechanics Scoring Guide Unit 5 Progress Check: MCQ 8. Four forces with the same magnitude are exerted on a uniform, regular hexagon as shown in the figure. Which of the labeled forces exerts the greatest torque about the center of the hexagon? (A) (B) (C) (D) Answer A Correct. This force acts with the longest lever arm, since the angle between the force and the position degrees. vector (from center to force location) is AP Physics C: Mechanics Page 7 of 16 Scoring Guide Unit 5 Progress Check: MCQ 9. Two identical cylinders of mass are attached to a rotating apparatus at equal distances from the rotation axis, as shown in the figure. The rotational inertia of the apparatus-cylinders system is experimentally determined for multiple values of . The solid line in the graph shows the best-fit line for as a function of for the identical cylinders with mass . The experiment is then repeated for two other identical cylinders of unknown mass. The dashed line in the graph shows the best-fit line for as a function of for the cylinders of unknown mass. Based ? on the graph, what is the unknown mass in terms of (A) (B) (C) (D) Answer C Correct. For cylinders of mass the rotational inertia of the system is , and the cylinder mass is proportional to the slope of the versus graph. The slope of the graph for the unknown mass is twice that of the graph for mass , so the unknown mass is . Page 8 of 16 AP Physics C: Mechanics Scoring Guide Unit 5 Progress Check: MCQ 10. A rigid rod of negligible mass is free to rotate on a pivot, as shown in the figures. In experimental Trial a uniform sphere of mass is attached to the rod with its center a distance to the left of the pivot, and a second is attached to the rod with its center a distance to the right of the pivot. The radii uniform sphere of mass of the spheres are negligible compared to . The rod is held horizontally and then released. In Trial the spheres are switched as shown, and the rod is again held horizontally and released. Compare the rotational motion of the rod-spheres system in the two trials. Trial 1 Trial 2 A Remains at rest Remains at rest B Remains at rest Begins to rotate C Begins to rotate Remains at rest D Begins to rotate Begins to rotate (A) (B) (C) (D) AP Physics C: Mechanics Page 9 of 16 Scoring Guide Unit 5 Progress Check: MCQ Answer B Correct. In Trial the net torque is zero, since each sphere’s weight results in the same torque of exerted in opposite directions. In Trial the net torque is not zero (and is in fact clockwise), since the larger sphere results in a torque of , which is greater than the torque of from the smaller sphere. 11. A force with magnitude is applied to the right end of a uniform rod of mass and length that lies on top of a horizontal surface, as shown in Figure . The rod is free to rotate with negligible friction about an axis through its , and center and perpendicular to the plane of the surface. The rotational inertia of the rod about the axis is the rod’s initial angular acceleration is . In a second scenario, the same force is applied to the same rod at the same location, but the axis of rotation is now a distance from the left end of the rod, as shown in Figure . This ? time the rod’s initial angular acceleration is . What is the ratio (A) (B) (C) (D) Page 10 of 16 AP Physics C: Mechanics Scoring Guide Unit 5 Progress Check: MCQ Answer B Correct. The angular acceleration is given by the rotational analog of Newton’s second law, torque changes from factor in Figure in Figure to . The in Figure . This is an increase by a . Using the parallel-axis theorem, the rotational inertia is found to change from to , in Figure . This is an increase by a factor of . Since the torque and rotational inertia change by the same factor, the angular acceleration does not change. 12. A disk of radius moves with negligible friction along a horizontal surface with a translational velocity as it rotates with an angular velocity . Which of the following statements must be true regarding the angular velocity of the disk? (A) (B) (C) (D) A relationship cannot be determined for in terms of and . Answer D Correct. Since friction is negligible, the disk can slip on the surface as it moves and rotates. The angular AP Physics C: Mechanics Page 11 of 16 Scoring Guide Unit 5 Progress Check: MCQ and translational velocities of the wheel have no clear relationship without additional information. 13. Three solid objects of equal mass, , , and , each hang from a pivot at the top end of the object, as shown in the figure. Object has its center of mass ( ) located midway between the top and bottom ends. Object has its located closer to the bottom end. Object is identical to Object , except that it is hung upside-down and is closer to the top end. Which object(s) has (have) the greatest rotational inertia about the pivot? so its (A) Object only (B) Object only (C) Object only (D) Objects and Answer B Correct. The mass of Object is distributed farther from the pivot point than it is for the other two objects. , the disk starts to spin about an axis through its center. The disk spins with a A disk is initially at rest. At time constant angular acceleration, and after second the disk has an angular displacement of radian. 14. The disk continues spinning with the same angular acceleration. At what time does the disk have an angular displacement of radians? (A) seconds (B) seconds (C) seconds (D) seconds Page 12 of 16 AP Physics C: Mechanics Scoring Guide Unit 5 Progress Check: MCQ Answer A . In order for the angle to increase Correct. This result can be determined from the equation by a factor of , the time must increase by a factor of . 15. Now the disk is given an initial angular velocity at time angular acceleration . The disk rotates through an angle the following is true for a disk with an initial angular velocity (A) The disk stops in a time greater than (B) The disk stops in a time less than . This time, the disk decelerates with a constant in a time before coming to a stop. Which of and constant angular acceleration ? . . (C) The disk rotates through an angle greater than (D) The disk rotates through an angle less than before stopping. before stopping. Answer C Correct. This result can be determined from the equation , which for the original disk becomes . and results in a larger angular displacement. Alternatively, From this we see that doubling both both disks must come to a stop in the same time , with the second disk spinning faster . Since the second disk spins faster during the same amount of time, it rotates for all times through a greater angle. AP Physics C: Mechanics Page 13 of 16 Scoring Guide Unit 5 Progress Check: MCQ 16. A bar of length and negligible mass is free to rotate about a pivot located a distance from its right end, as shown in the figure. Two applied forces, and , are exerted on the bar at the locations and in the directions shown. Which force results in a torque of greater magnitude about the pivot, and what reasoning supports this claim? (A) , because the lever arm is zero for . (B) , because the product (C) , because is greater than (D) , because is directed along the length of the rod. is greater than . . Answer A Correct. The line of action for both zero. Page 14 of 16 AP Physics C: Mechanics goes through the pivot point, so the lever arm and torque for are Scoring Guide Unit 5 Progress Check: MCQ 17. A beam of uniform mass density can rotate freely about a support pivot located under its midpoint, as shown in the figure. Three non-zero forces, , , and , are exerted downward on the beam. Forces and are applied at is applied somewhere between the pivot and . When all forces are applied either end of the beam, and force the beam stays in rotational equilibrium. Which of the following claims about the force magnitudes must be correct, and what reasoning supports the claim? (A) (B) , because has a longer lever arm than torque of equal magnitude. , because , so it must be smaller in order to contribute a contributes a torque to counteract the torques from both and the weight of the beam. (C) , because the net force on each side of the pivot must be equal for the beam to be in rotational equilibrium. (D) , because and same direction as . are the same distance from the pivot, and contributes a torque in the Answer D and contributing torques in the same Correct. For the beam to be in rotational equilibrium with direction, the torques from and must each be less than that from . Since and are equidistant from the pivot, must be less than . AP Physics C: Mechanics Page 15 of 16 Scoring Guide Unit 5 Progress Check: MCQ 18. Two pulleys of different radii are rigidly attached together at their centers, as shown in the figure. The smaller pulley, of radius , has a lightweight string wrapped around it. A force of pulls downward on the left pulls downward on the right section of string. The larger pulley, of radius section of string, and a force of , also has a lightweight string wrapped around it. A single force of pulls downward on this string. Which of the following gives the correct direction of the two-pulley system’s angular acceleration, and provides appropriate supporting evidence? (A) Clockwise. The sum of the torques from the two (B) Clockwise. The sum of the lever arms for the two force. (C) Counterclockwise. The torque from the forces. (D) Counterclockwise. The forces is greater than torque from the force. forces is greater than the lever arm for the force is greater than each of the torques from the force is greater than the sum of the two forces. Answer A Correct. The angular acceleration, and therefore the direction of rotation after being at rest, is in the same direction as the net torque. Net torque is found by adding the torques in one direction, and subtracting the torques in the opposite direction. Taking the clockwise direction as positive, the two 4-newton forces . The -newton force provides a torque give a torque of of . The net torque is then , which is in the clockwise direction. Page 16 of 16 AP Physics C: Mechanics
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