Worksheet Problem 1

1. Worksheet Problem 1 Rest an object on your separated horizontal index fingers. Slowly bring your fingers together. Where does the object end up? Explain the object’s final position and the path it followed to get there.

2. Work in Rotation § 10.3–10.4

3. v = velocity of the system’s center of mass = Smivi /M • Kcm = system’s K in its center-of-mass frame of reference = S1/2 mi (vi - v)2 K and the c.o.m. Frame • For any system: K = 1/2Mv2 + Kcm where • M = total mass of the system = Smi

4. K of a Rolling Body • Krot (edge axis) = 1/2 I||w2 = 1/2 (MR2 + Icm)w2 = 1/2 Mv2 + 1/2 Icmw2 • Ktrans + Krot (center of mass) = 1/2 Mv2 + 1/2 Icmw2 w w = vcm/R vcm R

5.  •  •  •  Group CPS Question w = dq/dt. What is the direction of dq? w vcm R

6. dq W = F · (dq R) Rotational Work W = F·ds ds F R

7. volume of parallelepiped defined by A, B, C Equivalent to • (B C)  A • (C A)  B C B A Aside: Scalar Triple Product • (A  B)  C

8. dq dW = F · (dq R) dW = (dq R)· F dW = (RF)· dq dW = t ·dq q2 W =  t ·dq q1 Rotational Work dW = F·ds ds F R

9. = t ·w • t ·dq/dt Power • The rate of doing work • dW/dt

10. t t I 2I CPS Question You apply equal torques to two different cylinders at rest, one of which has a moment of inertia twice as large as the other. After one complete rotation, which cylinder rotated the farthest? • The cylinder with the larger I. • The cylinder with the smaller I. • Both rotated through the same angle.

11. t t I 2I CPS Question You apply equal torques to two different cylinders at rest, one of which has a moment of inertia twice as large as the other. After one complete rotation, on which cylinder was the most work done? • The cylinder with the larger I. • The cylinder with the smaller I. • Both had the same amount of work done.

12. t t I 2I CPS Question You apply equal torques to two different cylinders at rest, one of which has a moment of inertia twice as large as the other. After one complete rotation, which cylinder has the greatest kinetic energy? • The cylinder with the larger I. • The cylinder with the smaller I. • Both have the same K.

13. t t I 2I CPS Question You apply equal torques to two different cylinders at rest, one of which has a moment of inertia twice as large as the other. After one complete rotation, which cylinder has the greatest angular speed? • The cylinder with the larger I. • The cylinder with the smaller I. • Both have the same w.

14. t t I 2I CPS Question You apply equal torques to two different cylinders at rest, one of which has a moment of inertia twice as large as the other. Which completes one rotation in the shortest time? • The cylinder with the larger I. • The cylinder with the smaller I. • Both took the same time.

15. t t I 2I CPS Question You apply equal torques to two different cylinders at rest, one of which has a moment of inertia twice as large as the other. To which cylinder was the greatest power applied? • The cylinder with the larger I. • The cylinder with the smaller I. • Both received the same power.

16. Worksheet Problem 2 A solid ball released from rest rolls down an incline with angle 65° below horizontal. r b • What must be the minimum coefficient of static friction between the ball and the slope for no slipping? • What is its total kinetic energy after 2 s if m =4 kg and r =0.10 m?