Question

# A child pushes her friend (m = 25 kg) located at a radius r = 1.5...

A child pushes her friend (m = 25 kg) located at a radius r = 1.5 m on a merry-go-round (rmgr = 2.0 m, Imgr = 1000 kg*m2) with a constant force F = 90 N applied tangentially to the edge of the merry-go-round (i.e., the force is perpendicular to the radius). The merry-go-round resists spinning with a frictional force of f = 10 N acting at a radius of 1 m and a frictional torque τ = 15 N*m acting at the axle of the merry-go-round, and the merry-go-round is initially at rest.  (Hint: Watch the following videos: "Session 9/Lecture/Rotational Motion Example Problems/Net Torque Example Problem 3")

1. What is the magnitude of the torque due to the 90 N force at 2 m?
2. In what direction is the torque due to the 90 N force at 2 m?
3. What is the magnitude of the torque due to the 10 N force at 1 m?
4. In what direction is the torque due to the 10 N force at 1 m?
5. What is the magnitude of the torque due to the friction at the merry-go-round axle?
6. In what direction is the torque due to the friction at the merry-go-round axle?
7. (1A) What is the magnitude of the net torque acting on the merry-go-round about its axle?
8. What is the moment of inertia of merry-go-round apparatus itself?
9. What is the moment of inertia of the child on the merry-go-round?
10. (1B) What is moment of inertia of the merry-go-round with the child on it?
11. Is the merry-go-round speeding up or slowing down?
12. (1C) What is the magnitude of the rotational acceleration of the merry-go-round?
13. In what direction is the merry-go-round accelerating?
14. (1D) If the child pushes for 15 s, what is the magnitude of rotational velocity of the merry-go-round at 15 s?
15. If the child stops pushing her friend on the merry-go-round with the 90 N force, will the merry-go-round speed up or slow down?
16. If the child stops pushing her friend on the merry-go-round with the 90 N force, what is the new net torque acting on the merry-go-round?
17. If the child stops pushing her friend on the merry-go-round with the 90 N force, in what direction is the new net torque acting on the merry-go-round?
18. If the child stops pushing her friend on the merry-go-round with the 90 N force, what is the magnitude of the deceleration of the merry-go-round?
19. (1E) Once the child stops pushing at 15 s, how long does it take for the merry-go-round to come to a stop?

A child (m = 25 kg) is riding on a frictionless merry-go-round at 1.5 m from the center axle. The merry-go-round (Imgr = 1000 kg*m2) is traveling at an angular velocity of 2 rad/s. (Hint: Watch the following videos: "Session 9/Lecture/Rotational Motion Example Problems/HW Problem 1 F-H Hints")

20. What is the moment of inertia of the merry-go-round with the child on it?
21. What is the angular momentum of the merry-go-round with the child on it?
22. If the child moves from 1.5 m from the center to 2.0 m from the center, what does the moment of inertia of the merry-go-round with the child on it become?
23. If the child moves from 1.5 m from the center to 2.0 m from the center, is angular momentum conserved?
24. If the child moves from 1.5 m from the center to 2.0 m from the center, does the angular momentum of the merry-go-round change?
25. If the child moves from 1.5 m from the center to 2.0 m from the center, does the angular velocity of the merry-go-round change?
26. (1F) If the merry-go-round were traveling at a rotational velocity of 2 rad/s and the child riding on the merry-go-round moves from a radius of 1.5 m outward to a radius of 2.0 m in 1 second, what is the magnitude of the rotational velocity of the merry-go-round immediately after the child moves?
27. (1G) In the process of the child moving outward, what is the magnitude of the rotational acceleration of the merry-go-round?

Question for thought: In the process of the child moving outward, is there a torque accelerating the system? If yes, what is the source of the accelerating torque?

A child (m = 25 kg) walks along a 4 m plank of uniform density (m = 8 kg) that is laying with one end extending 1 m off the end of a dock (i.e., 3 m of the plank are on the dock, and 1 m of the plank extends off the dock).  (Hint: Watch the following videos: "Session 9/Lecture/Lesson on Rotational Motion/Center of Mass" and "Session 9/Lecture/Rotational Motion Example Problems/Net Torque Example Problem 1" and "Session 9/Lecture/Rotational Motion Example Problems/Net Torque Example Problem 2")

28. (2A) How far (in m) is the center of mass of the plank from the left end of the plank?
29. How far (in m) is the center of mass of the plank from the end of the dock?
30. (2B) What is the magnitude of the torque of the plank about the edge of the dock (i.e. torque due to plank only, not including the boy)?
31. In what rotational direction is the torque about the edge of the dock due to the mass of the plank acting (i.e. the torque due to the plank only, not including the boy)?
32. As the boy walks out on the plank past the edge of the dock, in what rotational direction is the torque about the edge of the dock due to the boy acting?
33. At the maximum distance the boy can walk out on the plank, will the magnitude of the torque due to the boy be less than, equal to, or greater than the magnitude of the torque due to the plank only?
34. (2C) How far from the edge of the dock can the boy walk on the plank before the plank starts tipping into water?

Question for thought: How could the boy test this conclusion without taking the risk of falling in the water?

A person sits on a frictionless stool that is free to rotate but is initially at rest. The person is holding a bicycle wheel (I = 3 kg*m2) that is rotating at 8 rev/s in the clockwise direction as viewed from above, and the moment of inertia of the person-wheel-stool system is 9 kg*m2. For this problem, all answers involving a rotational component will be expressed in revolutions rather than radians. (Hint: Watch the following videos: "Session 9/Lecture/Rotational Motion Example Problems/Conservation of Angular Momentum HW Problem Hints")

35. What is the magnitude of the angular momentum of the rotating wheel?
36. What is the direction of the angular momentum of the rotating wheel?
37. What is the magnitude of the angular velocity of the PWS?
38. What is the magnitude of the angular momentum of the PWS?
39. What is the direction of the angular momentum of the PWS?
40. (3A) What is magnitude of the angular momentum of the system?
41. (3B) What is direction of the angular momentum of the system?
42. The person flips the rotating wheel over 180 degrees. What is the magnitude of the angular momentum of the rotating wheel?
43. After the person flips the rotating wheel over 180 degrees, what is the direction of the angular momentum of the rotating wheel?
44. After the person flips the rotating wheel over 180 degrees, what is the magnitude of the angular momentum of the system?
45. After the person flips the rotating wheel over 180 degrees, what is the direction of the angular momentum of the system?
46. (3C) The person flips the rotating wheel over 180 degrees, taking 0.5 seconds to complete this action. What is the magnitude of the angular velocity of the person-wheel-stool system about the axis of the stool after the wheel is flipped?
47. (3D) What is the direction of the angular velocity of the person-wheel-stool system about the axis of the stool after the wheel is flipped?
48. (3E) What is the magnitude of the angular acceleration of the person-wheel-stool system?

Question for thought: (3F) What is the source of the torque that accelerates the student-wheel-stool system?

Concept Question

49. T/F: The angular momentum of a system only changes with the application of an external net torque; otherwise the angular momentum of a system will never change.

1. The torque due to a force F acting tangentially at a distance r from the center is given by,

Lets say that the applied force is rotating the merry go round in counter clockwise direction and the torque is positive.

(a) Substituting values we get,

(b) The direction of torque vector is vertically upwards (counter clockwise motion) hence it is positive.

(c) Substituting values we get,

(d) Here this is friction hence the torque vector is vertically downwards (friction tries to rotate in clockwise motion) hence it is negative.