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Angular momentum
Announcements:
•  CAPA Set #10 is due today
Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/
Pulling the yo-yo
Yo-yo with inner radius R1 and outer radius R2 is
on a table. String is pulled with force F as shown.
There is enough static friction to prevent the yo-yo
from sliding. Which way does the yo-yo move?
Rolling without slipping:
R2
and
but you need to make sure you get the signs right!
so
Substituting
Rearranging:
gives
Pulling the yo-yo
Using the extended free body diagram and
Newton’s 2nd law we found the equations:
Applying rolling without slipping:
After some math:
and
(which is to the left)
then F exerts no torque and
Clicker question 1
Set frequency to BA
Two 0.2 kg spools of thread are on a frictionless table. Each
is pulled with 0.4 N of force. A is free to rotate (and let out
string) while in B the string is attached to the spindle. After 2
seconds, which will be moving faster (largest center of mass
speed)?
A.  A
A
B.  B
C.  same speed
B
Top view
Clicker question 1
Set frequency to BA
Two 0.2 kg spools of thread are on a frictionless table. Each
is pulled with 0.4 N of force. A is free to rotate (and let out
string) while in B the string is attached to the spindle. After 2
seconds, which will be moving faster (largest center of mass
speed)?
A.  A
A
B.  B
C.  same speed
B
Top view
Clicker question 2
Set frequency to BA
Two 0.2 kg spools of thread are on a frictionless table. Each
is pulled with 0.4 N of force. A is free to rotate (and let out
string) while in B the string is attached to the spindle. After 2
seconds, which one will have the most kinetic energy?
A.  A
B.  B
C.  same kinetic energy
A
B
Top view
Clicker question 2
Set frequency to BA
Two 0.2 kg spools of thread are on a frictionless table. Each
is pulled with 0.4 N of force. A is free to rotate (and let out
string) while in B the string is attached to the spindle. After 2
seconds, which one will have the most kinetic energy?
A.  A
B.  B
C.  same kinetic energy
A
B
Top view
Spool of thread
Two 0.2 kg spools on frictionless table are pulled with F=.4 N. A is free
to rotate while the B string is attached. After 2 seconds what’s going on?
Only 1 force in x direction so
gives
for a center of mass
velocity of
A
B
Distance spool travels:
Translational kinetic energy
Work done by force on spool B:
For spool B, all work goes into translational kinetic energy
Spool of thread
Previous slide used
about the rotational part?
A
. What
B
For B no torque because r and F are parallel.
so Let’s set
For A:
and
After 2 seconds
Rotational KE
Amount of string unwound:
Work done unwinding string:
For spool A, extra 3.2 J of work goes into rotational kinetic energy
Question about frictionless surface
For spool A (R=2 cm) we learned the following (after 2 seconds):
Spool moves 4 m
Spool rotates 400 rad
Is this an example of rotating without slipping?
A.  Yes
B.  No
If this were rolling without slipping then
A
Top view
None of these are true. There is no friction between table and
spool causing rotation without slipping.
The thread coming off the spool is sort of rolling without slipping
Quick word on work and power
In the linear case we have
The equivalent for angular motion is
The force causing the torque must be
tangential to the rotation. That is,
torque and angular displacement must
be in the same (or opposite) direction.
Remember power is work per time:
For linear motion also have
For rotational motion also have
Angular momentum
We again have another angular analog to the linear case.
Linear momentum gives an idea about how hard it would be to
slow something down. Angular momentum tells us how hard it
is to slow down something which is spinning.
Linear momentum:
Newton’s 2nd law
Angular momentum:
becomes
Can also talk about the angular momentum of a particle
(like a satellite orbiting the Earth). In this case, angular
momentum is
so
Clicker question 3
Set frequency to BA
A planet in elliptical orbit about the Sun is in the position
shown. With the origin located at the Sun, the torque on the
planet…
A.  is 0
B.  is in the +z direction
C.  is in the –z direction
D.  is in the x-y plane
E.  is none of these
planet
y
x
S
z
Clicker question 3
Set frequency to BA
A planet in elliptical orbit about the Sun is in the position
shown. With the origin located at the Sun, the torque on the
planet…
A.  is 0
B.  is in the +z direction
C.  is in the –z direction
D.  is in the x-y plane
E.  is none of these
planet
y
x
S
z
For any central force like gravity, the force vector is parallel
(or antiparallel) to the vector so no torque can be applied.
Thus, a central force cannot affect the angular momentum
Conservation of angular momentum
Previously found if no external force acts on a system, momentum
is constant. We called this conservation of momentum.
Similarly, if no external torque acts, angular momentum is
constant. This is conservation of angular momentum.
Since
, if
then
so
is constant
Just like internal forces inside a system cannot produce a net
force on the system due to Newton’s 3rd law, internal torques
cannot produce a net torque so only external torques can
change the total angular momentum of the system.
Conservation of angular momentum
To solve problems in which momentum is conserved
we generally used the equation
or
Similarly, for angular momentum conservation problems
we will usually use the equation
or
The spinning professor
Consider a person with arms outstretched holding weights and the
same person with arms tucked in holding weights.
Could estimate moment of inertia for both
situations using a cylinder for the body, rods for
the arms, and point masses for the weights.
Clearly
so if there is an initial angular velocity
then since there is no external torque
so
Note if angular momentum is conserved but
then rotational kinetic energy is not conserved:
Where does the work to change the kinetic energy come from?
Force over distance is required to pull the weight in.
Clicker question 4
Set frequency to BA
A disk of mass M and radius R is rotating with angular velocity ω0.
Consider the effect on the angular velocity if a small object of mass
m is dropped onto the disk. If the object is dropped at a radius of R
the final angular velocity is ω1 and if the object is dropped at a radius
of R/2 the final angular velocity is ω2. What is the relationship
between the various angular velocities?
A. 
B. 
C. 
D. 
E. 
Clicker question 4
Set frequency to BA
A disk of mass M and radius R is rotating with angular velocity ω0.
Consider the effect on the angular velocity if a small object of mass
m is dropped onto the disk. If the object is dropped at a radius of R
the final angular velocity is ω1 and if the object is dropped at a radius
of R/2 the final angular velocity is ω2. What is the relationship
between the various angular velocities?
A. 
B. 
C. 
Conservation of angular
D. 
momentum gives us:
E. 
so
More on clicker question 4
Since there is no external
torque, the overall angular
momentum of the system
remains the same which is
how we solved the problem.
Note that if we just consider the disk, the angular momentum
decreased. Where did the torque come from?
When the mass lands, friction acts to cause an acceleration
of the mass and the equal and opposite force slows the disk.
Need to carefully determine what is part of the system
on which conservation of momentum can be applied.
New types of collision problems
If an object comes in with some initial
angular momentum then we need to
account for that as well.
Initial angular momentum about the disk axis is
Angular momentum is conserved in the collision.
If the axis is fixed in place then momentum is not conserved (it is
conserved when you consider the Earth as part of the system).
If the axis is not fixed then momentum is also conserved.
Angular momentum is a vector
If I get on the rotating platform, initially at rest and start spinning a
bicycle wheel clockwise whose axis points up, what will happen?
I will start rotating in the opposite direction.
Initial angular momentum is 0 and no external torque is
applied so final angular momentum must also be 0.
Bicycle wheel has angular momentum pointing
down (clockwise with vertical axis upward)
Something else in the system must have angular momentum
pointing in the opposite direction to have a total angular
momentum of 0. (“Something else” is a rotating professor.)
Clicker question 5
Set frequency to BA
Three identical wheels are all spinning with the same angular
velocity . The total angular momentum of the 3-wheel system has
magnitude L. One of the three wheels is flipped upside-down,
while the magnitude of its angular velocity remains constant. The
new angular momentum of the 3-wheel system has magnitude…
A.  L (same as before)
B.  L/3
C.  2L/3
D.  L/2
E.  Some other value
Clicker question 5
Set frequency to BA
Three identical wheels are all spinning with the same angular
velocity . The total angular momentum of the 3-wheel system has
magnitude L. One of the three wheels is flipped upside-down,
while the magnitude of its angular velocity remains constant. The
new angular momentum of the 3-wheel system has magnitude…
A.  L (same as before)
B.  L/3
C.  2L/3
D.  L/2
Angular momentum is a vector.
E.  Some other value
Initial:
Final: