angular momentum lecturer: professor stephen t. thornton
TRANSCRIPT
Reading Quiz:Can an object moving in a straight line ever have a nonzero angular momentum?
A) Always B) Never C) Sometimes
Answer: C
Sometimes, because it depends upon the axis of rotation around which you want to find the angular momentum. There is no angular momentum when the object passes through the rotation axis, because the moment arm is zero. There is angular momentum when the moment arm is nonzero (see left sketch).
Last TimeRotational kinetic energyObjects rolling – energy, speedRotational free-body diagramRotational work
Angular Momentum of Circular Motion
This particle has linear momentum. We can also say it has an angular momentum with respect to a given point, in this case the center of the circle.
2
unit: kg m / s
L I
In the case of the particle moving around the circle, let’s look more carefully at the angular momentum.
2( )
But note that ,
so we have
vL I mr mvr
r
mv p
L pr
This is another way to
determine angular momentum.
Angular Momentum in Linear and Circular Motion
The L in each view is constant. If are the same, then L is the same.
,v r
L mv r̂=
Change in angular momentum
L = IL = I, divide by t
torque
LI I
t t
dLI
dt
This equation looks similar to Newton’s 2nd law. It is sometimes called Newton’s 2nd law for rotation.
looks like dp
F madt
Conservation of angular momentum
Note what happens when there is no torque. L = 0, and angular momentum is constant.
L dL
t dt
net, ext if 0i fL L
Note similarity to conservation of linear momentum when Fnet,ext = 0.
net, ext if 0i fp p F
Conceptual Quiz:A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, how do her rotational inertia, her angular momentum and her rotational kinetic energy change? A) They all increase.B) They all remain the same.C)They all decrease.D) Rot inertia decreases, angular momentum remains constant, and her KE increases.E)Rotational inertia and angular momentum decrease, KE decreases.
Answer: DAngular momentum must be conserved. No torque. Rotational inertia decreases, because radius decreases. Only D is possible.
How does KE increase?
I goes down, goes up, L constant.
But K will increase because of .
2 1 1( )2 2
12
I LK I
http://www.youtube.com/watch?v=AQLtcEAG9v0
Vector Cross Product; Torque as a Vector
The vector cross product is defined as:
The direction of the cross product is defined by a right-hand rule:
sinC AB q= ´ =A B
= ´C A B
Conceptual QuizThe direction of the vector cross product is along the direction
A)
B)
C)
D)
E)
j i
i
i
j
k
k
For a particle, the torque can be defined around a point O:
Here, is the position vector to the point of application of force relative to O.
r
t = ´r F
·
Torque can be defined as the vector product of the position vector from the axis of rotation to the point of action of the force with the force itself:
t = ´r F
Angular Momentum of a Particle
The angular momentum of a particle about a specified axis (or point) is given by:
L r p
Angular Momentum of a Particle
If we take the derivative of , we find:
Since
we have:
L
( )dL d dr dp dp
r p p r rdt dt dt dt dt
0
dp dLr F r
dt dt´ = ´ =å
dL
dtt =å
Opposite Particles. Two identical particles have equal but opposite momenta, and , but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.
p
p
Conceptual Quiz:When a large star burns up its fuel, the gravitational force contracts it to a small size, even a few km. This is called a neutron star. When neutron stars rotate at high speed, even 100 rev/sec, they are called pulsars. They have more mass than our sun. What causes the high rotational angular velocity?
A) Friction of gas particles B) Conservation of angular
momentum
C) The dark force
D) Conservation of energy