physics 151: lecture 24, pg 1 physics 151: lecture 24 today’s agenda l topics çangular...

Physics 151: Lecture 24, Pg 1

Physics 151: Lecture 24Physics 151: Lecture 24Today’s AgendaToday’s Agenda

TopicsAngular Momentum Ch. 11.3-5More fun demos !!!Gyroscopes Ch. 11.6


Angular Momentum:Angular Momentum:Definitions & DerivationsDefinitions & Derivations

We have shown that for a system of particles

Momentum is conserved if

Fp

EXTddt

FEXT 0

See text: 11.3

• With angular momentum L = r x p

p=mv

Now we also know,

Angular momentum is conserved if

EXT = 0

EXT dL

dt

Animation


What does it mean?What does it mean?

where andEXTddt

L EXT EXT r FL r p

EXTddt

L0 In the absence of external torquesIn the absence of external torques

Total angular momentum is conservedTotal angular momentum is conserved

See text: 11.5

L I

Also, remember thatAlso, remember thatLL of a rigid body about of a rigid body abouta fixed axis is:a fixed axis is:

Analogue of p = mv !!


Angular momentum of a rigid bodyAngular momentum of a rigid bodyabout a fixed axis:about 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

See text: 11.4


Demonstration ofDemonstration ofConservation of Angular MomentumConservation of Angular Momentum

Figure Skating :

A

z

B

z

Arm Arm

IA < IB

A > B

LA = LB


Lecture 23, Lecture 23, ACT 2ACT 2Angular momentumAngular momentum

Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2.Which one has the biggest moment of inertia ?

(a) disk 1 (b) disk 2 (c) not enough info

I1 < I2

L I1 1

disk 2

L I2 2

disk 1

K1 > K2


Example: Two DisksExample: Two Disks

A disk of mass M and radius R rotates around the z axis with angular velocity 0. 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. What is F ?

0

z

F

z



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

L MRINI I1 12

01

2

2

1



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

L MRFIN F I I1 1 2 22 2

1



Since LINI = LFIN

0

z

F

z

LINI LFIN

1

22

02MR MR F

F 1

2 0



Now let’s use conservation of energy principle:

0

z

F

z

EINI EFIN

1/2 I 02 = 1/2 (I + I) F

2

F2 = 1/2 0

2

F = 0 / 21/2

EINI = EFIN


So we got a different answers !


F’ = 0 / 21/2Conservation of energy !

Conservation of momentum !

Which one is correct ?

F = 0 / 2

Because this is an inelastic collision, since E is not conserved (friction) !


Lecture 25, Lecture 25, Act 2Act 2

A mass m=0.1kg is attached to a cord passing through a small hole in a frictionless, horizontal surface as in the Figure. The mass is initially orbiting with speed i = 5rad/s in a circle of radius ri = 0.2m. The cord is then slowly pulled from below, and the radius decreases to r = 0.1m. How much work is done moving the mas from ri to r ?

(A) 0.15 J (B) 0 J (C) - 0.15 J

ri

i

animation


Lecture 25, Lecture 25, Act 3Act 3

A particle whose mass is 2 kg moves in the xy plane with a constant speed of 3 m/s along the direction r = i + j. What is its angular momentum (in kg · m2/s) relative to the origin?

a. 0 k

b. 6 (2)1/2 k

c. –6 (2)1/2 k

d. 6 k

e. –6 k


Example: Bullet hitting stickExample: 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

M

F

before after

mD

D/4

See text: Ex. 11.11


Angular Momentum ConservationAngular Momentum Conservation

A freely moving particle has a definite angular momentum about any given axis.

If no torques are acting on the particle, its angular momentum will be conserved.

In the example below, the direction of LL is along the z axis, and its magnitude is given by LZ = pd = mvd.

y

x

vv

d

m

See text: 11.5


Example: Bullet hitting stick...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

before

D/4m

4D

mv)approachclosestofcetandis(xpL 1BEFORE

See text: Ex. 11.11

L r p



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

after

D/4

L mvD

AFTER F 2 4I

See text: Ex. 11.11



Set LBEFORE = LAFTER using

v1 v2

M

F

before after

mD

D/4

I 1

122MD

mvD

mvD

MD F1 22

4 4

1

12 F

m

MDv v

31 2

See text: Ex. 11.11


Example: Throwing ball from stoolExample: 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: before after

d

vM

I I

F

See example 13-9

See text: Ex. 11.11


Example: Throwing ball from stool...Example: Throwing ball from stool...

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

LAFTER = 0 = IF - Mvd

top view: before after

d

vM

I I

F

FMvd

I

See example 13-9

See text: Ex. 11.11


Gyroscopic Motion:Gyroscopic Motion:

Suppose you have a spinning gyroscope in the configuration shown below:

If the left support is removed, what will happen ??

pivotsupport

g

See text: 11.6


Gyroscopic Motion...Gyroscopic Motion...

Suppose you have a spinning gyroscope in the configuration shown below:

If the left support is removed, what will happen ?The gyroscope does not fall down !

pivot

g

See text: 11.6



... instead it precessesprecesses around its pivot axis !

pivot

See text: 11.6

This rather odd phenomenon can be easily understood using the simple relation between torque and angular momentum we derived in a previous lecture.



The magnitude of the torque about the pivot is = mgd. The direction of this torque at the instant shown is out of the page

(using the right hand rule).The change in angular momentum at the instant shown must

also be out of the page!

LL pivot

d

mg

ddtL

See text: 11.6



Consider a view looking down on the gyroscope. The magnitude of the change in angular momentum in a time dt

is dL = Ld.

So

where is the “precession frequency”

top viewLL(t)

LL(t+dt)

dLL d pivot

dL

dtL

d

dtL

See text: 11.6



So

In this example = mgd and L = I:

The direction of precession is given by applying the right hand rule to find the direction of and hence of dLL/dt.

dL

dtL

LL pivot

d

mg

mgd

I

L

See text: 11.6


Lecture 24, Lecture 24, Act 1Act 1StaticsStatics

Suppose you have a gyroscope that is supported on a gymbals such that it is free to move in all angles, but the rotation axes all go through the center of mass. As pictured, looking down from the top, which way will the gyroscope precess?

(a) clockwise (b) counterclockwise (c) it won’t precess


Lecture 24, Lecture 24, Act 1Act 1StaticsStatics

Remember that /L.

So what is ?

= r x F

r in this case is zero. Why?

Thus is zero.

It will not precess. At All. Even if you move the base.

This is how you make a direction finder for an airplane.

Answer (c) it won’t precess


Recap of today’s lectureRecap of today’s lecture

Chapter 11, Conservation of Angular MomentumSpinning Demos

physics 151: lecture 24, pg 1 physics 151: lecture 24 today’s agenda l topics çangular...

Documents

z ff z slide

total angular momentum

conservation of momentum

b slide

z subscript

angular velocity

z ff z e ini e fin

fixed z axis