Chapter 11

Angular Momentum

Introduction

• When studying angular motion, angular momentum plays a key role.

• Newton’s 2nd for rotation can be expressed in terms of angular momentum.

• When the net torque is zero, angular momentum is conserved. (similar to net force and linear momentum).

11.1 The Vector Product and Torque

• We have seen the product of two vectors result in a scalar value.

• The product of two vectors can also be a vector (as with Torque, τ = r F )

• Vector (Cross) Product-The product of two vectors A and B, defined as a

third vector C.and magnitudeBAC sinABC

11.1

• The direction of vector C is found by the right hand rule (curl fingers from A to B)

• Incidentally, the magnitude of the cross product is equal to the area of a parallelogram created by the parent vectors.

11.1

• Properties of the Cross Product– NOT commutative (order matters, changes the

direction of vector C)

– If A and B are parallel/antiparallel then

– If A and B are perpendicular then

0BA

ABBA

ABBA

11.1

• Properties cont’d– Cross Products are distributive

– The derivative of a cross product with respect to a variable like time, follows the derivative product rule. (maintaining the multiplicative order)

CABACBA

dt

d

dt

d

dt

d BAB

ABA

11.1

• Cross products with unit vectors0ˆˆˆˆˆˆ kkjjii

kij

kji

ˆˆˆ

ˆˆˆ

ijk

ikj

ˆˆˆ

ˆˆˆ

jik

jki

ˆˆˆ

ˆˆˆ

11.1

• Vector A x B is given by

(See Board Work for Proof)

kjiBA ˆˆˆxyyxzxxzyzzy BABABABABABA

11.1

• Quick Quizzes p. 339• Examples 11.1-11.2

11.2 Angular Momentum

• Developing Angular Momentum– We know Newton’s 2nd Law in terms of changing

momentum of a particle (mass m, position r, momentum p)

– Lets cross product both sides with position vector r to find the net torque on the particle

dt

dpF

dt

dprFr

11.2

• Now lets add to the right side a term equal to zero

• Product Rule

prp

r dt

d

dt

d

dt

d pr

11.2

• Angular Momentum– Dimensions of ML2T-1, units kg.m2/s– Magnitude of an object’s angular momentum

(Following cross product magnitude eqn)

• Net Torque- time rate of change of angular momentum

prL

dt

dL

sinmvrL

11.2

• Quick Quizzes p 341• Ex 11.3

11.3 Angular Momentum of a Rotating Rigid Object

• For a rotating object, every particle moves about the axis of rotation with angular velocity. (ω)

• That particle’s angular momentum is

• But rememberso

mvrL

rv 2mrL

11.3

• We can now define angular momentum of a rotating object as

• And remember

IL

I

dt

dI

dt

dL

11.3

• Quick Quiz p. 344• Examples 11.5, 11.6

11.4 Conservation of Angular Momentum

• Just with linear systems where the net force is zero and linear momentum is conserved, Angular momentum is conserved with zero net torque.

• Therefore L is a constant and Li = Lf

(both magnitude and direction)

0dt

dext

L

11.4

• Since angular momentum is conserved with zero net torque, a spinning object is considered to be very stable.

• Applications- – Gyroscopes– Motorcycle/Bicycle Wheels– Rifling/Arrow Fletching– Football Spiral

11.4

• More on the football, with zero net torque the axis of rotation should remain fixed in space.

• Sometimes the axis of rotation remains tangent to the trajectory.

11.4

• While gravity provides no net torque, air resistance can (depends on v2

and shape do)• The faster its thrown the more likely the ball is

to orient itself to reduce air resistance. (Rotation Axis follows the trajectory)

11.4

• Now angular momentum is conserved, what will happen to a rotating object if the M.o.I changes.

• I and ω are inversely proportional to each other.

• Figure skating is a prime example.

fi LL ffii II

11.4

11.4

• Quick Quizzes p 346• Examples 11.7-11.9

• End of CH 11