1 Class #6 Center of Mass Defined Relation to momentum Worked problems DVD Demonstration on momentum cons. and CM motion Angular Momentum And moment of inertia And torque And Central force Moment of Inertia Difference between it and CM Worked examples

2 Review of last class Impulse is a useful concept in the study of collisions. (e.g. Balls and bats, automobiles, comets and planets) Impulse is the average force acting over a time period multiplied by the time period. It may also be written as an integral Note the difference between impulse and work W.

3 Center of Mass Center of Mass and Center of gravity happen to be equivalent For a multi-particle discrete mass-distribution For a continuous mass-distribution.

4 Linear Momentum and CM

5 Worked Example L6-1 – Discrete masses Given m 1 to m 10 m  = m m  = 3m y x y x O1O1 O2O2 1 unit 2 units Calculate Given origin O1O1 For homework given origin O 1 and O 2

6 Worked Example L6-2 – Continuous mass Given quarter disk with uniform mass-density  and radius R: Calculate M total Write r in polar coords Write out double integral, in r and phi Solve integral  r O1O1 R Calculate Given origin O1O1 REPEAT for 3/4 disk

7 Analogies between Linear and Angular motion equations

8 Deriving Rotational Form of Newton’s 2 nd Law

9 Angular Momentum and Central Forces

10 Derivation of moment of inertia and “B” Completely general For object perpendicular to rotation axis

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

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

13 Derivation of parallel axis theorem O’ O

14   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   L Worked Example L6-3 – Moment of Inertia