Chapter 5
Work, Energy, and Power
Work

W = F x


This equation
applies when the
force is in the
same direction as
the displacement
F and x are in
the same direction
A linebacker pushes against the
blocker but the blocker does not
move. The work is:
0%
0%
No
t
en
ou
gh
in
fo
Ze
ro
0%
e
0%
tiv
4.
Ne
ga
3.
ve
2.
Positive
Negative
Zero
Not enough info
Po
sit
i
1.
When doing a bench press, you
gradually lower the bar down to
your chest. The work done by
you is:
0%
0%
No
t
en
ou
gh
in
fo
Ze
ro
0%
e
0%
tiv
4.
Ne
ga
3.
ve
2.
Positive
Negative
Zero
Not enough info
Po
sit
i
1.
When doing a bench press, you
gradually lower the bar down to
your chest. The work done by
gravity is:
0%
0%
No
t
en
ou
gh
in
fo
Ze
ro
0%
e
0%
tiv
4.
Ne
ga
3.
ve
2.
Positive
Negative
Zero
Not enough info
Po
sit
i
1.
When doing a curl, you exert a
force to raise the dumbbell to
your shoulder. The work done
by you is:
0%
0%
No
t
en
ou
gh
in
fo
Ze
ro
0%
e
0%
tiv
4.
Ne
ga
3.
ve
2.
Positive
Negative
Zero
Not enough info
Po
sit
i
1.
Because your two-year old cousin
refuses to move, you pull him along
the ground while tugging at an
angle. The work done is:
0%
0%
No
t
en
ou
gh
in
fo
Ze
ro
0%
e
0%
tiv
4.
Ne
ga
3.
ve
2.
Positive
Negative
Zero
Not enough info
Po
sit
i
1.
If there exists a force on
an object an the object
moves, work must have
been done.
0%
se
0%
Fa
l
2.
True
False
Tr
ue
1.
Work Can Be Positive or
Negative


Work is positive
when lifting the
box
Work would be
negative if
lowering the box

The force would
still be upward,
but the
displacement
would be
downward
When Work is Zero



Displacement is
horizontal
Force is vertical
cos 90° = 0
Units of Work

SI

Newton • meter = Joule


N•m=J
J = kg • m2 / s2
Work

W = (F cos )x

 is the angle between
F and x


If  = 0, cos  = 1, and
W = F Δx
If  = 90o, cos = 0,
and W = 0
More About Work

The work done by a force is zero
when the force is perpendicular to
the displacement

cos 90° = 0
Limitations of Work

This gives no information about:
the time it took for the displacement
to occur
-or the velocity or acceleration of the
object

Work is a Scalar

Even though the sign matters (like
vectors), the sign does not indicate
the direction if travel.
Let’s try some practice
problems:
Kinetic Energy



Energy associated with the motion
of an object
1
KE  mv 2
2
Scalar quantity measured in Joules
Work-Kinetic Energy
Theorem

The net work done on an object is equal
to the change in the object’s kinetic
energy
Wnet  KEf  KEi  KE


Speed will increase if work is positive
Speed will decrease if work is negative
Gravitational Potential
Energy




Gravitational potential energy is
associated with the vertical
position of the object
PEgrav = mgy
y = vertical position (relative to a
reference point – usually ground)
g = acceleration due to gravity
Conservation of
Mechanical Energy

Total mechanical energy is the
sum of the kinetic and potential
energies in the system and is stays
constant (if closed system)
Ei  E f
KEi  PEi  KE f  PE f
On which track does the
marble have the largest
initial potential energy?
0%
0%
sa
m
e
ue
th
e
Al
l
0%
Bl
5.
0%
Re
d
4.
0%
w
3.
Ye
llo
2.
Green
Yellow
Red
Blue
All the same
Gr
ee
n
1.
On which track will the
marble have the largest
final velocity?
0%
0%
sa
m
e
ue
th
e
Al
l
0%
Bl
5.
0%
Re
d
4.
0%
w
3.
Ye
llo
2.
Green
Yellow
Red
Blue
All the same
Gr
ee
n
1.
On which track does the
marble have the largest
total mechanical energy at
the beginning?
0%
0%
sa
m
e
ue
th
e
Al
l
0%
Bl
5.
0%
Re
d
4.
0%
w
3.
Ye
llo
2.
Green
Yellow
Red
Blue
All the same
Gr
ee
n
1.
If a steel marble is released down
the green track and a plastic marble
goes down the blue track, which will
have the greater velocity at the end
of the track?
0%
e
0%
Sa
m
0%
st
ic
3.
Pl
a
2.
Steel
Plastic
Same
St
ee
l
1.
If a steel marble is released down
the green track and a plastic marble
goes down the blue track, which will
have the greater kinetic energy at
the end of the track?
0%
e
0%
Sa
m
0%
st
ic
3.
Pl
a
2.
Steel
Plastic
Same
St
ee
l
1.
On which track did the
marble have the largest
average velocity?
0%
0%
sa
m
e
ue
th
e
Al
l
0%
Bl
5.
0%
Re
d
4.
0%
w
3.
Ye
llo
2.
Green
Yellow
Red
Blue
All the same
Gr
ee
n
1.
On which track did the
marble have the second
largest average velocity?
0%
0%
sa
m
e
ue
th
e
Al
l
0%
Bl
5.
0%
Re
d
4.
0%
w
3.
Ye
llo
2.
Green
Yellow
Red
Blue
All the same
Gr
ee
n
1.
Notes About Conservation
of Energy

We can neither create nor destroy
energy


Another way of saying energy is
conserved
If the total energy of the system does
not remain constant, the energy must
have crossed the boundary by some
mechanism (friction, heat, sound, …)
When graphing F vs. –x,
what was the relationship?
100%
50%
40%
30%
20%
0%
10%
0%
0%
0%
0%
...
t io
ns
hi
re
la
No
er
Fu
n
In
ve
rs
e
ct
io
n
ic
0%
ra
t
5.
60%
Po
w
4.
70%
Qu
ad
3.
80%
ea
r
2.
Linear
Quadratic
Power Function
Inverse
No relationship
Lin
1.
90%
Springs – Hooke’s Law
One of the simplest type
of simple harmonic
motion is called
Hooke's Law. This is
primarily in reference to
SPRINGS.
Fs  x
k  Constant of Proportion ality
k  Spring Constant(U nit : N/m)
Fs  kx or  kx
The negative sign
only tells us that
“F” is what is called
a RESTORING
FORCE, in that it
works in the
OPPOSITE
direction of the
displacement.
Hooke’s Law
Felas = -kx
Felas = Elastic force of the spring (force points
back to equilibrium position.) (N)
k = Spring constant (N/m)
x = displacement from equilibrium (m)
(note:opposite to direction of the elastic force)
Hooke’s Law from a Graphical Point of View
Fs  kx
Suppose we had the following data:
x(m)
Force(N)
Fs
x
k  Slope of a F vs. x graph
k
Force vs. Displacement
0
0
0.1
12
y = 120x + 1E-14
R2 = 1
80
70
0.2
24
0.3
36
0.4
48
0.5
60
Force(Newtons)
60
k =120 N/m
50
40
30
20
10
0
0.6
72
0
0.1
0.2
0.3
0.4
Displacement(Meters)
0.5
0.6
0.7
We have seen F vs. x Before!!!!
Force vs. Displacement
Work or ENERGY = Fx
y = 120x + 1E-14
R2 = 1
80
Since WORK or ENERGY
is the AREA, we must
get some type of energy
when we compress or
elongate the spring. This
energy is the AREA
under the line!
70
Force(Newtons)
60
50
Area = ELASTIC
POTENTIAL
ENERGY
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Displacement(Meters)
Since we STORE energy when the spring is compressed and
elongated it classifies itself as a “type” of POTENTIAL ENERGY,
Us. In this case, it is called ELASTIC POTENTIAL ENERGY.
Elastic Potential Energy
The graph of F vs.x for
a spring that is
IDEAL in nature will
always produce a
line with a positive
linear slope. Thus
the area under the
line will always be
represented as a
triangle.
NOTE: Keep in mind that this can be applied to WORK
or can be conserved with any other type of energy.
Elastic potential energy
W   F ( x)dx   (kx)dx
x
x
x 0
x 0
W   (kx)dx  k  xdx
x2 x
W  k | | x 0  W  U spring  1 kx2
2
2
Elastic “potential” energy is a fitting term as springs
STORE energy when there are elongated or
compressed.
Conservation of Energy in
Springs
Strain PE vs. Gravitational PE
Strain
Force (N)
0



½Fx
Gravitational
0
Force (N)
“mg”
x (m)
Fx
or
mgh
x (m)
“h”
The area under the curve on the left equals the
energy stored in a linear spring, or the amount of
work required to deform the spring.
The area under the curve on the right equals the
potential energy due to the constant force of gravity
(mg), or the work required to lift an object x m.
Note that one area is square and the other
triangular.
Power


Often also interested in the rate at
which the energy transfer takes place
Power is defined as this rate of energy
transfer


W

 Fv
t
SI units are Watts (W)

J kg m2
W  
s
s2
Power, cont.

US Customary units are generally hp

Need a conversion factor
ft lb
1 hp  550
 746 W
s
Work is done when
e.
.
di
sp
la
nd
ea
fo
rc
0%
ce
m
er
o.
sz
ei
fo
rc
th
e
ac
em
sp
l
di
0%
th
e
en
ti
sz
ot
...
sn
en
ti
th
e
0%
er
o.
0%
ac
em
4.
sp
l
3.
di
2.
the displacement
is not zero.
the displacement
is zero.
the force is zero.
the force and
displacement are
perpendicular.
th
e
1.
0%
0%
0%
0%
27
.
4.
9.
3.
6.
2.
3.
1.
If both the mass and the velocity of
a ball are tripled, the kinetic energy
of the ball is increased by a factor of
3.
6.
9.
27.
Of the following examples, the one
that represents work as defined by a
scientist is:
w
he
av
y
bo
x
ns
ta
d
ry
in
ga
ar
ng
h
pu
sh
i
0%
on
...
...
0%
ag
ai
ve
lw
as
ho
go
n
le
an
in
ca
r
ur
...
yo
ro
m
0%
hi
l..
.
0%
ok
f
4.
bo
3.
ga
2.
lifting a book from
your desk.
leaning on a shovel
while others labor.
pushing hard against
a wall for an hour.
carrying a heavy box
on your head.
lif
tin
1.
ya
nd
...
in
m
ea
s
ea
su
re
d
m
nd
er
a
0%
ur
ed
...
in
m
ea
s
nd
po
w
en
er
g
ed
ya
en
er
g
0%
in
.. .
0%
ur
ed
in
j..
.
0%
m
ea
su
r
4.
nd
3.
ea
2.
in joules.
energy and measured
in watts.
power and measured
in watts.
energy and measured
in joules.
fo
rc
1.
The ability to do work is
defined
as:
force
and measured
A rolling wagon has 50 joules of kinetic
energy. If the wagon’s velocity is
doubled, the kinetic energy of the wagon:
is
e.
0%
sa
m
th
e
ns
re
m
ai
20
0
in
cr
ea
se
d
to
to
10
0
25
in
cr
ea
se
d
0%
jo
ul
es
.
0%
jo
ul
es
.
jo
ul
es
.
0%
to
4.
is
3.
re
du
ce
d
2.
is reduced to 25
joules.
is increased to 100
joules.
is increased to 200
joules.
remains the same.
is
1.