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Angular 2

of a

Text sections 11.2 - 11.4

Practice Problems: Chapter 11, problems 11, 13, 15, 19, 53

Physics 1D03 Review Quiz Two astronauts are held together by a long rope and rotate about their common center of . One has twice the mass of other. One astronaut gathers in 1/3 of the rope separating them.

Which of the following remains constant? A) Kinetic B) Angular C) Angular momentum D) Tension in the rope

By what factor do each of the others change?

Physics 1D03 Angular momentum of a particle

L = r ×p = r ×(mv) z L This is the fundamental definition of L. y • L is a vector . O r v • Like , it depends on the choice x m φ of origin (or “pivot”).

• If the particle is all in the x-y , L is parallel to the z axis.

Physics 1D03 Angular momentum of a particle (2-D):

v r ⊥ |L| = mrv ⊥ v = mvr sin φ , etc m φ

For a particle travelling in a circle (constant | r|), v⊥ = r ω, and 2 L = mrv ⊥ = mr ω = Iω

Physics 1D03 A hockey puck slides in a straight line at constant past a physicist at O. How does its angular momentum about O change with ?

A) increases, then decreases B) decreases, then increases C) remains constant, but not zero O D) is zero unless the puck is spinning

v

Physics 1D03 General motion: “orbital” and “” angular momentum

Angular momentum of a particle: L = r ×p = r ×(mv) of a rotating : L = I ω.ω.ω.

In general, for a moving, rotating rigid body, ω L = r ×(mvCM ) + ICM

The first term is called the “orbital” angular momentum and the term is the “spin” angular momentum.

Example: angular momentum of a about the .

Physics 1D03 Example:

Uniform thin hoop (mass M, R); axis to hoop. Calculate its angular momentum (about P) when it rotates about P at angular speed ω.

P CM

Physics 1D03 Review Exercise: Compare this with the Parallel-Axis Theorem D

CM

2 I = ICM + MD

Physics 1D03 Quiz

Angular momentum provides an elegant approach to Atwood’s Machine. We will find the acclerations of the using “external torque = rate of change of L”.

For now: assume equal masses , so that m 1 descends at constant speed (after an initial push), and we calculate the angular momentum about the centre of the pulley (point O). As the mass ω descends, the angular momentum L1 (of m 1 only) O will R A) increase B) remain constant C) decrease v m2 v m1

Physics 1D03 Atwoods Machine, frictionless, massive pulley L For m1 : 1 = | r1 x p 1|= Rp 1 L so 1 = +m 1vR L 2 = +m2vR R O ω Lpulley = Iω = Iv/R R Thus L = ( m + m + I/R 2)vR 1 2 r

2 so dL/dt = ( m1 + m2 + I/R )aR

v m2 v m1 Torque, τ = m1gR − m2gR p = (m 1 − m2 )gR 1 p1

Write τ = dL/dt , and complete the calculation to solve for a.

Note that we only consider the external on the entire system.

Physics 1D03 Summary

Particle: L = r ×p = r ×(mv)

ω Rigid body: L = r ×(mvCM ) + ICM

Physics 1D03