Torque & Rotational Inertia Lecturer: Professor Stephen T. Thornton

1. Torque & Rotational Inertia Lecturer: Professor Stephen T. Thornton

2. Reading Quiz A)F1 B) F3 C) F4 D) all of them E) none of them In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? For all cases the magnitude of the applied force is the same.

3. Reading Quiz A)F1 B) F3 C) F4 D) all of them E) none of them In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? For all cases the magnitude of the applied force is the same. The torque is t=F d sin q,and so the force that is at 90° to the lever arm is the one that will have the largest torque. Clearly, to close the door, you want to push perpendicularly!!

4. Last Time Began angular motion Angular position, displacement Angular speed, velocity Angular acceleration Similarities between translation and rotation

5. Today Torque Rotational inertia (moment of inertia) Rotational kinetic energy

6. What is torque? We recognize there is a relationship between tangential force and making something rotate.

7. Only the Tangential Component of a Force Causes a Torque

8. The Moment Arm moment arm

9. Sign convention for torque according to most textbooks: > 0 if the torque causes a CCW acceleration. < 0 if the torque causes a CW acceleration.

10. Conceptual Quiz:You are using a wrench to loosen a rusty nut. Which of the arrangements below is least effective in loosening the nut? Force is proportional to length of vector. A. B. C. D. E. not possible to determine B A C D

11. Answer: C The force vectors are all the same. The arrangement that is the least effective is the one with the shortest moment arm. That is C.

12. Conceptual Quiz:A mechanic is finding it very difficult to muster enough torque to twist a stubborn bolt with a wrench, and she wishes she had a length of pipe to place over the wrench handle to increase her leverage. Will torque be increased if the mechanic pulls just as hard on a length of rope tied to the wrench handle? A) Yes B) No C) Only in space. D) Not enough information given.

13. Answer: B (no) The rope placed in this position neither increases the force or the moment arm (length of application of the force causing the torque).

14. Angular Quantities If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration:

15. Torque and Angular Acceleration

16. Torque and angular acceleration

17. Linear and angular quantities LinearAngular m  I a  F  

18. Similarities between linear and angular motion quantities ***

19. Look at system of particles

20. Kinetic Energy of a Rotating Object massless rod

21. Kinetic Energy of a Rotating Object of Arbitrary Shape

22. Rotational InertiaMoment of Inertia Rotational kinetic energy I appears to be quite useful!!

23. The Rotational Inertia (Moment of Inertia) of a Hoop M

24. The Rotational Inertia (Moment of Inertia) of a Disk This is almost certainly an example in textbook. Use calculus to find this value.

25. Rotational Dynamics; Torque and Rotational Inertia The quantity is called the rotationalinertia of an object. The distribution of mass matters here—these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.

26. Rotational Inertia for Uniform, Rigid Objects of Various Shapes and Total Mass M Do not memorize!!

27. Rotational Inertia for Uniform, Rigid Objects of Various Shapes and Total Mass M

28. Demos: Rotational inertia rods Moment of Inertia wheel

29. If a physical object is available, the rotational inertia (moment of inertia) can be measured experimentally. Otherwise, if the object can be considered to be a continuous distribution of mass, the rotational inertia may be calculated:

30. The parallel-axis theorem gives the rotational inertia about any axis parallel to an axis that goes through the center of mass of an object:

31. Falling Rod. A thin rod of length stands vertically on a table. The rod begins to fall, but its lower end does not slide. (a) Determine the angular velocity of the rod as a function of the angle it makes with the tabletop. (b) What is the speed of the tip of the rod just before it strikes the table?

32. Conceptual Quiz:A figure skater spins around with her arms extended. When she pulls in her arms, her rotational inertia A) increases. B) decreases. C) stays the same.

33. Answer: B, decreases The mass stays the same, but the radius decreases for the mass in her arms. The I must decrease.

34. hollow solid same mass & radius Conceptual Quiz Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the bigger rotational inertia about an axis through its center? A) solid aluminum B) hollow gold C) same

35. hollow solid same mass & radius Conceptual Quiz Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the bigger rotational inertia about an axis through its center? A) solid aluminum B) hollow gold C) same Rotational inertia depends on mass and distance from axis squared. It is bigger for the shell because its mass is located farther from the center.

36. Conceptual Quiz:Two wheels with fixed hubs, each having a mass of 1 kg, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is I = mR2. In order to impart identical angular accelerations, how large must F2 be? A) 0.25 N B) 0.5 N C) 1.0 N D) 2.0 N E) 4.0 N

37. Answer: D The hint on the figure should help. You want Fr/I to be the same ratio. Fr/mr2 = F/mr, so F/r must have the same ratio = 2.