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Transcript
The law of Conservation of Energy
Monday, October 31, 11
Force as the derivative of potential energy
We often encounter situations where we know the
potential energy as a function of position and we need
to find the corresponding force.
We recall that for a conservative force:
W = −ΔU
So if we now apply this to a small displacement Δx, the work done by
the force Fx(x) during this displacement approaches Fx(x) Δx as Δx
approaches zero.
ΔU
Fx (x)Δx = −ΔU ⇒ Fx (x) = −
Δx
So in the limit as Δx → 0
dU(x)
Fx (x) = −
dx
Monday, October 31, 11
Force as the derivative of potential energy
ΔU
Fx (x)Δx = −ΔU ⇒ Fx (x) = −
Δx
So in the limit as Δx → 0
dU(x)
Fx (x) = −
dx
Monday, October 31, 11
Energy diagrams give us insight
• Energy diagrams plot
energy as a function of
position.
• These diagrams give
useful information
about limits and zeros
for the physical
properties involved.
Monday, October 31, 11
The potential energy curve for motion of a particle
dU
Fx = −
dx
Monday, October 31, 11
Energy diagrams give us insight
Monday, October 31, 11
Summary
Monday, October 31, 11
Summary
Monday, October 31, 11
Summary
Monday, October 31, 11
Q7.1
A piece of fruit falls straight down. As it falls,
A. the gravitational force does positive work on it and the
gravitational potential energy increases.
B. the gravitational force does positive work on it and the
gravitational potential energy decreases.
C. the gravitational force does negative work on it and the
gravitational potential energy increases.
D. the gravitational force does negative work on it and the
gravitational potential energy decreases.
Monday, October 31, 11
A7.1
A piece of fruit falls straight down. As it falls,
A. the gravitational force does positive work on it and the
gravitational potential energy increases.
B. the gravitational force does positive work on it and the
gravitational potential energy decreases.
C. the gravitational force does negative work on it and the
gravitational potential energy increases.
D. the gravitational force does negative work on it and the
gravitational potential energy decreases.
W = −ΔU
Monday, October 31, 11
Q7.2
You toss a 0.150-kg baseball
straight upward so that it leaves
your hand moving at 20.0 m/s. The
ball reaches a maximum height y2.
What is the speed of the ball when
it is at a height of y2/2? Ignore air
resistance.
A. 10.0 m/s
v2 = 0
v1 = 20.0 m/s
m = 0.150 kg
B. less than 10.0 m/s but more than zero
C. more than 10.0 m/s
D. not enough information given to decide
Monday, October 31, 11
y2
y1 = 0
A7.2
You toss a 0.150-kg baseball
straight upward so that it leaves
your hand moving at 20.0 m/s. The
ball reaches a maximum height y2.
What is the speed of the ball when
it is at a height of y2/2? Ignore air
resistance.
A. 10.0 m/s
v2 = 0
v1 = 20.0 m/s
m = 0.150 kg
B. less than 10.0 m/s but more than zero
C. more than 10.0 m/s
D. not enough information given to decide
Monday, October 31, 11
y2
y1 = 0
Q7.4
The two ramps shown are both frictionless. The heights y1 and y2 are
the same for each ramp. A block of mass m is released from rest at
the left-hand end of each ramp. Which block arrives at the righthand end with the greater speed?
A. the block on the curved track
B. the block on the straight track
C. Both blocks arrive at the right-hand end with the same speed.
D. The answer depends on the shape of the curved track.
Monday, October 31, 11
A7.4
The two ramps shown are both frictionless. The heights y1 and y2 are
the same for each ramp. A block of mass m is released from rest at
the left-hand end of each ramp. Which block arrives at the righthand end with the greater speed?
A. the block on the curved track
B. the block on the straight track
C. Both blocks arrive at the right-hand end with the same speed.
D. The answer depends on the shape of the curved track.
Monday, October 31, 11
Q7.5
A block is released from rest on a
frictionless incline as shown. When the
moving block is in contact with the spring
and compressing it, what is happening to
the gravitational potential energy Ugrav and
the elastic potential energy Uel?
A. Ugrav and Uel are both increasing.
B. Ugrav and Uel are both decreasing.
C. Ugrav is increasing, Uel is decreasing.
D. Ugrav is decreasing, Uel is increasing.
E. The answer depends on how the block’s speed is changing.
Monday, October 31, 11
A7.5
A block is released from rest on a
frictionless incline as shown. When the
moving block is in contact with the spring
and compressing it, what is happening to
the gravitational potential energy Ugrav and
the elastic potential energy Uel?
A. Ugrav and Uel are both increasing.
B. Ugrav and Uel are both decreasing.
C. Ugrav is increasing, Uel is decreasing.
D. Ugrav is decreasing, Uel is increasing.
E. The answer depends on how the block’s speed is changing.
Monday, October 31, 11
Q7.6
The graph shows the potential
energy U for a particle that moves
along the x-axis.
The particle is initially at x = d
and moves in the negative xdirection. At which of the labeled
x-coordinates does the particle
have the greatest speed?
A. at x = a B. at x = b
C. at x = c
E. more than one of the above
Monday, October 31, 11
D. at x = d
A7.6
The graph shows the potential
energy U for a particle that moves
along the x-axis.
Potential energy lowest,
kinetic energy highest
The particle is initially at x = d
and moves in the negative xdirection. At which of the labeled
x-coordinates does the particle
have the greatest speed?
A. at x = a B. at x = b
C. at x = c
E. more than one of the above
Monday, October 31, 11
D. at x = d
Q7.7
The graph shows the potential
energy U for a particle that moves
along the x-axis.
The particle is initially at x = d
and moves in the negative xdirection. At which of the labeled
x-coordinates is the particle
slowing down?
A. at x = a B. at x = b
C. at x = c
E. more than one of the above
Monday, October 31, 11
D. at x = d
A7.7
The graph shows the potential
energy U for a particle that moves
along the x-axis.
The slope is positive so
the acceleration is negative
The particle is initially at x = d
and moves in the negative xdirection. At which of the labeled
x-coordinates is the particle
slowing down?
A. at x = a B. at x = b
C. at x = c
E. more than one of the above
Monday, October 31, 11
dU
Fx = −
= ma x
dx
D. at x = d
Q7.8
The graph shows the potential
energy U for a particle that moves
along the x-axis. At which of the
labeled x-coordinates is there zero
force on the particle?
A. at x = a and x = c B. at x = b only
C. at x = d only
D. at x = b and d
E. misleading question — there is a force at all values of x.
Monday, October 31, 11
A7.8
The graph shows the potential
energy U for a particle that moves
along the x-axis. At which of the
labeled x-coordinates is there zero
force on the particle?
The slope is zero
so the force is zero
dU
Fx = −
dx
A. at x = a and x = c B. at x = b only
C. at x = d only
D. at x = b and d
E. misleading question — there is a force at all values of x.
Monday, October 31, 11
Monday, October 31, 11
Q7.9
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx = 0 at x = a. Which statement
about the associated potential energy function U
at x = a is correct?
Fx
0
A. U = 0 at x = a
B. U is a maximum at x = a.
C. U is a minimum at x = a
D. U is neither a minimum or a maximum at x = a, and
its value at x = a need not be zero.
Monday, October 31, 11
x
a
A7.9
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx = 0 at x = a. Which statement
about the associated potential energy function U
at x = a is correct?
Fx
0
A. U = 0 at x = a
B. U is a maximum at x = a.
C. U is a minimum at x = a.
D. U is neither a minimum or a maximum at x = a, and
its value at x = a need not be zero.
Monday, October 31, 11
x
a
Monday, October 31, 11
Q7.10
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx = 0 at x = a. Which statement
about the associated potential energy function U
at x = a is correct?
Fx
0
A. U = 0 at x = a
B. U is a maximum at x = a.
C. U is a minimum at x = a.
D. U is neither a minimum or a maximum at x = a, and
its value at x = a need not be zero.
Monday, October 31, 11
x
a
A7.10
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx = 0 at x = a. Which statement
about the associated potential energy function U
at x = a is correct?
Fx
0
A. U = 0 at x = a
B. U is a maximum at x = a.
C. U is a minimum at x = a.
D. U is neither a minimum or a maximum at x = a, and
its value at x = a need not be zero.
Monday, October 31, 11
x
a
Monday, October 31, 11
Q7.11
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx > 0 and dFx/dx < 0 at x = a.
Which statement about the associated potential
energy function U at x = a is correct?
A. dU/dx > 0 at x = a
B. dU/dx < 0 at x = a
C. dU/dx = 0 at x = a
D. Any of the above could be correct.
Monday, October 31, 11
Fx
0
x
a
A7.11
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx > 0 and dFx/dx < 0 at x = a.
Which statement about the associated potential
energy function U at x = a is correct?
Fx
0
x
a
A. dU/dx > 0 at x = a
B. dU/dx < 0 at x = a
C. dU/dx = 0 at x = a
D. Any of the above could be correct.
dU
Fx = −
dx
At a Fx>0 so dU/dx<0
Monday, October 31, 11
Momentum, Impulse and
Angular Momentum
Monday, October 31, 11
Introduction
• If you watch a football game,
you’ll see collisions, tackles,
many men colliding at once,
maybe just two in an open
area. Are these situations
different?
• Newton told us the forces



d
v
d
d
p


result in acceleration of a
∑ F = ma = m dt = dt (mv) = dt
mass. We will now study two
new points of view—
momentum and impulse.
Monday, October 31, 11
How does momentum relate to mass and velocity?
Understanding
momentum begins with
the simple relationship
that momentum is equal
to mass multiplied by
velocity.
Monday, October 31, 11
Impulse




 dp p 2 − p1
∑ F = ma = dt = t − t
2
1




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
Impulse
Monday, October 31, 11
The meaning of the area of a ∑F versus t graph
The impulse is the area under the ∑F versus t graph
Monday, October 31, 11
Impulse
Although impulse and momentum are both vector quantities, it is
often easier to deal with the components.
Monday, October 31, 11
Compare momentum and kinetic energy
The work-energy theorem
tells us that a change in a particle’s
kinetic energy is due to work
done on the particle
Wtot = K 2 − K1
The impulse-momentum theorem
tells us that a change in a particle’s
momentum is due to an impulse
 

J = p 2 − p1
Monday, October 31, 11
Compare momentum and kinetic energy
The work-energy theorem
focuses on the distance of force
application.
The impulse–momentum
relationship depends on
duration of an impact.
Both rest on the foundation of
Newton’s laws and are integral
principles, relating the motion
at two different times separated
by a finite interval.
Newton’s second law is a
differential principle, relating
forces to the rate of change of
either velocity or momentum.
Monday, October 31, 11
A ball hits a wall
Monday, October 31, 11
A ball hits a wall
(a)
 

J = p 2 − p1
Monday, October 31, 11
A ball hits a wall
(a)
 

J = p 2 − p1
Monday, October 31, 11
A ball hits a wall
(a)
 

J = p 2 − p1
Monday, October 31, 11
A ball hits a wall
(a)
 

J = p 2 − p1
Monday, October 31, 11
A ball hits a wall
Monday, October 31, 11
A ball hits a wall
(b)




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
A ball hits a wall
(b)




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
A ball hits a wall
(b)




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
A ball hits a wall
(b)




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
Duration of an impact
J
Fav =
Δt




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
Duration of an impact
The impulse–momentum relationship depends on the duration of an impact.
Golf ball hitting a steel plate at 150 mph shot at 70,000 fps
J
Fav =
Δt




∑ F(t2 − t1 ) = p 2 − p1 = J
Monday, October 31, 11
Kicking a soccer ball
Monday, October 31, 11
Kicking a soccer ball
Monday, October 31, 11
Kicking a soccer ball
Monday, October 31, 11
Kicking a soccer ball
Monday, October 31, 11
Kicking a soccer ball
Monday, October 31, 11
Like energy, momentum also has conservation rules
Monday, October 31, 11
Like energy, momentum also has conservation rules
Monday, October 31, 11
Like energy, momentum also has conservation rules
Monday, October 31, 11
Conservation of Momentum means Conservation of its components
If the vector sum of the
external forces on the system
is zero, then the components
of momentum are all constant.
Monday, October 31, 11
Recoil of a rifle
Monday, October 31, 11
Recoil of a rifle
Conservation of momentum in the x-direction gives:
Monday, October 31, 11
Recoil of a rifle
Monday, October 31, 11
Recoil of a rifle
Wtot = K 2 − K1
The Kinetic Energy gained is: ∑F·s, where s is the displacement.
During the period of interaction between the bullet and the rifle, ∆t,
the bullet travelled much farther than the rifle, so the bullet acquires
much greater kinetic energy than the rifle.
Work is ∑F·s, so the bullet does more work than the rifle.
The ratio of the two kinetic energies is 600:1, this is equal to the
inverse ratio of their masses (get this from conservation of
momentum).
This always happens, if the rifle was powered by a stiff spring instead
of an explosion we would have got the same result.
Monday, October 31, 11
Objects colliding along a straight line
Monday, October 31, 11
Objects colliding along a straight line
Conservation of momentum in the
x-direction gives:
Monday, October 31, 11
Objects colliding along a straight line
Conservation of momentum in the
x-direction gives:
Monday, October 31, 11
Objects colliding along a straight line
Conservation of momentum in the
x-direction gives:
Same momentum after the collision, so:
Monday, October 31, 11
Objects colliding along a straight line
Conservation of momentum in the
x-direction gives:
Same momentum after the collision, so:
Monday, October 31, 11
Objects colliding along a straight line
Conservation of momentum in the x-direction gives:
Monday, October 31, 11
Objects colliding along a straight line
Conservation of momentum in the x-direction gives:
Same momentum after the collision, so:
Monday, October 31, 11
Objects colliding along a straight line
Monday, October 31, 11
Objects colliding along a straight line
Monday, October 31, 11
Objects colliding along a straight line
Monday, October 31, 11
Now, consider a two-dimensional collision
Monday, October 31, 11
Now, consider a two-dimensional collision
Conservation of momentum in x & y directions gives:
Monday, October 31, 11
Now, consider a two-dimensional collision
Conservation of momentum in x & y directions gives:
Monday, October 31, 11
Now, consider a two-dimensional collision
Monday, October 31, 11
Elastic compared to inelastic
Monday, October 31, 11
Elastic compared to inelastic
Monday, October 31, 11
Elastic compared to inelastic
Monday, October 31, 11
Completely (or nearly) inelastic collisions
Monday, October 31, 11
Completely (or nearly) inelastic collisions
Cars are designed to crumple and absorb as much energy as
possible so the passengers do not need to.
Monday, October 31, 11
Completely (or nearly) inelastic collisions
Cars are designed to crumple and absorb as much energy as
possible so the passengers do not need to.
Monday, October 31, 11
The ballistic pendulum
A ballistic pendulum is a system
for measuring the speed of a
bullet.
This can be demonstrated with a 4
× 4 block and a .22 caliber rifle.
The bullet of mass mB, is fired into
a wood block of mass mw,
suspended like a pendulum and
makes a completely inelastic
collision with it.
After the bullet’s impact the block
swings up to a maximum height, y.
If we know the values of y, mB,
and mw, what is the initial speed of
the bullet vi?
Monday, October 31, 11
The ballistic pendulum
Monday, October 31, 11
The ballistic pendulum
Stage 1) Conservation of momentum
Monday, October 31, 11
The ballistic pendulum
Stage 1) Conservation of momentum
Stage 2) Conservation of energy
Monday, October 31, 11
The ballistic pendulum
Stage 1) Conservation of momentum gives:
Monday, October 31, 11
The ballistic pendulum
Stage 1) Conservation of momentum gives:
Stage 2) Conservation of energy gives:
Final potential energy
Initial kinetic energy
Monday, October 31, 11
The ballistic pendulum
Stage 1) Conservation of momentum gives:
Stage 2) Conservation of energy gives:
Final potential energy
Initial kinetic energy
Monday, October 31, 11
Classifying Collisions
Monday, October 31, 11
A possible simple model for automobile accidents
Monday, October 31, 11
A possible simple model for automobile accidents
Monday, October 31, 11
A possible simple model for automobile accidents
Components of momentum before the collision
An Inelastic Collision
Momentum Conserved
Kinetic Energy NOT conserved
Monday, October 31, 11
A possible simple model for automobile accidents
Components of momentum before the collision
An Inelastic Collision
Momentum Conserved
Kinetic Energy NOT conserved
Monday, October 31, 11
A possible simple model for automobile accidents
Components of momentum before the collision
An Inelastic Collision
Momentum Conserved
Kinetic Energy NOT conserved
Monday, October 31, 11
A possible simple model for automobile accidents
Components of momentum before the collision
An Inelastic Collision
Momentum Conserved
Kinetic Energy NOT conserved
By conservation of momentum during the collision
Monday, October 31, 11
Elastic collisions
Billiard balls are a very
good example of objects
that collide elastically.
Monday, October 31, 11
Q8.1
A ball (mass 0.40 kg) is
initially moving to the
left at 30 m/s. After
hitting the wall, the ball
is moving to the right at
20 m/s. What is the
impulse of the net force
on the ball during its
collision with the wall?
A. 20 kg • m/s to the right
B. 20 kg • m/s to the left
C. 4.0 kg • m/s to the right
D. 4.0 kg • m/s to the left
E. none of the above
Monday, October 31, 11
A8.1
A ball (mass 0.40 kg) is
initially moving to the
left at 30 m/s. After
hitting the wall, the ball
is moving to the right at
20 m/s. What is the
impulse of the net force
on the ball during its
collision with the wall?
A. 20 kg • m/s to the right
B. 20 kg • m/s to the left
C. 4.0 kg • m/s to the right
 

E. none of the above
J = p 2 − p1
= 0.40 × 20 − 0.40 × −30 = 20
D. 4.0 kg • m/s to the left
Monday, October 31, 11
Q8.2
You are testing a new car using crash test dummies. Consider two
ways to slow the car from 90 km/h (56 mi/h) to a complete stop:
(i) You let the car slam into a wall, bringing it to a sudden stop.
(ii) You let the car plow into a giant tub of gelatin so that it comes to
a gradual halt.
In which case is there a greater impulse of the net force on the car?
A. in case (i)
B. in case (ii)
C. The impulse is the same in both cases.
D. not enough information given to decide
Monday, October 31, 11
A8.2
You are testing a new car using crash test dummies. Consider two
ways to slow the car from 90 km/h (56 mi/h) to a complete stop:
(i) You let the car slam into a wall, bringing it to a sudden stop.
(ii) You let the car plow into a giant tub of gelatin so that it comes to
a gradual halt.
In which case is there a greater impulse of the net force on the car?
A. in case (i)
B. in case (ii)
C. The impulse is the same in both cases.
D. not enough information given to decide
 

J = p 2 − p1 , in both cases the car comes to a complete stop so both p1 & p 2 are the same
Monday, October 31, 11
Q8.3
A 3.00-kg rifle fires a 0.00500-kg bullet at a speed of 300
m/s. Which force is greater in magnitude:
(i) the force that the rifle exerts on the bullet; or
(ii) the force that the bullet exerts on the rifle?
A. the force that the rifle exerts on the bullet
B. the force that the bullet exerts on the rifle
C. both forces have the same magnitude
D. not enough information given to decide
Monday, October 31, 11
A8.3
A 3.00-kg rifle fires a 0.00500-kg bullet at a speed of 300
m/s. Which force is greater in magnitude:
(i) the force that the rifle exerts on the bullet; or
(ii) the force that the bullet exerts on the rifle?
A. the force that the rifle exerts on the bullet
B. the force that the bullet exerts on the rifle
C. both forces have the same magnitude
D. not enough information given to decide
Newton’s third law, for every action there
is an equal and opposite reaction
Monday, October 31, 11
Q8.4
Two objects with different
masses collide and stick to
each other. Compared to
before the collision, the
system of two objects after
the collision has
A
B
A. the same total momentum and the same total kinetic energy.
B. the same total momentum but less total kinetic energy.
C. less total momentum but the same total kinetic energy.
D. less total momentum and less total kinetic energy.
E. not enough information given to decide
Monday, October 31, 11
A8.4
Two objects with different
masses collide and stick to
each other. Compared to
before the collision, the
system of two objects after
the collision has
A
B
A. the same total momentum and the same total kinetic energy.
B. the same total momentum but less total kinetic energy.
C. less total momentum but the same total kinetic energy.
D. less total momentum and less total kinetic energy.
E. not enough information given to decide
Monday, October 31, 11
An Inelastic Collision
Momentum Conserved
Kinetic Energy NOT conserved
Q8.5
Two objects with different
masses collide and bounce
off each other. Compared to
before the collision, the
system of two objects after
the collision has
A
B
A. the same total momentum and the same total kinetic energy.
B. the same total momentum but less total kinetic energy.
C. less total momentum but the same total kinetic energy.
D. less total momentum and less total kinetic energy.
E. not enough information given to decide
Monday, October 31, 11
A8.5
Two objects with different
masses collide and bounce
off each other. Compared to
before the collision, the
system of two objects after
the collision has
A
B
A. the same total momentum and the same total kinetic energy.
B. the same total momentum but less total kinetic energy.
C. less total momentum but the same total kinetic energy.
D. less total momentum and less total kinetic energy.
E. not enough information given to decide
Do not know if elastic or inelastic collision
Monday, October 31, 11
Q8.6
Block A has mass 1.00 kg and block B has mass 3.00 kg. The
blocks collide and stick together on a level, frictionless surface.
After the collision, the kinetic energy (KE) of block A is
A. 1/9 the KE of block B.
B. 1/3 the KE of block B.
C. 3 times the KE of block B.
D. 9 times the KE of block B.
E. the same as the KE of block B.
Monday, October 31, 11
A8.6
Block A has mass 1.00 kg and block B has mass 3.00 kg. The
blocks collide and stick together on a level, frictionless surface.
After the collision, the kinetic energy (KE) of block A is
A. 1/9 the KE of block B.
B. 1/3 the KE of block B.
C. 3 times the KE of block B.
D. 9 times the KE of block B.
E. the same as the KE of block B.
The ratio of the two kinetic energies is equal to the inverse ratio of
their masses.
Monday, October 31, 11
Q8.7
Block A on the left has mass 1.00 kg. Block B on the right
has mass 3.00 kg. The blocks are forced together,
compressing the spring. Then the system is released from rest
on a level, frictionless surface. After the blocks are released,
the kinetic energy (KE) of block A is
A. 1/9 the KE of block B.
C. 3 times the KE of block B.
E. the same as the KE of block B.
Monday, October 31, 11
B. 1/3 the KE of block B.
D. 9 times the KE of block B.
A8.7
Block A on the left has mass 1.00 kg. Block B on the right
has mass 3.00 kg. The blocks are forced together,
compressing the spring. Then the system is released from rest
on a level, frictionless surface. After the blocks are released,
the kinetic energy (KE) of block A is
A. 1/9 the KE of block B.
C. 3 times the KE of block B.
E. the same as the KE of block B.
Conservation of momentum
Monday, October 31, 11
B. 1/3 the KE of block B.
D. 9 times the KE of block B.
Q8.8
An open cart is rolling to the left on a
horizontal surface. A package slides down a
chute and lands in the cart. Which
quantities have the same value just before
and just after the package lands in the cart?
A. the horizontal component of total momentum
B. the vertical component of total momentum
C. the total kinetic energy
D. two of A., B., and C.
E. all of A., B., and C.
Monday, October 31, 11
A8.8
An open cart is rolling to the left on a
horizontal surface. A package slides down a
chute and lands in the cart. Which
quantities have the same value just before
and just after the package lands in the cart?
A. the horizontal component of total momentum
B. the vertical component of total momentum
C. the total kinetic energy
D. two of A., B., and C.
E. all of A., B., and C.
Monday, October 31, 11
Angular Momentum
Every rotational
quantity we have
encountered has a
translational analog.
The analog of
momentum, p=mv, is
angular momentum,
Monday, October 31, 11
Angular Momentum
Monday, October 31, 11
Conservation of Angular Momentum
The conservation of angular momentum follows
directly from

 dL
As ∑ τ =
dt



dL
If ∑ τ = 0 then
= 0, and so L remains constant
dt
Monday, October 31, 11
A Falling Cat
Monday, October 31, 11
The professor as ballerina?
Angular momentum is conserved.
Monday, October 31, 11
This is how a car’s clutch works!
Monday, October 31, 11
This is how a car’s clutch works!
This is analogous to a completely inelastic collision
Monday, October 31, 11
Q10.11
A spinning figure
skater pulls his arms
in as he rotates on the
ice. As he pulls his
arms in, what happens
to his angular
momentum L and
kinetic energy K?
A. L and K both increase.
B. L stays the same, K increases.
C. L increases, K stays the same.
D. L and K both stay the same.
Monday, October 31, 11
A10.11
A spinning figure
skater pulls his arms
in as he rotates on the
ice. As he pulls his
arms in, what happens
to his angular
momentum L and
kinetic energy K?
A. L and K both increase.
B. L stays the same, K increases.
C. L increases, K stays the same.
D. L and K both stay the same.
Monday, October 31, 11