Advanced Placement
PHYSICS 1
Angular Momentum
Student
2014-2015
Angular Momentum and its Conservation
What I Absolutely Have to Know to Survive the AP* Exam
If you have a problem involving a collision or an explosion, the first approach you should consider is
conservation of either momentum or angular momentum. Linear momentum is not conserved if an object
that is part of the system is fixed in place by a hinge. Angular momentum can be conserved if the point
about which angular momentum is taken is the hinge where external forces are applied. Whenever, you
have a collision or an explosion, never assume that mechanical energy is conserved.
In a system in which there is both rotation and translation, you must include both rotational and
translational kinetic energy in the same conservation of energy expressions. If you are looking for a velocity
or angular velocity, always think first of Newton’s Second Law. If you are looking for a velocity or angular
velocity, and if there is motion in which potential energy is changing, always first think about energy
considerations. Newton’s Second Law is possible if the body is moving in a circular path, but energy is
usually the best approach.
The temptation is to memorize formulas, but know the relationships and understand the concepts.
Key Formulas and Relationships
Linear and Angular analogs - variables
Linear
x Linear Distance (m)
Δx
v
a
m
F
p
Angular
Rotational distance
(radians)
Linear Displacement (m)
Rotational
displacement (radians)
Linear Velocity (m/s)
Rotational velocity
(radians/s)
2
Linear Acceleration (m/s ) Rotational acceleration
(radians/s2)
Mass (kg)
Rotational inertia
(kg.m2)
Force (N)
Torque (N.m or mN)
Linear momentum
Angular momentum
.
(kg m/s)
(kg.m2/s)
θ
Δθ
ω
α
Ι
τ
L
AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not
involved in the production of this material.
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
Linear and Angular analogs – kinematics and energy equations
Linear
Angular
Constant Motion
x = xo + vt
θ = θo + ω t
Motion with Constant Acceleration
v = vo + at
ω = ω o + αt
1
x = (vo + v)t
2
1
x = xo + vot + at 2
2
2
2
vo = v + 2ax
1
θ = (ω o + ω )t
2
1
θ = θ o + ω ot + α t 2
2
2
2
ω o = ω + 2αθ


 ΣF Fnet
a=
=
m
m
2nd Law of Motion
 
 Στ τ net
α=
=
I
I
Linear and Angular Momentum
Linear momentum


p = mv
Angular momentum
for a particle
L = rmv sin θ
for a system rotating about an axis
L = Iω
ΔL = τΔt
Conservation of Linear and Angular Momentum
psystem = constant
Lsystem = constant
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
Multiple Choice
1. A stunt man is jumping several cars with a motorcycle. While the rider and the
motorcycle are in the air, he uses the throttle to cause the rear wheel to spin faster.
What happens to the rotation of the rider-motorcycle system when the rear wheel
spins faster?
a.
b.
c.
d.
Front of motorcycle rotates upward
Rear of motorcycle rotates upward
System does not rotate
System rotates to the left
2. The angular velocity of an object increases, ω initial < ω final . Which of the
following would explain why the angular momentum would not increase?
a.
b.
c.
d.
The object’s rotational inertia remains constant.
The object had mass added to it.
The object’s rotational inertia decreased.
The object began to roll down hill.
3. A child on a spinning merry-go-round walks from the center to the outside edge
of the merry-go-round. What happens to the angular momentum, angular speed
and kinetic energy of the child-merry-go-round system?
Angular Momentum
a. constant
b. constant
c. decrease
d. increase
Angular Speed
constant
decrease
constant
increase
Kinetic Energy
constant
decrease
decrease
increase
4. A 50-kg student runs with an initial speed of 5 m/s along a path tangential to the
rim of a 2-m radius merry-go-round. The merry-go-round has a rotational inertia
of I = 1000 kg.m2, and is initially at rest. The student jumps on to the merry-goround at its rim. What is the final angular speed of the student - merry-go-round
system?
a.
b.
c.
d.
1/100 radians/sec
1/20 radians/sec
2/5 radians/sec
5/12 radians/sec
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Angular Momentum and its Conservation
5. A spherical spinning star has a uniform density, mass M, radius R and an angular
speed ω1. It begins to collapse and shrinks to ½ its original radius. What will be the ratio
of the initial angular speed ω1 to the final angular speed ω2? The rotational inertia for a
2
uniform sphere is MR 2 .
5
ω1
ω2
ω1
b.
ω2
ω1
c.
ω2
ω1
d.
ω2
a.
= 0.25
= 0.50
= 1.0
= 2.0
6. A hockey puck of mass M is moving in a straight line at a constant speed v across the
ice rink. The puck is a distance x from the center of the rink at its nearest approach to the
center. What is the puck’s angular momentum about the center of the rink?
a. Iω cos θ
b. Mvx
c. Mv
d. Mvx sin 60 0
7. A pendulum bob is released and swings downward. The pendulum string strikes a peg
and begins to wrap around the peg, with the pendulum bob spiraling inward. Neglecting
air resistance, what happens to the mechanical energy and angular momentum as the
string with the pendulum bob wraps around the pole?
Mechanical Energy
a. remains constant
b. increases
c. decreases
d. remains constant
Angular Momentum
conserved
conserved
conserved
decreases
8. Two identical spoked-wheels have a common axis and are initially not touching one
another. Wheel # 1 is initially spinning and wheel # 2 is not. When the two wheels are
brought into contact, they stick together. Which combination of the total kinetic energy
and total angular momentum is true for the two-wheel system?
a.
b.
c.
d.
Total Kinetic Energy
¼ original value
½ original value
value is unchanged
value is unchanged
Total Angular Momentum
value is unchanged
½ original value
value is unchanged
¼ original value
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
1
MR 2 is rotating about a radial
2
axis as shown in the diagram. Its angular momentum is L. What is its kinetic energy?
9. A solid disk of mass M, radius R and rotational inertia
L2
MR 2
1
b. MR 2
2
1
c. MR 2ω
2
d. 2MRL
a.
10. A student holding two textbooks sits on a spinning lab stool with her arms extended
outward. What happens to her rotational kinetic energy as she pulls her arms and books
inward?
a.
b.
c.
d.
remains constant
increases
decrease
doubles
11. A student holding two textbooks sits on a spinning lab stool with her arms extended
outward. What happens to her angular momentum if she drops the books?
a. remains constant
b. increases
c. decrease
d. doubles
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
Free Response
Question 1
An ice skater starts her performance-ending spin with her arms outstretched, rotating at
2.5 rev/second. As she pulls her arms inward, her rotational inertia decreases to 0.70 of
its original value.
A. What is her final angular speed?
B. What is the ratio of her final kinetic energy to initial kinetic energy?
C. If the ratio between the final kinetic energy and the initial kinetic energy is less
than 1, where did the energy go? If the ratio is greater than 1, where did the extra
energy come from? Justify your answer.
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
Question 2
The two wheels shown above are connected by a belt. The belt does not slip as the
wheels rotate. Wheel #2 has a radius that is 2.5 times larger than the radius of wheel #1.
I
A. What would the ratio of 1 be equal to if the both wheels had the same angular
I2
momentum?
B. What would the ratio of
I1
be equal to if the both wheels had the same rotational
I2
kinetic energy?
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
Question 3
A metal rod of mass 5.0 kg and length of 2.0 meters is attached to a pivot at the top of
1
the rod. The rotational inertia about this pivot is ML2 . A 0.35-kg rubber ball hits
3
the rod with a speed of 15.0 m/s at 1.5 meters from the pivot and bounces off with a
speed of 10.0 m/s in the opposite direction.
A. What is the initial angular momentum of the ball-rod system?
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.
Angular Momentum and its Conservation
B. What is the angular speed of the rod immediately after the collision?
C. Is this an elastic collision? Justify your answer.
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Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.