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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