conservation of angular momentum -...

Department of Physics and Applied Physics95.141, Fall 2013, Lecture 20

Lecture 20

Chapter 11

Conservation of Angular momentum

11.25.2013Physics I

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

Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html


Chapter 11

Conservation of Angular Momentum Kepler’s 2nd Law revisited

Outline


Exam III Info

Exam III Wed Dec. 4, 9:00-9:50am, OH 150.Exam III covers Chapters 9-12

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)Ch. 12: Static Equilibrium of Rigid Bodies (no sections 3-7)

Exam Review Session TBA, Ball 210


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


Rotational N.2nd law

netdtLd

Relationship between angular momentum and torque for a rigid body:

Inet dt

dI

dtId )(

dtLd

Torque causes angular momentum to change

amF

dtvdm

dt

vmd )(

dtpd

It is similar to translational N. 2nd law


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.

0,0 dtLdthenIf net

constL


Example: Bullet strikes cylinder edgeConservation of angular momentum

A bullet of mass m moving with velocity v strikes and becomes embedded at the 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 cylinder after this collision? Is kinetic energy conserved?


Exa

mpl

e:

Bul

let s

trik

es a

cyl

inde

r

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:

A) remain stationary

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

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


Bicycle wheel/turntable as a demo of Angular Momentum Conservation


Bicycle wheel/Ang. Mom. Conservation

xLx

z

y

wzL

ADzL

initial final

finz

inz LL

Angular Momentum Conservation (z comp):

ADz

wz

finz LLL w

zADz LL wwcc II

wc

wc 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


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 )( II


Angular Momentum Conservation. Examples

I11 I22

IL For a rigid body

21 LL

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


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.


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 vdtrdA

dAdt

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


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


Bicycle wheel precession.

r

gm

dtLd

Apply Rotational N.2nd law:

dtLd

mg produces torque in +y direction

)ˆ( jrmggmr

Ld

inL

finL

LdLL infin

z

x

y

dtLin

Torgue changes direction of angular momentum

Since L I

then angular velocity changes its direction

It is called precession

wir

e

TF

ConcepTest 3 traffic light/car

A car of mass 1000 kg drives away from a traffic light h=10 m high, as shown below, at a constant speed of v=10 m/s. What is the angular momentum of the car with respect to the light?

A) B) C)

skgmk 2 )ˆ(000,10

x

y

z

h

skgmi 2 ˆ000,100

v

)ˆ(000,10)ˆ()ˆ)(( kkmvhkrSinmvprL

skgmk 2 )ˆ(000,10

r


Thank youSee you on Wednesday

conservation of angular momentum -...

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