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Chapter 7
Work and
Kinetic Energy
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Work Done by a Constant Force
The definition of work, when the force is parallel to
the displacement:
SI unit: newton-meter (N·m) = joule, J
2
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Convenient notation: the dot product
The work can also be written as the dot product
of the force and the displacement:
vector “dot” operation: project
one vector onto the other
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Play Ball!
In a baseball game, the
catcher stops a 90-mph
a) catcher has done positive work
pitch. What can you say
b) catcher has done negative work
about the work done by the
c) catcher has done zero work
catcher on the ball?
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Play Ball!
In a baseball game, the
catcher stops a 90-mph
a) catcher has done positive work
pitch. What can you say
b) catcher has done negative work
about the work done by the
c) catcher has done zero work
catcher on the ball?
The force exerted by the catcher is opposite in direction to the
displacement of the ball, so the work is negative. Or using the
definition of work (W = F (Δr)cos  ), because  = 180º, then W <
0. Note that the work done on the ball is negative, and its speed
decreases.
Follow-up: What about the work done by the ball on the catcher?
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Tension and Work
A ball tied to a string is
a) tension does no work at all
being whirled around in
b) tension does negative work
a circle with constant
c) tension does positive work
speed. What can you
say about the work done
by tension?
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Tension and Work
A ball tied to a string is
a) tension does no work at all
being whirled around in
b) tension does negative work
a circle with constant
c) tension does positive work
speed. What can you
say about the work done
by tension?
No work is done because the force
acts in a perpendicular direction to the
displacement.
Or using the
definition of work (W = F (Δr)cos  ),
T
 = 90º, then W = 0.
v
Follow-up: Is there a force in the direction of the velocity?
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Work by gravity
Fg
A ball of mass m drops a distance h. What is the
total work done on the ball by gravity?
a
W = Fd = Fgx h
h
W = mgh
Path doesn’t matter when asking “how
much work did gravity do?” Only the
change in height!
A ball of mass m rolls down a ramp of height h at an
angle of 45o. What is the total work done on the ball by gravity?
a
Fgx = Fg sinθ
N
h = L sinθ
Fg
W = Fd = Fgx L = (Fg sinθ) (h / sinθ)
h
W = Fg h = mgh
θ
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Motion and energy
When positive work is done on an object, its speed
increases; when negative work is done, its speed
decreases.
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Kinetic Energy
As a useful word for the quantity of work we have
done on an object, thereby giving it motion, we
define the kinetic energy:
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Work-Energy Theorem
Work-Energy Theorem: The total work done on an
object is equal to its change in kinetic energy.
(True for rigid bodies that remain intact)
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Kinetic Energy I
By what factor does the
a) no change at all
kinetic energy of a car
b) factor of 3
change when its speed is
c) factor of 6
tripled?
d) factor of 9
e) factor of 12
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Kinetic Energy I
By what factor does the
a) no change at all
kinetic energy of a car
b) factor of 3
change when its speed is
c) factor of 6
tripled?
d) factor of 9
e) factor of 12
Because the kinetic energy is
mv2, if the speed increases by
a factor of 3, then the KE will increase by a factor of 9.
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Work Done by a Variable Force
We can interpret the work done graphically:
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Work Done by a Variable Force
We can then approximate a continuously varying
force by a succession of constant values.
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Work Done by a Variable Force
The force needed to stretch a spring an amount x is
F = kx.
Therefore, the work
done in stretching the
spring is
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Chapter 8
Potential
Energy and
Conservation
of Energy
Recall
Work is the force directed along a displacement:
The total work done on an object is equal to its
change in kinetic energy:
Lets wrap up our discussion of work and kinetic
energy...
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Force and Work
a) one force
A box is being pulled up a rough
b) two forces
incline by a rope connected to a
c) three forces
pulley. How many forces are
doing work on the box?
d) four forces
e) no forces are doing work
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Force and Work
a) one force
A box is being pulled up a rough
b) two forces
incline by a rope connected to a
c) three forces
pulley. How many forces are
doing work on the box?
Any force not perpendicular
to the motion will do work:
N does no work
d) four forces
e) no forces are doing work
N
T
T does positive work
f
f does negative work
mg does negative work
mg
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Free Fall I
Two stones, one twice the mass of
the other, are dropped from a cliff.
Just before hitting the ground,
what is the kinetic energy of the
heavy stone compared to the light
one?
a) quarter as much
b) half as much
c) the same
d) twice as much
e) four times as much
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Free Fall I
Two stones, one twice the mass of
the other, are dropped from a cliff.
Just before hitting the ground,
what is the kinetic energy of the
heavy stone compared to the light
one?
a) quarter as much
b) half as much
c) the same
d) twice as much
e) four times as much
Consider the work done by gravity to make the stone fall
distance d:
KE = Wnet = F d cos 
KE = mg d
Thus, the stone with the greater mass has the greater
KE, which is twice as big for the heavy stone.
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Power
Power is a measure of the rate at which
work is done:
SI unit: J/s = watt, W
1 horsepower = 1 hp = 746 W
if work is energy, then you would think of “energy flow”
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Power
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Power
If an object is moving at a constant speed in the
face of friction, gravity, air resistance, and so forth,
the power exerted by the driving force can be
written:
Question: what is the total work per unit
time done on this object (by all forces)?
This expression, P = Fv, gives the instantaneous power
applied, even if the object is not moving at constant speed
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Electric Bill
a) energy
When you pay the electric
b) power
company by the kilowatt-hour,
c) current
what are you actually paying for?
d) voltage
e) none of the above
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Electric Bill
a) energy
When you pay the electric
b) power
company by the kilowatt-hour,
c) current
what are you actually paying for?
d) voltage
e) none of the above
We have defined: Power = energy / time
So we see that: Energy = power × time
This means that the unit of power × time
(watt-hour) is a unit of energy !!
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A block rests on a horizontal frictionless surface. A string is attached
to the block, and is pulled with a force of 45.0 N at an angle above the
horizontal, as shown in the figure. After the block is pulled through a
distance of 1.50 m, its speed is 2.60 m/s, and 50.0 J of work has been
done on it. (a) What is the angle (b) What is the mass of the block?
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The pulley system shown is used
to lift a 52 kg crate. Note that one
chain connects the upper pulley
to the ceiling and a second chain
connects the lower pulley to the
crate. Assuming the masses of
the chains, pulleys, and ropes
are negligible, determine
(a) the force F required to lift the
crate with constant speed, and
(b) the tension in two chains
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(a) the force F required to lift the crate
with constant speed, and
(b) the tension in two chains
(a) constant velocity, a=0, so net
force =0.
2T - (52kg)(9.8m/s2) = 0
T = 250 N
F = -250 Ny
(b) massless pully!
Upper:
Tch - 2Trope = 0
Tch = 500 N
Mechanical
Advantage!
Lower:
Tch -2Trope =0 Tch = 500 N
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What about work?
(a) how much power is applied to the
box by the chain?
(b) how much power is applied on the
rope by the applied force?
Trope = 250 N
Tchain = 500 N
F = -250 Ny
(a) P = Fv = 500 N * vbox
(b) P = Fv = 250 N * vhand
vhand = 2 vbox!
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You can explain what is about to
happen, in terms of force (elastic
band is about to pull on the rock,
accelerating it toward our camera
lens)...
...or using work/energy: work has
been done on the elastic band, and it
now “contains” energy.
Energy wants to be “free” - in fact, physics can be
described in terms of the rules that govern stored energy
Also: energy really is a “thing”. E = mc2 relates mass to energy,
and stored energy counts... it’s not just an accounting rule!
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Work by gravity
Fg
A ball of mass m drops a distance h. What is the
total work done on the ball by gravity?
a
W = Fd = Fgx h
h
W = mgh
Path doesn’t matter when asking “how
much work did gravity do?” Only the
change in height!
A ball of mass m rolls down a ramp of height h at an
angle of 45o. What is the total work done on the ball by gravity?
a
Fgx = Fg sinθ
N
h = L sinθ
Fg
W = Fd = Fgx L = (Fg sinθ) (h / sinθ)
h
W = Fg h = mgh
θ
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Application: ball on a track
how high must I place the ball so
that it can complete a loop?
Condition: Fcp > mg at top of loop
Fcp = mv2/r = mg
v2 = gr
KE = mv2 / 2 = mgr/2
Gravity must provide
this energy
Wg = mgh = KE
h = r/2 above the top of the loop!
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Conservative and Nonconservative Forces
Conservative force:
- the work it does is stored in the form of energy that
can be released at a later time
- the work done by a conservative force moving an
object around a closed path is zero
- Force depends upon position only
Example of a conservative force: gravity
Example of a nonconservative force: friction
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Work done by gravity on a closed path is zero
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Work done by friction on a closed path is not zero
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The work done by a conservative force is zero on
any closed path
Go A-B on path 1, the back B-A.
Wt = W1 + -W1
Go A-B on path 1, the B-A on
path 2.
Wt = W1 + -W2
So the work must be reversible (opposite when taking the
same path) AND path independent (same amount of work
for any two different paths connecting two points)
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Potential Energy
If we pick up a ball and put it on the shelf, we have
done work on the ball. We can get that energy back if
the ball falls back off the shelf (gravity does positive
work on the ball, “releasing” the work that we put in
before).
Until that happens, we say the energy is stored as
potential energy.
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Potential Energy
Consider the process
in which the book goes
from h=0 to h=0.50 m
Work done by gravity:
W = - (mg)h = -13.5 J
For the book to go up against gravity, another force
must be applied to overcome the weight. This other
force did a (minimum) work of 13.5 J
If I lft the book steadily, the “external force” is provided by
my hand with F~mg, work done by me: W=(mg)h = 13.5 J
The book’s potential energy changed by: 13.5 J
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Potential Energy
The work done against a conservative force is stored
in the form of (potential) energy that can be released
at a later time.
Note the minus sign:
•positive Wc (work by the conservative force) is negative
potential energy (energy is released)
•negative Wc is positive potential energy (another force as
done work against the conservative force)
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Gravitational Potential Energy
Q: What does “UG = 0” mean?
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Work Done by a Variable Force
on the spring
The force needed to stretch a
spring an amount x is F = kx.
Therefore, the work done in
stretching (or compressing)
the spring is
with positive work applied
leading to a positive change
in potential: W = Uf - Ui
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Potential energy in a spring
The corresponding conservative force is the force of the spring
acting on the hand: positive work by the spring releases
potential energy Wc = - ΔU
So, taking U=0 at x=0:
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Up the Hill
Two paths lead to the top of a big
hill. One is steep and direct, while
the other is twice as long but less
steep. How much more potential
energy would you gain if you take
the longer path?
a) the same
b) twice as much
c) four times as much
d) half as much
e) you gain no PE in either
case
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Up the Hill
Two paths lead to the top of a big
hill. One is steep and direct, while
the other is twice as long but less
steep. How much more potential
energy would you gain if you take
the longer path?
a) the same
b) twice as much
c) four times as much
d) half as much
e) you gain no PE in either
case
Because your vertical position (height) changes by
the same amount in each case, the gain in potential
energy is the same.
Follow-up: How much more work do you do in taking the steeper path?
Follow-up: Which path would you rather take? Why?
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Sign of the Energy
Is it possible for the
a) yes
gravitational potential
b) no
energy of an object to
be negative?
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Sign of the Energy
Is it possible for the
a) yes
gravitational potential
b) no
energy of an object to
be negative?
Gravitational PE is mgh, where height h is measured relative to some
arbitrary reference level where PE = 0. For example, a book on a table
has positive PE if the zero reference level is chosen to be the floor.
However, if the ceiling is the zero level, then the book has negative PE
on the table. Only differences (or changes) in PE have any physical
meaning.
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KE and PE
You and your friend both solve a
problem involving a skier going down a
slope, starting from rest. The two of
you have chosen different levels for y
= 0 in this problem. Which of the
following quantities will you and your
friend agree on?
A) skier’s PE
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
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KE and PE
You and your friend both solve a
problem involving a skier going down a
slope, starting from rest. The two of
you have chosen different levels for y
= 0 in this problem. Which of the
following quantities will you and your
friend agree on?
A) skier’s PE
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
The gravitational PE depends upon the reference level, but the
difference  PE does not! The work done by gravity must be
the same in the two solutions, so PE and KE should be the
same.
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Mechanical Energy
Consider the total amount of work done on a body by
the conservative and the non-conservative forces. This
is the change in kinetic energy (work-energy theorem)
It is useful to define the mechanical energy:
Then:
The work done by all non-conservative forces is the
change in the mechanical energy of a body
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Conservation of Mechanical Energy
The work done by all non-conservative forces is the
change in the mechanical energy of a body
If there are only conservative forces doing work during
a process, we find:
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Work-Energy Theorem vs. Conservation of
Energy?
Work-Energy Theorem
total work done (by both conservative and nonconservative forces) = change in kinetic energy
Conservation of mechanical energy
total work done by non-conservative forces
= change in mechanical energy
These two are completely equivalent. The
difference is only how to treat conservative forces.
Do NOT use both potential energy AND work by
the conservative force... that’s double-counting!
In general, energy conservation makes
kinematics problems much easier to solve...
54
Runaway Truck
A truck, initially at rest, rolls
a) half the height
down a frictionless hill and
attains a speed of 20 m/s at the
bottom. To achieve a speed of
40 m/s at the bottom, how many
times higher must the hill be?
b) the same height
c) < 2 times the height
d) twice the height
e) four times the height
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Runaway Truck
A truck, initially at
rest, rolls down a
frictionless hill and
attains a speed of 20
m/s at the bottom.
UseTo
energy
conservation:
achieve
a speed
 initial energy: Ei = PEg = mgH
of 40 m/s at the 2
 final energy: Ef = KE = mv
bottom,
how many
Conservation
of Energy:
2
E
=
mgH
=
E
=
mv
i
f
times
higher
must
therefore: gH = v2
the
hill
be?
So if v doubles, H quadruples!
a) half the height
b) the same height
c) < 2 times the height
d) twice the height
e) four times the height
56
Cart on a Hill
A cart starting from rest rolls down a hill and
at the bottom has a speed of 4 m/s. If the
cart were given an initial push, so its initial
speed at the top of the hill was 3 m/s, what
would be its speed at the bottom?
a) 4 m/s
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
57
Cart on a Hill
A cart starting from rest rolls down a hill and
at the bottom has a speed of 4 m/s. If the
cart were given an initial push, so its initial
speed at the top of the hill was 3 m/s, what
would be its speed at the bottom?
a) 4 m/s
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
When starting from rest, the
cart’s PE is changed into KE:
 PE =  KE = m(4)2
When starting from 3 m/s, the
final KE is:
KEf = KEi +  KE
= m(3)2 + m(4)2
= m(25)
= m(5)2
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Potential Energy Curves
The curve of a hill or a roller coaster is itself
essentially a plot of the gravitational potential
energy:
Q: at what point is speed maximized?
Q: where might apparent weight be minimized?
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Potential Energy for a Spring
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Potential Energy Curves and
Equipotentials
Contour maps are also a form of potential energy curve:
Each contour is an equal
height, and so an
“equipotential” for
gravitational potential
energy
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Question 8.5 Springs and Gravity
A mass attached to a vertical
spring causes the spring to
stretch and the mass to move
downwards. What can you
say about the spring’s
potential energy (PEs) and the
gravitational potential energy
(PEg) of the mass?
a) both PEs and PEg decrease
b) PEs increases and PEg decreases
c) both PEs and PEg increase
d) PEs decreases and PEg increases
e) PEs increases and PEg is constant
62
Question 8.5 Springs and Gravity
A mass attached to a vertical
spring causes the spring to
stretch and the mass to move
downwards. What can you
say about the spring’s
potential energy (PEs) and the
gravitational potential energy
(PEg) of the mass?
a) both PEs and PEg decrease
b) PEs increases and PEg decreases
c) both PEs and PEg increase
d) PEs decreases and PEg increases
e) PEs increases and PEg is constant
The spring is stretched, so its elastic PE increases,
because PEs =
kx2. The mass moves down to a
lower position, so its gravitational PE decreases,
because PEg = mgh.
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8-4 Work Done by Nonconservative Forces
In this example, the
nonconservative force
is water resistance:
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