Angular momentum
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Angular momentum – concepts & definition
- Linear momentum:
p = mv
- Angular (Rotational) momentum:
L = moment of inertia x angular velocity = I
linear
rotational
inertia
m
I
speed
v

p=mv
L=I
linear
momentum
rigid body
angular
momentum
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Angular momentum of a bowling ball
 6.1. A bowling ball is rotating as shown about its mass center
axis. Find it’s angular momentum about that axis, in kg.m2/s
A) 4 B) ½
C) 7
D) 2
E) ¼
 = 4 rad/s
M = 5 kg
R = 0.5 m
I = 2/5 MR2
L  I
Angular momentum of a point particle
L  I   mr 2  mvT r  mvr sin   mvrp =pT r  prp
O
rp
vT   r

r
tangential
vT

 
v
radial
Note: L = 0 if v is parallel to r (radially in or out)
| L | I   mvT r  mvr sin   mvrp
= rpT  rp sin   rp p
  rFT  rF sin   rp F
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Angular momentum for car
 5.2. A car of mass 1000 kg moves with a speed of 50 m/s on a circular track of
radius 100 m. What is its angular momentum (in kg • m2/s) relative to the center
of the race track (point “P”) ?
A) -5.0  102
A
B) -5.0  106
C) -2.5 
B
104
P
D) 5.0  106
E)
5.0  102
 5.3. What would the angular momentum about point “P” be if the car leaves the
track at “A” and ends up at point “B” with the same velocity ?
A) Same as above
B) Different from above | L |
C) Not Enough Information
I   mvT r  mvr sin   mvrp
= rpT  rp sin   rp p
Net angular momentum
Lnet  L1  L2  L3  ...
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Example: calculating angular momentum for particles
PP10602-23*: Two objects are moving as shown in the figure . What is their total
angular momentum about point O?
m2
m1
| L | I   mvT r  mvr sin   mvrp
= rpT  rp sin   rp p
 net  I  I
 L

t
t
L   net t
Conservation of angular momentum :
Angular momentum,
L is consreved if  net  0
Conservation of net angular momentum for a system
Net angular momentum,
L net is consreved if  net,ext  0
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A puck on a frictionless air hockey table has a mass of 5.0 g and is
attached to a cord passing through a hole in the surface as in the
figure. The puck is revolving at a distance 2.0 m from the hole
with an angular velocity of 3.0 rad/s. The cord is then pulled from
below, shortening the radius to 1.0 m. The new angular velocity (in
rad/s) is ________.
Tethered Astronauts
 6.3. Two astronauts each having mass m are connected by a mass-less rope of
length d. They are isolated in space, orbiting their center of mass at identical
speeds v. By pulling on the rope, one of them shortens the distance between
them to d/2. What are the new net angular momentum L’ and speed v’ of the
astronaut?
A) L’ = mvd/2, v’= v/2
B) L’ = mvd, v’ = 2v
C) L’ = 2mvd, v’ = v
D) L’ = 2mv’d, v’ = v/2
E) L’ = mvd, v’= v/2
L  I
L  m v Tr
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Demonstration: Spinning Professor: Web link
Isolated
System
 net ,ext  0
 L  constant
Iii  I f  f
Moment of inertia changes
Another link
Example
A professor on a freely rotating platform extends his arms
horizontally, holding a 10-kg mass in each hand.
He is rotating about a vertical axis with an angular velocity of
2 rad/s. If he pulls the mass closer to his body, what will the
final angular velocity be if his body’s moment of inertia
(excluding the two masses, of course) remains approximately
constant at 10 kg m2, the moment of inertia of the platform is
5 kg m2, and the distance of the masses from the axis changes
from 1 m to 0.2 m?
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Example: A merry-go-round problem
A 40-kg child running at 4.0 m/s jumps tangentially onto a
stationary circular merry-go-round platform whose radius is
2.0 m and whose moment of inertia is 20 kg-m2.
Find the angular velocity of the platform after the child has
jumped on.
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