Title: Angular Momentum Lecturer: Professor Stephen T. Thornton
1Angular Momentum Lecturer Professor Stephen
T. Thornton
2Reading QuizCan an object moving in a straight
line ever have a nonzero angular momentum?
A) Always B) Never C) Sometimes
3Answer 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).
4Last Time
- Rotational kinetic energy
- Objects rolling energy, speed
- Rotational free-body diagram
- Rotational work
5Today
- Angular momentum
- Vector (cross) products
- Torque again with vectors
- Unbalanced torque
6Angular 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.
7In the case of the particle moving around the
circle, lets look more carefully at the angular
momentum.
This is another way to determine angular
momentum.
8The Angular Momentum of Non-Tangential Motion
9A particle moving in any direction can have
angular momentum about any point.
10Angular Momentum in Linear and Circular Motion
The L in each view is constant. If are
the same, then L is the same.
11Change in angular momentum
- L I?
- ?L I??, divide by ?t
This equation looks similar to Newtons 2nd law.
It is sometimes called Newtons 2nd law for
rotation.
12Conservation of angular momentum
- Note what happens when there is no torque. ?L
0, and angular momentum is constant.
Note similarity to conservation of linear
momentum when Fnet,ext 0.
13 Conceptual QuizA 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) Rotatio
nal inertia and angular momentum decrease,
KE decreases.
14Answer D
- Angular 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 ?.
http//www.youtube.com/watch?vAQLtcEAG9v0
15Vector 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
16The vector (cross) product can also be written in
determinant form
17Some properties of the cross product
18 Conceptual QuizThe direction
of the vector cross product is along the
direction
19Answer E
20For 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.
21Torque 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
22Torque
A right-hand rule gives the direction of the
torque.
23The Right-Hand Rule for Torque
24Yo-yo demo
torque in
torque out
O
O
25Angular Momentum of a Particle
The angular momentum of a particle about a
specified axis (or point) is given by
26The Right-Hand Rule for Angular Momentum
p
27 Lets do this demo!
28 Lets do this demo!
No torque L is conserved. I decreases,
therefore ? must increase.
29A Rotational Collision Angular momentum will be
conserved here.
30Angular Momentum of a Particle
If we take the derivative of , we find
0
Since
we have
31 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.
32Conceptual QuizWhen 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
33Answer B
- Just like our own sun, these stars rotate about
their own axis. As gravity contracts the
particles closer and closer, the density becomes
huge. There are no torques, so angular momentum
must be conserved. LI?, so as I decreases, ?
must increase.