1. Physics 1A, Section 2 November 15, 2010

2. Translation / Rotation translational motionrotational motion position x angular position  velocity v = dx/dt angular velocity  = d  /dt acceleration a = dv/dt = d 2 x/dt 2 angular acceleration  = d  /dt = d 2  /dt 2 mass mmoment of inertia I momentum p = mv angular momentum L = I  force F = ma torque  = I  kinetic energy ½ mv 2 kinetic energy ½ I  2

3. Moment of Inertia Moments:  0 th moment: ∫dm = mass  useful 1 st moment: ∫ r dm (1/M) ∫ r dm is the center of mass  useful 2 nd moment: ∫R 2 dm = moment of inertia  R is the distance from the axis of rotation  Other ways to write the integral: ∫dm = ∫  dV = ∫∫∫  dx dy dz Moment of inertia examples: – Frautschi et al. Table 14.1, page 379 Parallel axis theorem: I = I CM + Md 2

4. Quiz Problem 41

5. Answer: a)  1 = 8  da 4  2 = ½  da 4  1 /  2 = 16 b)  f = 16  /17 c) No, some lost to friction. K final /K initial = 16/17

6. Quiz Problem 24

7. Angular Momentum Angular momentum vector L is constant, in the absence of outside torques. If there are outside torques, dL/dt =  Single particle: L = r x p = m r x v r = distance (vector) from a fixed reference point System of particles: L = M tot r cm x v cm +  m j r´ j x v´ j orbital spin (about c.m.) r´ j, v´ j measured with respect to center of mass Rotating rigid body: L spin = I  (for principal axis) I =∫R 2 dm = moment of inertia R = distance from rotation axis

8. Quiz Problem 24

9. Answer: a) L = 2MRv b)  = 3MR 2 c)  = (2/3) v/R d) 1/3 is lost Quiz Problem 24

10. fluid mechanics Thursday, November 18: