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.

Work in Rotation § 10.3–10.4

K and the c.o.m. Frame For any system: K = 1/2 Mv 2 + K cm where M = total mass of the system =  m i K cm = system’s K in its center-of-mass frame of reference =  1/2 m i (v i - v) 2 v = velocity of the system’s center of mass =  m i v i /M

K of a Rolling Body K rot (edge axis) = 1/2 I ||  2 = 1/2 (MR 2 + I cm )  2 = 1/2 Mv 2 + 1/2 I cm  2 K trans + K rot (center of mass) = 1/2 Mv 2 + 1/2 I cm  2 R  v cm  = v cm /R

Group CPS Question  = d  /dt. What is the direction of d  ? A.  B.  C.  D.  E. F. R  v cm

Rotational Work W = F·ds ds F R dd W = F · (d   R)

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

Rotational Work dW = F·ds ds F R dd dW = F · (d   R) dW = (R  F) · d  dW =  ·d  dW = (d   R) · F W =   ·d  22 11

Power The rate of doing work dW/dt  ·d  /dt =  · 

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? A.The cylinder with the larger I. B.The cylinder with the smaller I. C.Both rotated through the same angle.   I 2I2I

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? A.The cylinder with the larger I. B.The cylinder with the smaller I. C.Both had the same amount of work done.   I 2I2I

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? A.The cylinder with the larger I. B.The cylinder with the smaller I. C.Both have the same K.   I 2I2I

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? A.The cylinder with the larger I. B.The cylinder with the smaller I. C.Both have the same .   I 2I2I

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? A.The cylinder with the larger I. B.The cylinder with the smaller I. C.Both took the same time.   I 2I2I

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? A.The cylinder with the larger I. B.The cylinder with the smaller I. C.Both received the same power.   I 2I2I

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