Energy, Work and Power
Energy
Energy: the currency of the universe.
Just like money, it comes in many
forms!
Everything that is accomplished has to
be “paid for” with some form of energy.
Energy can’t be created or destroyed, but
it can be transformed from one kind
into another and it can be transferred
from one object to another.
• Doing WORK is one way to transfer
energy from one object to another.
Work = Force x displacement
W = F∙d
• Unit for work is Newton x meter. One
Newton-meter is also called a Joule, J.
Work- the transfer of
energy
Work = Force x displacement
• Work is not done unless there is a displacement.
• If you hold an object a long time, you may get tired,
but NO work was done on the object.
• If you push against a solid wall for hours, there is
still NO work done on the wall.
• For work to be done, the displacement
of the object must be along the same
direction as the applied force. They
must be parallel.
• If the force and the displacement are
perpendicular to each other, NO work is
done by the force.
• For example, in lifting a book, the force
exerted by your hands is upward and the
displacement is upward- work is done. F
d
• Similarly, in lowering a book, the force
exerted by your hands is still upward, and
F
the displacement is downward.
d
• The force and the displacement are STILL
parallel, so work is still done.
• But since they are in opposite directions,
now it is NEGATIVE work.
• On the other hand, while carrying a
book down the hallway, the force
from your hands is vertical, and the
displacement of the book is
horizontal.
• Therefore, NO work is done by
your hands.
• Since the book is obviously moving,
what force IS doing work???
The static friction force between your
hands and the book is acting
parallel to the displacement and IS
doing work!
F
d
Example
How much work is done to push a 5 kg
cat with a force of 25 N to the top of
a ramp that is 7 meters long and 3
meters tall?
W = Force x displacement
Which measurement is parallel to the
force- the length of the ramp or the
height of the ramp?
W = 25 N x 7 m
W = 175 J
3m
Example
How much work is done to carry a 5 kg
cat to the top of a ramp that is
7 meters long and 3 meters tall?
W = Force x displacement
What force is required to carry the cat?
Force = weight of the cat
Which is parallel to the weight vectorthe length of the ramp or the height?
d = height NOT length
W = mg x h
W = 5 x 10 x 3
W = 150 J
3m
Your Force
Vertical component of d
• And,….while
carrying yourself
when climbing
stairs or walking
up an incline, only
the height is used
to calculate the
work you do to get
yourself to the
top!
• The force required
is your weight!
Horizontal component of d
How much work do you do on a
30 kg cat to carry it from one
side of the room to the other
if the room is 10 meters
long?
ZERO, because your Force is
vertical, but the displacement
is horizontal.
Pre-AP only…
Example
Displacement = 20 m
A boy pushes a
lawnmower 20 meters
across the yard. If he
pushed with a force of
200 N and the angle
between the handle
and the ground was
50 degrees, how
much work did he do?
F cos q
q
F
W = (F cos q )d
W = (200 cos 50˚) 20 m
W = 2571 J
A 5.0 kg box is pulled 6 m across a rough horizontal floor
(m = 0.4) with a force of 80 N at an angle of 35 degrees
above the horizontal. What is the work done by EACH
force exerted on it? What is the NET work done?
●Does the gravitational force do any work?
Normal
NO! It is perpendicular to the displacement.
● Does the Normal force do any work?
FA
No! It is perpendicular to the displacement.
q
f
● Does the applied Force do any work?
Yes, but ONLY its horizontal component!
WF = FAcosq x d = 80cos 35˚ x 6 m = 393.19 J
mg
● Does friction do any work?
Yes, but first, what is the normal force? It’s NOT mg!
Normal = mg – FAsinq
Wf = -f x d = -mN∙d = -m(mg – FAsinq)∙d = -7.47 J
● What is the NET work done?
393.19 J – 7.47 J = 385.72 J
Watch for those “key words”
NOTE: If while pushing an object, it is
moving at a constant velocity,
the NET force must be zero.
So….. Your applied force must be exactly
equal to any resistant forces like friction.
• Energy and Work have no direction
associated with them and are therefore
scalar quantities, not vectors.
YEAH!!
• Power is the rate at which
work is done- how fast you
do work.
Power = work / time
P=W/t
• You may be able to do a lot
of work, but if it takes you a
long time, you are not very
powerful.
• The faster you can do work,
the more powerful you are.
• The unit for power is Joule / seconds
which is also called a Watt, W
(just like the rating for light bulbs)
In the US, we usually measure power
developed in motors in “horsepower”
1 hp = 746 W
Example
A power lifter picks up a 80 kg barbell above his
head a distance of 2 meters in 0.5 seconds.
How powerful was he?
P=W/t
W = Fd
W = mg x h
W = 80 kg x 10 m/s2 x 2 m = 1600 J
P = 1600 J / 0.5 s
P = 3200 W
Another way of looking at Power:
work
power 
time
(force x displacement)
power =
time
 displacement 
power  force x 



time
power  force x velocity
Power = Force x velocity
Kinds of Energy
Kinetic Energy
the energy of motion
K = ½ mv2
Potential Energy
Stored energy
It is called potential
energy because it
has the potential to
do work.
Different kinds of Potential
(stored) Energy
• Example 1: Spring potential energy, SPE, in the
stretched string of a bow or spring or rubber
band. SPE = ½ kx2
k = spring force constant (N/m)
x = distance spring is stretched or compressed (m)
• Example 2: Chemical potential energy in fuelsgasoline, propane, batteries, food!
• Example 3: Gravitational potential energy, GPEstored in an object due to its position from a
chosen reference point.
Gravitational potential energy
GPE = weight x
height
GPE = mgh
Since you can
measure height
from more than
one reference
point, it is
important to
specify the
location from
which you are
measuring.
• The GPE may be negative. For
example, if your reference point is the
top of a cliff and the object is at its
base, its “height” would be negative, so
mgh would also be negative.
• The GPE only depends on the weight
and the height, not on the path that it
took to get to that height.
Many different forms of
Energy…
Thermal Energy
Solar Energy
Atomic Energy
Sound Energy
Electromagnetic Energy
Nuclear Energy
Electrical Energy
E = mc2
Work and Energy
Often, some force must do work
to give an object potential or
kinetic energy.
“Work” is the transfer of energy!!
You push a wagon and it starts
moving kinetic energy. You
stretch a spring and you
transform your work energy 
spring potential energy.
Or, you lift an object to a certain
height- you transfer your work
energy into the object in the
form of gravitational potential
energy.
Get a clicker and login!!
Get a calculator!
Get ready to take notes!
Don’t Forget!!
Rocket War project on Friday!!!
The Work-Kinetic Energy Theorem
NET Work done by all forces = D Kinetic Energy
Wnet = D ½ mv2
NET Work = D Kinetic Energy
F·d = D½ mv2
How much more distance is required to stop if a car is
going twice as fast (all other things remaining the
same)?
The work done by the forces stopping the car = the change in the kinetic energy
F·d = D½ mv2
With TWICE the speed, the car has
FOUR times the kinetic energy.
Therefore it takes FOUR times the stopping distance.
A car going 20 km/h will skid to a stop over a
distance of 7 meters.
If the same car was moving at 50 km/h, how many
meters would be required for it to come to a stop?
The velocity changed
50
20
by a factor of 2.5,
therefore the stopping distance is 2.52 times the
original distance:
7 meters x 6.25 = 43.75 meters
Example, Wnet = Fnetd = DK
A 500 kg car moving at 15 m/s skids 20 m to a
stop.
How much kinetic energy did the car lose?
DK = D½ mv2
DK = -½ (500 kg)(15 m/s) 2
DK = -56250 J
What force was applied to stop the car?
F·d = DK
F = DK / d
F = -56250 J / 20 m
F = -2812.5 N
F·d = D ½ mv2
A 0.02 kg bullet moving at 90 m/s strikes a block
of wood. If the bullet comes to a stop at a depth
of 2.5 cm inside the wood, how much force did
the wood exert on the bullet?
F = 3240 N
I love mrs. BRown
Mechanical Energy
Mechanical Energy = Kinetic Energy + Potential Energy
E
= ½ mv2
+
mgh
“Conservative” forces - mechanical energy is
conserved if these are the only forces
acting on an object.
The two main conservative forces are:
Gravity, spring forces
“Non-conservative” forces - mechanical
energy is NOT conserved if these forces
are acting on an object.
Forces like kinetic friction, air resistance
(which is really friction!)
2
mv
W=½
W = Fd
2
Fd = ½ mv
Conservation of Mechanical Energy
If there is no kinetic friction or air resistance, then the
total mechanical energy of an object remains the
same.
If the object loses kinetic energy, it gains potential
energy.
If it loses potential energy, it gains kinetic energy.
For example: tossing a ball upward
Conservation of Mechanical Energy
The ball starts with kinetic energy…
Which changes to potential energy….
Which changes back to kinetic energy
What about the
energy when it is
not at the top or
bottom?
PE = mgh
Energybottom = Energytop
½ mvb2 = mght
E = ½ mv2 + mgh
K = ½ mv2
K = ½ mv2
Examples
•
•
•
•
•
dropping an object
box sliding down an incline
tossing a ball upwards
a pendulum swinging back and forth
A block attached to a spring oscillating
back and forth
First, let’s look at examples where there
is NO friction and NO air resistance…..
Conservation of Mechanical Energy
If there is no friction or air resistance, set the mechanical
energies at each location equal. Remember, there may be
BOTH kinds of energy at any location.
E1
=
E2
mgh1 + ½mv12 = mgh2 + ½ mv22
A 3.0 kg Egg sits on top of a 12 m tall wall.
What is its potential energy?
PE = mgh
PE = 3 kg x 10 m/s2 x 12 m = 360 J
If the Egg falls, what is its kinetic energy just
before it hits the ground?
Potential energy at top = Kinetic energy at bottom
K = 360 J
What is its speed just before it strikes the ground?
360 J = K = ½ mv2
2(360 J) / 3 kg = v2
v = 15.49 m/s
Example of Conservation of Mechanical Energy
Rapunzel dropped her hairbrush
from the top of the castle where she
was held captive. If the window was
80 m high, how fast was the brush
moving just before it hit the ground?
(g = 10 m/s2)
mgh1 + ½ mv12 = mgh2 + ½ mv22
mgh = ½ mv2
gh = ½ v2
2gh = v2
Don’t forget to take the square root!
Enter your answer on #1
Now… do one on your own
#2 An apple falls from a tree that is 1.8 m
tall. How fast is it moving just before it hits
the ground? (g = 10 m/s2)
mgh1 + ½ mv12 = mgh2 + ½ mv22
mgh = ½ mv2
And another one…
#3 A woman throws a ball straight
up with an initial velocity of 12 m/s.
How high above the release point
will the ball rise? g = 10 m/s2
mgh1 + ½ mv12 = mgh2 + ½ mv22
½ mv2 = mgh
h = ½ v2 / g
And another one…
#4
Mario, the pizza man, tosses the
dough upward at 8 m/s. How high
above the release point will the
dough rise?
g = 10 m/s2
mgh1 + ½ mv12 = mgh2 + ½ mv22
Conservation of Mechanical Energy- another look
A skater has a kinetic energy of 57 J
at position 1, the bottom of the ramp
(and NO potential energy)
At his very highest position, 3, he
comes to a stop for just a moment so
that he has 57 J of potential energy
(and NO kinetic energy)
Mechanical energy = KE + PE
#5
What is his kinetic energy
at position 2, if his potential
energy at position 2 is 25.7 J?
E = 57 J
E = 57 J
PE = 25.7 J
KE = ??
Conservation of Mechanical
Energy… more difficult
A stork, at a height of 80 m
flying at 18 m/s, releases
his “package”. How fast
will the baby be moving
just before he hits the
ground?
Energyoriginal = Energyfinal
mgh + ½ mvo2 = ½ mvf2
Vf = 43.5 m/s
#6 Now you
do one …
The car on a roller coaster starts from rest at
the top of a hill that is 60 m high. How fast
will the car be moving at a height of 10 m?
(use g = 9.8 m/s2)
mgh1 + ½ mv12 = mgh2 + ½ mv22
mgh1
= mgh2 + ½ mv22
# 6 Enter your answer with ONE decimal place.
If there is kinetic friction or air resistance,
mechanical energy will not be conserved.
Mechanical energy will be lost in the form of
heat energy.
The DIFFERENCE between the
original energy and the final energy
is the amount of mechanical energy lost due
to heat.
Final energy – original energy = energy loss
Let’s try one…
#7
A 2 kg cannonball is shot straight up from
the ground at 18 m/s. It reaches a highest
point of 14 m. How much mechanical energy
was lost due to the air resistance? g = 10 m/s2
Final energy – original energy = Energy loss
mgh
–
½ mv2
=
Heat loss
2 kg(10 m/s2)(14 m) – ½ (2 kg)(18 m/s)2 = ??
And one more…
#8
A 1 kg flying squirrel drops from the top of a
tree 5 m tall. Just before he hits the ground,
his speed is 9 m/s. How much mechanical
energy was lost due to the air resistance?
g = 10 m/s2
Final energy – original energy = Energy loss
Sometimes, mechanical energy is actually
INCREASED!
For example: A bomb sitting on the floor
Don’t even think
explodes.
about it…
Initially:
Kinetic energy = 0
Gravitational Potential energy = 0
Mechanical Energy = 0
After the explosion, there’s lots of kinetic
and gravitational potential energy!!
Did we break the laws of the universe and
create energy???
Of course not! NO ONE, NO ONE, NO ONE
can break the laws!
The mechanical energy that now appears
came from the chemical potential energy
stored within the bomb itself!
According to the
Law of Conservation of Energy
energy cannot be created or destroyed.
But one form of energy may be transformed
into another form as conditions change.
The Work-Kinetic Energy Theorem
NET Work done by all forces = D Kinetic Energy
Wnet = ½ mv2f – ½ mv2o
Example Wnet = Fnet d = DK
A 500 kg car moving at 15 m/s slows to 10 m/s.
How much kinetic energy did the car lose?
DK = ½ mvf2 – ½ mvo2
DK = ½ (500 kg)(10 m/s)2 - ½ (500 kg)(15 m/s)2
DK = -31250 J
What force was applied to slow the car if the
distance moved was 12 m?
F·d = DK
F = DK / d
F = -31250 J / 12 m
F = -2604 N
Example Wnet = Fnetd = DK
A 500 kg car moving on a flat road at 15 m/s skids to
a stop.
How much kinetic energy did the car lose?
DK = ½ mvf2 – ½ mvo2
DK = -½ (500 kg)(15 m/s)2
DK = -56250 J
How far did the car skid if the effective coefficient of
friction was m  0.6?
Stopping force = friction = mN = mmg
F·d = DK
-(mmg)·d = DK
d = DK / (-mmg) *be careful to group in the denominator!
d = 56250 J / (0.6 · 500 kg · 9.8 m/s2) = 19.13 m
Back to Power…
Since Power = Work / time and
Net work = DK…
Power = DK / time
In fact, Power can be calculated in many
ways since Power = Energy / time, and
there are MANY forms of energy!
Conservation of Mechanical Energy
1. Draw a sketch and choose a reference point for height.
2. Look at the first position of your object. If it is moving, it
has Kinetic energy. If it has some height above or
below your reference point, it has Potential energy.
3. Repeat for the second location.
4. If there is no friction or air resistance, set the
mechanical energies at each location equal.
E1
=
E2
mgh1 + ½mv12 = mgh2 + ½ mv22
5. If there is friction or air resistance, use E1 – E2 to find
the energy lost.
Graphing Force vs. postion
• If you graph the applied force vs. the
position, you can find how much work
was done by the force.
Work = Fd = “area under the curve”.
Total Work = 2 N x 2 m + 3N x 4m = 16 J
Area UNDER the x-axis is NEGATIVE
work = - 1N x 2 m
Force, N
F
Position, m
d
Net work = 16 J – 2 J = 14 J
Back to the Work-Kinetic Energy
Theorem…
According to that theorem,
net work done = the change in the kinetic energy
Wnet = DK
But, if the work can be found by taking the “area
under the curve”, then it is also true that
Area under the curve = DK = ½ mvf2 – ½ mvo2
Therefore, the area can be used to predict the
final velocity of an object given its initial velocity
and its mass.
For example…
Suppose from the previous graph
(Area = Wnet = 14 J), the object upon which
the forces were exerted had a mass of 3
kg and an initial velocity of 4 m/s. What
would be its final velocity?
Area under the curve = ½ mvf2 – ½ mvo2
14 J = ½(3 kg)vf 2 – ½(3 kg)(4 m/s)2
vf = 5.0 m/s
The Spring Force
If you hang an object
from a spring, the
gravitational force
pulls down on the
object and the spring
force pulls up.
The Spring Force
The spring force is
given by
Fspring = kx
Where x is the amount
that the spring
stretched and k is the
“spring constant”
which describes how
stiff the spring is
The Spring Force
If the mass is hanging at
rest, then
Fspring = mg
Or
kx = mg
(this is called “Hooke’s Law)
The easiest way to
determine the spring
constant k is to hang a
known mass from the
spring and measure how
far the spring stretches!
k = mg / x
Graphing the Spring Force
Suppose a certain spring had a spring constant
k = 30 N/m.
Graphing spring force vs. displacement:
On horizontal axis- the displacement of the spring: x
On vertical axis- the spring force = kx = 30x
What would the graph look like?
Fs = kx
In “function” language: f(x) = 30x
Spring Force vs. Displacement Fs = 30x
Fs
How could you use the graph
To determine the work done by
The spring from some x1 to x2?
Take the AREA under the curve!
x1
x2
x
Analytically…
The work done by the spring is given by
Ws = ½ kxf2 – ½ kxo2
where x is the distance the spring is
stretched or compressed
(Which would yield the same result as taking
the area under the curve!)
Physics 1:
1. Get a Clicker and sign in to A240
2. You need a calculator
3. You need your notes
Simple Machines and Efficiency
Machine: A device that HELPS do work.
A machine cannot produce more WORK
ENERGY than the energy you put into itthat would break the Law of Conservation
of Energybut it can make your work easier to do.
• Some common “simple machines” include levers, pulleys,
wheels and axles, and inclined planes
• Ideally, with no friction, the work energy you get out of a
machine equals the work energy you put into it.
Ideally:
Work energy in = work energy out
Work = Force x distance
The work you put into a machine is called
EFFORT work.
The work you get out of the machine- is called
RESISTANCE work, so ideally
Effort Work = Resistance Work
Feffortdeffort = Fresistancedresistance
(if there’s no NON-conservative forces!)
Try one…
# 1. Hercules pushes a 500 kg
boulder up a hill a distance of
25 m using a force of 6000 N.
How much work did Hercules
do?
Effort Work = Resistance Work
Feffortdeffort = Fresistancedresistance
Levers
A
Effort force
B
Effort force
C
Effort force
• The RESISTANCE force is the weight
of the load being lifted.
#2 Which arrangement will require the least
EFFORT force?
Levers
Effort force
Effort force
Effort force
3. How do you “pay” for a small effort force?
A) You push harder
B) You push just the same
C) You push a smaller distance
D) You push a greater distance
Inclined Planes
A
B
C
Weight =
Resistance Force
Height =
Resistance
Distance
#4. Which arrangement will require the least
EFFORT force?
How do you “pay” for a smaller effort force?
Two pulleys
with a belt
A motor is attached to one of the
pulleys so that as it turns, the belt
causes the second pulley to turn.
To have the least effort force, the effort
distance must be the greatest. In this
case the effort distance is the number
of turns around – the ROTATIONS!
Which pulley will have to go around
more times? This is the pulley that the
motor should be attached to for the
least effort force.
A
B
# 5 Which pulley should
the motor be attached to so
that it requires the least
effort force from the motor?
Efficiency
No machine is perfect. That is reflected in
the “efficiency” of the machine. In the real
world, the efficiency will always be less
that 100%. It is found by
Energy out work out (resis tan ce)
Efficiency 

Energy in
work in (effort)
Some practice…
#6 While using a simple machine, you put in
4500 J of work energy. The machine puts
out 3690 J of energy. What was the percent
efficiency of the machine?
Energy out work out (resis tan ce)
Efficiency 

Energy in
work in (effort)
A man pushes a 48 kg box up a 12 m long incline that is 4.2
meters high by applying a force of 240 N. (g = 10 m/s2)
What is the effort (input) work?
Weffort = Feffortdeffort
#7 We = ?
What is the resistance (output) work?
Energy out work out (resis tan ce)
Wresistance = Frdr
Efficiency 

Energy in
work in (effort)
W = mg x h
#8 Wr = ?
#9 What is the percent efficiency of the incline?