1. 1 Class #7 Review previous class Kinetic Energy and moment of inertia Angular Momentum And moment of inertia And torque And Central force Moment of Inertia Difference between it and CM Worked examples HW discussion

2. 2 Last time Showed that for a multi-particle or extended body there is a special point called the center of mass such that: Where:.

3. 3 Analogies between Linear and Angular motion equations

4. 4 Imagine calculating kinetic energy for 3 masses rotating on a massless wheel The crux or insight is that Why is

5. 5 Relating Angular Momentum and Torque Differentiate both sides Product rule for cross product Eliminate one term Use Newton’s 2 nd law

6. 6 Angular Momentum and Central Forces Kepler’s 2 nd law: Planetary orbits sweep out equal areas in equal times.

7. 7 Derivation of moment of inertia from angular momentum Completely general For object perpendicular to rotation axis

8. 8 Moment of Inertia vs. Center of Mass For a multi-particle discrete mass-distribution For a continuous mass-distribution.

9. 9 Parallel Axis Theorem CM Axis 1 (through CM) Axis 2 (Parallel to axis 1) a

10. 10 Derivation of parallel axis theorem O’ O Crux of derivation is that 2 nd term on right=0. BUT WHY?

11. 11 Moment of Inertia worked problem I The solid block of mass M shown rotates about y axis. a) What is its kinetic energy of rotation about its CM? b) Answer “a” if its axis of rotation is moved to one edge parallel to the y axis (as indicated)

12. 12 Moment of Inertia worked problem II   Calculate the moment of inertia and kinetic energy of a wire of uniform mass-density lambda, mass M,  and length L. A) If rotated about axis at midpoint at angular velocity   B) If rotated about axis at endpoint at angular velocity   Solve “B” using parallel axis theorem. L

13. 13 Parallel axis thm worked problem A physicist owns a factory making spherical chocolate Easter eggs with marshmallow centers. The Easter egg has radius “R” and the marshmallow center has radius “R/3”. You may assume for simplicity that the “UltraFluff” marshmallow filling used is massless. a) What is the moment of inertia of a marshmallow filled Easter egg (about and axis through its center of mass)? b) If the machine malfunctions and offsets the marshmallow center by a distance of R/3 from the axis of rotation through the original center of the object, what is the new moment of inertia of the entire defective Easter egg about the center of the sphere?