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Conservation of Energy
and Energy Diagrams
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 11
The stretch of a spring and
the force that caused it
The force applied to
an ideal spring will be
proportional to its
stretch.
The graph of force on
the y axis versus
stretch on the x axis
will yield a slope of k,
the spring constant.
Monday, October 24, 11
Stepping on a scale
Whether you like the result or not, stepping on a
scale is an excellent example of applied force and
the work being done to compress that spring.
Monday, October 24, 11
Motion with a varying force
Monday, October 24, 11
Motion with a varying force
Monday, October 24, 11
Motion on a curved path
If you watch a child on a swing set, you can also
consider the motion of a particle along a curved
path.
Monday, October 24, 11
Watt about power?
Once work is calculated, dividing
by the time that passed
determines power.
The pun is credit to James Watt.
(You will see that scientists of
that era often were privileged to
leave their names on the topic of
their efforts.)
Also note the popular culture
power unit of horsepower.
The energy you use may be noted
from the meter the electric
company probably installed to
measure your consumption of
energy in kilowatt-hours.
Monday, October 24, 11
Power
Monday, October 24, 11
Power
Monday, October 24, 11
Power
•
•
Same mass...
Both reach 60 mph...
Same final kinetic
energy, but
different times mean
different powers.
Monday, October 24, 11
An example you might do
if the elevator is out
It’s interesting how a lighter stair climber and heavier stair
climber can expend the same power by using different
amounts of time.
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 11
Gravitational potential energy, U
Energy associated with position
The path of the basketball can demonstrate many
principles of potential and kinetic energy.
Monday, October 24, 11
Can express this as a function of gravitational potential energy, U
So the work done by the gravitational force during the
displacement from y1 to y2 is:
Monday, October 24, 11
The sign of ΔU
Monday, October 24, 11
The sign of ΔU
Monday, October 24, 11
Total Mechanical Energy of the System
Suppose that a body is falling freely with no air resistance, then the
body’s weight is the only force acting. If its speed at point y1 is v1,
and its speed at y2 is v2, then the work-energy theorem gives us
Wtot = ΔK = K 2 − K1
Wtot = Wgrav = −ΔU grav = U grav,1 − U grav,2
so
ΔK = −ΔU grav or K 2 − K1 = U grav,1 − U grav,2
Which we can rewrite as:
Monday, October 24, 11
Total Mechanical Energy of the System
The total mechanical energy of the system is called E
E = K + U grav
A quantity that always has the same value is called a conserved
quantity. When only the force of gravity does work, the total
mechanical energy is conserved.
Monday, October 24, 11
Athletes and the conservation of energy
Notice how potential and kinetic energy interchange as the
athlete jumps.
Monday, October 24, 11
Zero Height
Monday, October 24, 11
Athletes and energy
Notice how velocity changes as forms of energy
interchange.
Monday, October 24, 11
Height of a Baseball from Energy Conservation
You throw a 0.145 kg baseball straight up in the air, giving it an initial
upward velocity of magnitude 20.0 m/s. Find how high it goes,
ignoring air resistance.
Monday, October 24, 11
Height of a Baseball from Energy Conservation
You throw a 0.145 kg baseball straight up in the air, giving it an initial
upward velocity of magnitude 20.0 m/s. Find how high it goes,
ignoring air resistance.
y2
y1=0
Monday, October 24, 11
Height of a Baseball from Energy Conservation
You throw a 0.145 kg baseball straight up in the air, giving it an initial
upward velocity of magnitude 20.0 m/s. Find how high it goes,
ignoring air resistance.
y2
y1=0
Monday, October 24, 11
Height of a Baseball from Energy Conservation
You throw a 0.145 kg baseball straight up in the air, giving it an initial
upward velocity of magnitude 20.0 m/s. Find how high it goes,
ignoring air resistance.
y2
y1=0
Monday, October 24, 11
Height of a Baseball from Energy Conservation
You throw a 0.145 kg baseball straight up in the air, giving it an initial
upward velocity of magnitude 20.0 m/s. Find how high it goes,
ignoring air resistance.
y1 = 0 ⇒ U grav,1 = mgy1 = 0
y2
1 2
v2 = 0 ⇒ K 2 = mv2 = 0
2
K1 = U grav,2
y1=0
Monday, October 24, 11
Height of a Baseball from Energy Conservation
y2
y1 = 0 ⇒ U grav,1 = mgy1 = 0
1 2
v2 = 0 ⇒ K 2 = mv2 = 0
2
K1 = U grav,2
1 2 1
K1 = mv1 = × 0.145kg × 20.0m / s = 29.0J
2
2
U grav,2
29.0J
U grav,2 = mgy2 ⇒ y2 =
=
= 20.4m
2
mg
0.145kg × 9.80m / s
Monday, October 24, 11
y1=0
Forces other than gravity doing work
We already know that:
If we have forces additional to gravity at work then we have:
Wtot = Wother + Wgrav = K 2 − K1 = ΔK
As we also know that:
Wgrav = U grav,1 − U grav,2 = −ΔU grav
So substituting and rearranging:
K1 + U grav,1 + Wother = K 2 + U grav,2
Substituting the appropriate energy expressions we get
1 2
1 2
mv1 + mgy1 + Wother = mv2 + mgy2
2
2
Monday, October 24, 11
Forces other than gravity doing work
1 2
1 2
mv1 + mgy1 + Wother = mv2 + mgy2
2
2
When we are considering a problem using mechanical energy it is
often best to use the energy approach if we have varying forces, and/
or motion along a curved path.
Monday, October 24, 11
Forces other than gravity doing work
You throw a 0.145 kg baseball
straight up in the air, and while
you are throwing the ball your
hand moves up 0.50 m. The
ball leaves your hand with an
initial upward velocity of
magnitude 20.0 m/s. Ignore air
resistance.
a) assuming your hand exerts a
constant upward force on the
ball, find the magnitude of that
force.
b) Find the speed of the ball at
a point 15 m above the point
that it leaves your hand.
Monday, October 24, 11
Forces other than gravity doing work
a)
y1 = −0.50m ⇒ U grav,1 = 0.145kg × 9.80ms / s 2 × −0.50m = −0.71J
v1 = 0 ⇒ K1 = 0
y2 = 0 ⇒ U grav,2 = 0
1
2
v2 = 20m / s ⇒ K 2 = 0.145kg × (20.0m / s) = 29.0J
2
Monday, October 24, 11
Forces other than gravity doing work
a)
y1 = −0.50m ⇒ U grav,1 = 0.145kg × 9.80ms / s 2 × −0.50m = −0.71J
v1 = 0 ⇒ K1 = 0
y2 = 0 ⇒ U grav,2 = 0
1
2
v2 = 20m / s ⇒ K 2 = 0.145kg × (20.0m / s) = 29.0J
2
Wother = (K 2 − K1 ) + (U grav,2 − U grav,1 )
= (29.0J − 0) + (0 − (−0.71J )) = 29.7J
= F(y2 − y1 )
Wother
29.7J
F=
=
= 59N
(y2 − y1 ) 0.50m
Monday, October 24, 11
Forces other than gravity doing work
b) Note that between points 2 and 3 the force of your hand no longer
acts so mechanical energy is conserved as Wother=0
K 2 + U grav,2 = K 3 + U grav, 3
U grav, 3 = mgy3 = 0.145kg × 9.80m / s × 15.0m = 21.3J
2
K 3 = (K 2 + U grav,2 ) − U grav, 3
= (29.0J + 0) − 21.3J = 7.7J
1 2
2K 3
2 × 7.7J
= mv3y ⇒ v3y = ±
=±
= ±10m / s
2
m
0.145kg
Note that the ball passes point 3 twice, on the way up, and on the way
down.
Monday, October 24, 11
Work and energy along a curved path
Notice the same expression for
gravitational potential energy is
the same along either a curved

or straight path.
ii j = 0
 



W = FiΔs = −mg ji(Δx i + Δy j) = −mgΔy
Why the minus?
So the total work is:
Monday, October 24, 11
Consider projectile motion using energetics
Monday, October 24, 11
What’s the speed in a vertical circle?
Monday, October 24, 11
What’s the speed in a vertical circle?
Monday, October 24, 11
What’s the speed in a vertical circle?
Monday, October 24, 11
What’s the speed in a vertical circle?
Monday, October 24, 11
Speed in a vertical circle with friction
Suppose that the ramp was not
frictionless and his speed at the
bottom was only 6.00m/s. What
work was done by the frictional
force acting on him?
Monday, October 24, 11
Speed in a vertical circle with friction
Suppose that the ramp was not
frictionless and his speed at the
bottom was only 6.00m/s. What
work was done by the frictional
force acting on him?
Wother = (K 2 − K1 ) + (U grav,2 − U grav,1 )
Monday, October 24, 11
Speed in a vertical circle with friction
Suppose that the ramp was not
frictionless and his speed at the
bottom was only 6.00m/s. What
work was done by the frictional
force acting on him?
Wother = (K 2 − K1 ) + (U grav,2 − U grav,1 )
Monday, October 24, 11
Elastic Potential Energy
Monday, October 24, 11
Elastic Potential Energy
For an ideal spring we know that the magnitude of the force exerted
to stretch it a distance x is:
F = kx
Where k is the force constant of the spring.
X
X
1 2
W = ∫ Fx dx = ∫ kx dx = kX
2
0
0
Monday, October 24, 11
Elastic Potential Energy
So the work done on a spring to move it from elongation x1 to x2 is
From Newton’s third law, the work done by a spring from elongation
x1 to x2 is
“el” stands for elastic
Monday, October 24, 11
Elastic Potential Energy
Monday, October 24, 11
Work and energy in the motion of a mass on a spring
Monday, October 24, 11
Work and energy in the motion of a mass on a spring
Monday, October 24, 11
Monday, October 24, 11
Monday, October 24, 11
“Bungee jumping” requires we look at two potential energies
In this case we need to consider
gravitational potential energy, kinetic
energy, and elastic potential energy.
Monday, October 24, 11
Motion with elastic potential energy
Monday, October 24, 11
Motion with elastic potential energy
K 2 − K1 = U1 − U 2 ⇒ K 2 = K1 + U1 − U 2
K 2 = K1 + U1 − U 2 = 0 + 0.0250J − 0.0160J = 0.0090J
2K 2
2 × 0.0090J
v2 x = ±
=±
= ±0.30m / s
m
0.200kg
Monday, October 24, 11
Bring together two potential energies and friction
The fall of the elevator is stopped
by a spring and by a constant
friction force.
Monday, October 24, 11
Bring together two potential energies and friction
The elevator’s initial speed, v1, is 4.00m/s so the initial kinetic
energy, K1, is
Monday, October 24, 11
Bring together two potential energies and friction
Monday, October 24, 11
Conservative forces
The work by a conservative force like gravity does
not depend on the path your hiking team chooses,
only how high you climb.
Monday, October 24, 11
Conservative forces
Monday, October 24, 11
Does Friction Depend
on the Path Taken?
Monday, October 24, 11
Friction does depend on the path taken
Nonconservative frictional force changes with path.
Monday, October 24, 11
Friction does depend on the path taken
Nonconservative frictional force changes with path.
The magnitude of the frictional force is:
fk = µ k n = µ k mg
Monday, October 24, 11
Friction does depend on the path taken
Nonconservative frictional force changes with path.
The magnitude of the frictional force is:
fk = µ k n = µ k mg
Monday, October 24, 11
The law of Conservation of Energy
Monday, October 24, 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 24, 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 24, 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 24, 11
The potential energy curve for motion of a particle
dU
Fx = −
dx
Monday, October 24, 11
Energy diagrams give us insight
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 11
Summary
Monday, October 24, 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 24, 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.
Monday, October 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 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 24, 11
x
a
Monday, October 24, 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 24, 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 24, 11
x
a
Monday, October 24, 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 24, 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 24, 11