Physics 111: Lecture 22 Today’s Agenda

Physics 111: Lecture 22, Pg 1

Physics 111: Lecture 22

Today’s Agenda Angular Momentum:

Definitions & DerivationsWhat does it mean?

Rotation about a fixed axisL = IExample: Two disksStudent on rotating stool

Angular momentum of a freely moving particleBullet hitting stickStudent throwing ball


Lecture 22, Act 1Rotations

A girl is riding on the outside edge of a merry-go-round turning with constant . She holds a ball at rest in her hand and releases it. Viewed from above, which of the paths shown below will the ball follow after she lets it go?

(c)

(b)(a)

(d)


Lecture 22, Act 1 Solution

Just before release, the velocity of the ball is tangent to the circle it is moving in.



After release it keeps going in the same direction since there are no forces acting on it to change this direction.


Angular Momentum:Definitions & Derivations

We have shown that for a system of particles

Momentum is conserved if

What is the rotational version of this??

F pEXT

ddt

FEXT 0

L r p

r F The rotational analogue of force F is torque

Define the rotational analogue of momentum p to be

angular momentum

p = mv


Definitions & Derivations...

First consider the rate of change of L: ddt

ddt

L r p

ddt

ddt

ddt

r p r p r p

v vm0

So ddt

ddt

L r p (so what...?)


Definitions & Derivations...

ddt

ddt

L r p

F pEXT

ddt

EXTFdtd rL Recall that

Which finally gives us: EXTddt

L

Analogue of !! F pEXT

ddt

EXT


What does it mean?

where andEXTddt

L EXT EXT r FL r p

EXTddt

L 0 In the absence of external torques

Total angular momentum is conserved


i

j

Angular momentum of a rigid bodyabout a fixed axis:

k̂vrmmi

iiii

iiiii

i vrprL

Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle:

r1

r3

r2

m2

m1

m3

v2

v1

v3

We see that L is in the z direction.

Using vi = ri , we get

LI

(since ri and vi are perpendicular)

Analogue of p = mv!!

krmLi

2ii

ˆ

Rolling chain


Angular momentum of a rigid bodyabout a fixed axis:

In general, for an object rotating about a fixed (z) axis we can write LZ = I

The direction of LZ is given by theright hand rule (same as ).

We will omit the Z subscript for simplicity,and write L = I

LZ I

z


Example: Two Disks

A disk of mass M and radius R rotates around the z axis with angular velocity i. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity f.

i

z

f

z


Example: Two Disks

First realize that there are no external torques acting on the two-disk system.Angular momentum will be conserved!

Initially, the total angular momentum is due only to the disk on the bottom:

0

z

f2

11i MR21L I

2

1


Example: Two Disks

First realize that there are no external torques acting on the two-disk system.Angular momentum will be conserved!

Finally, the total angular momentum is dueto both disks spinning:

f

z

f2

2211f MRL II21


Example: Two Disks

Since Li = Lf

f

z

f

z

Li Lf

f2

i2 MRMR

21

if 21

An inelastic collision,since E is not

conserved (friction)!

Wheel rimdrop


Example: Rotating Table

A student sits on a rotating stool with his arms extended and a weight in each hand. The total moment of inertia is Ii, and he is rotating with angular speed i. He then pulls his hands in toward his body so that the moment of inertia reduces to If. What is his final angular speed f?

i

Ii

f

If


Example: Rotating Table...

Again, there are no external torques acting on the student-stool system, so angular momentum will be conserved.Initially: Li = Iii

Finally: Lf = If f f

i

i

fII

i

Ii

f

If

LiLf

Drop mass from stool

Student on stool


Lecture 22, Act 2Angular Momentum

A student sits on a freely turning stool and rotates with constant angular velocity 1. She pulls her arms in, and due to angular momentum conservation her angular velocity increases to 2. In doing this her kinetic energy:

(a) increases (b) decreases (c) stays the same

1 2

I2 I1

L L



I2

LI21K

22 (using L = I)

L is conserved:

I2 < I1 K2 > K1 K increases!

1 2

I2 I1

L L



Since the student has to force her arms to move toward her body, she must be doing positive work!

The work/kinetic energy theorem states that this will increase the kinetic energy of the system!

1

I1

2

I2

L L


Angular Momentum of aFreely Moving Particle

We have defined the angular momentum of a particle about the origin as

This does not demand that the particle is moving in a circle!We will show that this particle has a constant angular

momentum!

y

x

v

L r p


Angular Momentum of aFreely Moving Particle...

Consider a particle of mass m moving with speed v along the line y = -d. What is its angular momentum as measured from the origin (0,0)?

x

v

md

y



We need to figure out The magnitude of the angular momentum is:

Since r and p are both in the x-y plane, L will be in the z direction (right hand rule):

y

xp=mvd

r

approachclosestofcetandisxp

pdsinrpsinrp prL

L pdZ

L r p



So we see that the direction of L is along the z axis, and its magnitude is given by LZ = pd = mvd.

L is clearly conserved since d is constant (the distance of closest approach of the particle to the origin) and p is constant (momentum conservation).

y

x

p

d


Example: Bullet hitting stick

A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v1, and the final speed is v2. What is the angular speed F of the stick after the

collision? (Ignore gravity)

v1 v2

MF

initial final

m D D/4


Example: Bullet hitting stick...

Conserve angular momentum around the pivot (z) axis! The total angular momentum before the collision is due

only to the bullet (since the stick is not rotating yet).

v1

D

M

initial

D/4m

4DmvapproachclosestofcedisxpL 1i )tan(



Conserve angular momentum around the pivot (z) axis! The total angular momentum after the collision has

contributions from both the bullet and the stick.

where I is the moment of inertia of the stick about the

pivot.

v2

F

final

D/4

F2f 4DmvL I



Set Li = Lf using

v1 v2

MF

initial final

m D D/4

I 1

122MD

mv D mv D MD F1 22

4 41

12 F

mMD

v v 3

1 2


Example: Throwing ball from stool

A student sits on a stool which is free to rotate. The moment of inertia of the student plus the stool is I. She throws a heavy ball of mass M with speed v such that its velocity vector passes a distance d from the axis of rotation. What is the angular speed F of the student-stool

system after she throws the ball?

top view: initial final

d

vM

I I

F


Example: Throwing ball from stool...

Conserve angular momentum (since there are no external torques acting on the student-stool system):Li = 0Lf = 0 = IF - Mvd

top view: initial final

d

vM

I I

F

FMvd

I


Lecture 22, Act 3Angular Momentum

1

A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she releases the ball, the angular velocity of the merry-go-round will:

(a) increase (b) decrease (c) stay the same

2



The angular momentum is due to the girl, the merry-go-round and the ball. LNET = LMGR + LGIRL + LBALL

Initial: L mR vR

mvRBALL

I 2

v

Rm

Final: L mvRBALL

vm

same



Since LBALL is the same before & after, must stay the same to keep “the rest” of LNET unchanged.


Lecture 22, Act 3Conceptual answer

Since dropping the ball does not cause any forces to act on the merry-go-round, there is no way that this can change the angular velocity.

Just like dropping a weight from a level coasting car does not affect the speed of the car.

2


Recap of today’s lecture

Angular Momentum: (Text: 10-2, 10-4)Definitions & DerivationsWhat does it mean?

Rotation about a fixed axis (Text: 10-2, 10-4)L = IExample: Two disksStudent on rotating stool

Angular momentum of a freely moving particle(Text: 10-2, 10-4)

Bullet hitting stickStudent throwing ball

Look at textbook problems Chapter 10: # 21, 29, 35, 41, 47, 49

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Physics 111: Lecture 22 Today’s Agenda

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