Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Lecture 20

Chapter 11

Conservation of Angular momentum

04.14.2014Physics I

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture: http://echo360.uml.edu/danylov2013/physics1spring.html

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Exam III Info

Exam III Friday April 25, 9:00-9:50am, OH 150.Exam III covers Chapters 9-11

Same format as Exam IIPrior Examples of Exam III posted

Ch. 9: Linear Momentum (no section 10)Ch. 10: Rotational Motion (no section 10)

Ch. 11: Angular Momentum; General Rotation (no sections 7-9)

Exam Review Session TBA

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Chapter 11

Conservation of Angular Momentum Kepler’s 2nd Law revisited

Outline

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Two definitions of Angular Momentum

L

L I

L r p

Rigid symmetrical bodySingle particle

x

z

y

O

r pm

L r p

L is perpendicular to the “plane of motion” created by r and p

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Rotational N.2nd law

netdtLd

Torque causes angular momentum to change

It is similar to translational N. 2nd law

amF

dtpdF

I

Translational N.2nd law Rotational N.2nd law

dtLd

Total angular momentum of a system

Net Torque of a system

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Conservation of Angular Momentum

netdtLd

Angular momentum is an important concept because, under certain conditions, it is conserved.

If the net external torque on an object is zero, then the total angular momentum is conserved.

,0extnetIf

constL

Rotational N.2nd law

0dtLdthen

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Example: Bullet strikes cylinder edge

A bullet of mass m moving with velocity v strikes and becomes embedded atthe edge of a cylinder of mass M and radius R0. The cylinder, initially at rest,begins to rotate about its symmetry axis, which remains fixed in position.Assuming no frictional torque, what is the angular velocity of the cylinderafter this collision? Is kinetic energy conserved?

Application of Conservation of angular momentum

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Exa

mpl

e:

Bul

let s

trik

es a

cyl

inde

r

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Bicycle wheel/turntable as a demo of Angular Momentum Conservation

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Bicycle wheel/Ang. Mom. Conservation

xLx

z

y

wzL

ADzL

initial final

finz

inz LL Angular Momentum Conservation (z comp):

ADz

wz

finz LLL

wz

ADz LL wwADAD II

wAD

wAD I

I )( counterclockwiseclockwise

0inzL

0

ConcepTest 1 Spinning Bicycle Wheel

You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:

The total angular momentum of the system is L upward, and it is conserved. So if the wheel has−L downward, you and the table must have +2L upward.

1) remain stationary

2) start to spin in the same direction as before flipping

3) start to spin in the same direction as after flipping

finz

inz LL

tablemeLLL zLL tableme 2

L

L2L

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Angular Momentum Conservation. Examples

I11 I22 IL For a rigid body

21 LL

Launch: leg and hands are out to make I large

Flight: leg and hands are in to make large

Landing: leg and hands are out to “dump” large

1212 )( IIsmall-large

1

1

I

large-s

2

2

mallI

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Bicycle wheel precession.A bicycle wheel is spun up to high speed and is suspended from the ceiling by a wire attached to one end of its axle.

Now mg try to tilt the axle downward.

You expect the wheel to go down (and it would if it weren’t rotating), but it unexpectedly swerves to the left and starts rotating!Let’s explain that.

gm

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Bicycle wheel precession.

r

gm

dtLdapply Rotational N.2nd law:

dtLd

mg produces torque in +y directiongmr

Ld

inL

finL

LdLL infin

z

x

y

dtLin

Torgue changes direction of angular momentum.

Now, since

ILthen angular momentum rotates with angular velocity It is called precession

wir

e

TF

Top view of the precession

So L rotates.

)ˆ( jrmgLet’s calculate torque on the wheel

jrmgdt ˆ)( in +y direction

Notice, angular momentum isn’t conserved. There is an external torque produced by mg. So

ConcepTest 2 Figure Skater

A) the same

B) larger because she’s rotating faster

C) smaller because her rotational inertia is smaller

A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:

KErot = I 2 = (I ) = L (used L = I ). Because L is conserved, larger means larger KErot. The “extra” energy comes from the work she does on her arms.

12

12

12

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Conservation of Angular Momentum. Kepler’s second law

Kepler’s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times. Use

conservation of angular momentum to show this.

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Kepler’s second law

Kepler’s 2nd law is a consequence of conservation of ang. momentum!So if L is constant, dA/dt is a constant

)sin()(21 vdtrdAdAdt

L2m

r

pv,

morigin

Let’s find ang. moment. of the Earth(treat the Earth as a point):

L r psinrpL

sinmrvNext step. Let’s find area swept by r in dt:

vdtdl

sin21 rvdt

mmrv

dtdA

2sin

)ˆ( rFr gext Let’s show that L is conserved:

gF

,0sin netce

netdtLd

0constLthen

)sin( vdt

Department of Physics and Applied Physics95.141, Spring 2014, Lecture 20

Thank youSee you on Wednesday