Download gravitational potential energy

Chapter 5
Energy
Introduction

Energy is one of the most
fundamental and farreaching concepts in the
physics world view. It took
nearly 300 years to fully
develop the ideas of
energy and energy
conservation—the idea
that the total energy of an
isolated system does not
change.
Windmills transform the energy in the
wind to electric energy.
Forms of Energy

Mechanical



Position and Motion
focus for now
NonMechanical




chemical
Electromagnetic
Nuclear- Fusion, Fission
Thermal
Using Energy
Considerations

Energy can be transformed
from one form to another


Essential to the study of
physics, chemistry, biology,
geology, astronomy
Can be used in place of
Newton’s laws to solve
certain problems more
simply
A First Equation
Energy of Motion

The most obvious form of energy is the one
an object has because of its motion.


We call this quantity of motion the kinetic
energy of the object.
kinetic energy depends on the mass and the
motion of the object. But the kinetic energy K
of an object has its own equation.
K=1/2 mv2
Units of Energy
Energy of Motion

The units for kinetic energy, and
therefore for all types of energy is a
joule (J).

Kinetic energy is not a vector
quantity.


An object has the same kinetic energy
regardless of its direction as long as its
speed does not change.
A typical textbook dropped from a
height of 10 centimeters (about 4
inches) hits the floor with a kinetic
energy of about 1 J.
Details of the Equation
Energy of Motion


The factor of ½ makes the
kinetic energy compatible with
other forms of energy, which
we will study later.
Notice that the kinetic energy
of an object increases with the
square of its speed.

This means that if an object has
twice the speed, it has 4× the
kinetic energy; if it has 3× the
speed, it has 9× the kinetic
energy; and so on.
Doing the Math
Energy of Motion

The kinetic energy of a 70kilogram (154-pound)
person running at a speed
of 8 meters per second is:
KE = ½mv 2
= ½(70 kg)(8 m/s)2
= (35 kg)(64 m2/s2) =
2240 J
Arlo
Changing Kinetic Energy


A cart rolling along a frictionless, horizontal
surface has a certain kinetic energy because it
is moving.
A net force on the cart can change its speed
and thus its kinetic energy.
If you push in the
direction the cart is
moving, you increase
the cart’s kinetic
energy.
Pushing in the
opposite direction
slows the cart and
decreases its
kinetic energy.
Changing Kinetic
Energy
.

The displacement through which the force acts determines how
much the kinetic energy changes.



The product of the net force F in the direction of motion and the
displacement moved d is known as the work W:
From the definition of work, we conclude that the units of work
are newton-meters.
If we substitute our previous expression for a newton (kg ·
m/s2), we find that a newton-meter (N · m) equals a kg · m2/s2,
which is the same as the units for energy—that is, a joule.
Therefore, the units of work are the same as those of energy.
More About Work


Scalar quantity
The work done by a force is zero
when the force is perpendicular to
the displacement


cos 90° = 0
If there are multiple forces acting
on an object, the total work done
is the algebraic sum of the amount
of work done by each force
Changing Kinetic Energy
If the net force is in the same
direction as the velocity, the
work is positive, and the
kinetic energy increases.
 If the net force and velocity
are oppositely directed, the
work is negative, and the
kinetic energy decreases.

Practice
1.

In the figures below, identical boxes of mass 10 kg are moving at the same
initial velocity to the right on a flat surface. The same magnitude force, F, is
applied to each box for the distance indicated in the figure.
Rank these situations in order of the work done on the box by F while the
box moves the indicated distance to the right.
F
F
F
d=5m
d=10m
F
d=5m
F
F
d=10m
d=5m
d=5m

Newton’s 2nd law and our expressions
for acceleration and distance traveled
can be combined with the definition of
work to show that the work done on an
object is equal to the change in its
kinetic energy:
Forces That Do No Work

The meaning of work in physics is different from the
common usage of the word.


Commonly, people talk about “playing” when they throw a
ball and “working” when they study physics.
The physics definition of work is quite precise—work
occurs when the product of the force and the
distance is nonzero.


When you throw a sphere, you are actually doing work on
the ball; its kinetic energy is increased because you apply a
force through a distance.
Although you may move pages and pencils as you study
physics, the amount of work is quite small.
Holding Something Up
Forces That Do No Work

Similarly, if you hold a suitcase above your head for 30
minutes, you would probably claim it was hard work.
According to the physics definition, however, you did not
do any work on the suitcase;

the 30 minutes of straining and groaning did not change the
suitcase’s kinetic energy.
It takes no work to hold a
cheerleader in the air.
Forces That
Do No Work


There are other situations in which a net force does
not change an object’s kinetic energy.
If the force is applied in a direction perpendicular to
its motion, the velocity of the object changes, but its
speed doesn’t.


Therefore, the kinetic energy does not change.
The definition of work takes this into account by
stating that it is only the force that acts along the
direction of motion that can do work.
Pushing At An Angle
Forces That Do No Work

Often, a force is neither parallel nor perpendicular to
the displacement of an object. Because force is a
vector, we can think of it as having two components,
one that is parallel and one that is perpendicular to the
motion as illustrated. The parallel component does
work, but the perpendicular one does not do any work.
Any force can be replaced by two
perpendicular component forces. Only the
component along the direction of motion
does work on the box.
Circular Motion
Forces That Do No Work


Consider an air-hockey puck moving in a circle on the
end of a string attached to the center of the table
shown in Figure 7-5.
Because the speed is constant, the kinetic energy is
also constant.

The force of gravity is balanced by the upward force
of the table. These vertical forces cancel and do no
work.

The tension that the string exerts on the puck is not
canceled but does no work because it always acts
perpendicular to the direction of motion.
When the force is perpendicular to the velocity, the
force does no work.
Perfect Orbits
Forces That Do No Work

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If Earth’s orbit were a circle with the Sun at the center, the
gravitational force the Sun exerts on Earth would do no work.
However, because the orbit is an ellipse, the force is not always
perpendicular to the direction of motion.


During one-half of each orbit, a small component of the force acts in
the direction of motion, increasing Earth’s kinetic energy and speed.
During the other half of each orbit, the component is opposite the
direction of motion, and Earth’s kinetic energy and speed decrease.
The gravitational force of the Sun
does work on Earth whenever the
force is not perpendicular to
Earth’s velocity. The elliptical
nature of the orbit has been
exaggerated to show the parallel
component.
Gravitational Potential Energy

When a sphere is thrown vertically upward,

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it has a certain amount of kinetic energy that
disappears as it rises.
At the top of its flight, it has no kinetic energy,
but as it falls, the kinetic energy reappears.
If we believe that energy is an invariant, we
must be missing one or more forms of
energy.
Gravitational Potential Energy

The loss and subsequent reappearance of the ball’s kinetic energy
can be understood by examining the work done on the ball.



Rather than simply saying that the kinetic energy temporarily
disappears, we can retain the idea of the conservation of energy
by defining a new form of energy.


As the ball rises, the force of gravity performs negative work on the
ball, reducing its kinetic energy until it reaches zero.
On the way back down, the force of gravity increases the ball’s
kinetic energy by the same amount it lost on the way up.
Kinetic energy is then transformed into this new form and later
transformed back.
This new energy is called gravitational potential energy.
Gravitational Potential Energy


Its change must also be given by the work done by the force of
gravity, and it must increase when the kinetic energy decreases, and
vice versa.
Therefore, we define the gravitational potential energy of an
object at a height h above some zero level as equal to the
work done by the force of gravity on the object as it falls to
height zero.
The gravitational potential energy of an object near Earth’s
mgh
The units are the same as the units for work.
surface is then given by
gravitational potential energy
=
force of gravity x height
Reference Levels for
Gravitational Potential Energy

A location where the gravitational potential
energy is zero must be chosen for each
problem


The choice is arbitrary since the change in the
potential energy is the important quantity
Choose a convenient location for the zero
reference height


often the Earth’s surface
may be some other point suggested by the problem
The only thing that has any physical significance is the change in
gravitational potential energy.


If a ball gains 20 joules of kinetic energy as it falls, it must
lose 20 joules of gravitational potential energy.
It does not matter how much gravitational potential energy
it had at the beginning; it is only the amount lost that has
any meaning in physics.
Gravitational Potential Energy

As an example, we can calculate the gravitational
potential energy of a 6-kg ball located 0.5 meter above
the level that we choose to call zero:
GPE = mgh
= (6 kg)(10 m/s2)(0.5 m)
= 30 J

Notice that only the vertical height is important.
Moving an object 100 meters sideways does not
change the gravitational potential energy because the
force of gravity is perpendicular to the motion and
therefore does no work on the object.
Conceptual Question
Gravitational Potential Energy

Question: What is the change in gravitational
potential energy of a 50-kg person who climbs
a flight of stairs with a height of 3 meters and
a horizontal extent of 5 meters?
Answer: The change in gravitational potential
energy is
GPE = mgh = (50 kg)(10 m/s2)(3 m) = 1500 J
The horizontal extent has no effect on the
answer.
Flawed Reasoning
Gravitational Potential Energy



Bill and Will are calculating the gravitational potential
energy of a 5-newton ball held 2 meters above the
floor of their classroom.
Bill: “This is easy. Gravitational potential energy is
mgh, where mg is the weight of the ball. We just
multiply the 5 newtons by the 2 meters to get the
gravitational potential energy of 10 joules.”
Will: “You are forgetting that our classroom is on the
second floor. We are going to have to find out how
high the ball is relative to the ground.”
Do you agree with either of these students?
Flawed Reasoning
Gravitational Potential Energy

Answer:
It is only the difference in gravitational potential
energy that matters. We can either say that:

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
the ball fell from a height of 2 meters to a height of zero,
or we can say that it fell from a height of 5 meters (relative
to the ground) to a height of 3 meters.
Either way, we get the same decrease in gravitational
potential energy and the same increase in kinetic
energy. Both students would get correct answers.

In general, it is usually easiest to take the lowest point in
each problem to be the zero for height.
Conservation of Mechanical Energy

The sum of the gravitational potential
and kinetic energies is conserved in
some situations. This sum is called the
mechanical energy of the system:
Total energy at an instant= Total energy at another instant
½ mv12+mgh1= ½ mv22+mgh2
Conservation of Mechanical Energy


When frictional forces can be ignored
and the other nongravitational forces
do not perform any work, the
mechanical energy of the system does
not change.
The simplest example of this
circumstance is free fall (Figure 7-7).
Any decrease in the gravitational
potential energy shows up as an
increase in the kinetic energy, and vice
versa.
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
Applies to areas other than physics
Galileo’s Ramps
Conservation of Mechanical Energy

Galileo released a ball that rolled down a ramp, across a
horizontal track, and up another ramp, as shown in. He
remarked that the ball always returned to its original
height.

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This result is independent of the slopes of the ramps.
During the horizontal portion of the ball’s trip, the gravitational
potential energy remained constant. Hence, the kinetic energy
did not change and the speed remained constant, in agreement
with Newton’s 1st law.
The Pendulum Bob
Conservation of Mechanical Energy

The pendulum bob shown in Figure 7-9 cyclically gains and
loses kinetic and gravitational potential energy.


Suppose that at the beginning the bob has zero speed at
point A and a gravitational potential energy of 10 joules.


Note that the tension exerted by the string does no work.
Therefore, if we ignore frictional forces, the total mechanical
energy is conserved.
(We have chosen the zero for gravitational potential energy to
be at the lowest point of the bob’s path.)
Because the kinetic energy is zero, the
total mechanical energy is the same as
the gravitational potential energy—
that is, 10 joules.
The Pendulum Bob
Conservation of Mechanical Energy



Even when the bob is someplace between the
highest and lowest points of the swing, the total
mechanical energy is still 10 joules.
If at this point we determine from the height of
the bob that it has 6 joules of gravitational
potential energy, we can immediately declare that
it has 4 joules of kinetic energy.
Because we know the expression
for the kinetic energy, we can
calculate the speed of the bob
at this point.
The Pendulum Bob
Conservation of Mechanical Energy

Question: Suppose the bob is released
at twice the height. What is the
maximum kinetic energy?
Answer: The initial gravitational
potential energy is now twice as big, so
the maximum kinetic energy will also be
twice as big—that is, 20 joules.
Roller Coasters
Conservation

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Imagine trying to determine the speed of a
roller coaster inside a thrilling loop-the-loop as it traverses the track.
Determining the speed at any spot using Newton’s 2nd law is difficult
because the forces are continually changing magnitude and direction.
We can, however, use conservation of mechanical energy to determine the
speed of an object without knowing the details of the net forces acting on
it, providing we can ignore the frictional forces.


If we know the mass, speed, and height of the roller coaster at some spot, we
can calculate its mechanical energy.
Now determining its speed at any other spot is greatly simplified. The
height gives us the gravitational potential energy. Subtracting this from the
total mechanical energy yields the kinetic energy from which we can obtain
the roller coaster’s speed.

Notice that we don’t need to worry about all the energy transformations that
occurred earlier in the ride.
Roller Coasters
Conservation of Mechanical Energy



Suppose the roller-coaster ride was designed like the one in,
and upon reaching the top of the lower hill, your car almost
comes to rest.
Assuming that there are no frictional forces to worry about (not
true in real situations), is there any possibility that you can get
over the higher hill?
The answer is no. Your gravitational potential energy at the top
of the hill is nearly equal to the mechanical energy.


This energy is not enough to get you over the higher hill.
You will gain speed and thus kinetic energy
as you coast down the hill, but as you start
up the other hill, you will find that you
cannot exceed the height of the original hill.
Roller Coasters
Conservation of Mechanical Energy

Is there any way that the roller-coaster car
can make it over the second hill when
starting on top of the first hill?


If your car has some kinetic energy at the top of
the first hill, it might be possible to make it over
the second hill.
You would need enough kinetic
energy to equal or exceed the
extra gravitational potential
energy required to climb the next hill.
Flawed Reasoning
Conservation of Mechanical Energy




The following question appears on the final exam: “Three bears are throwing
identical rocks from a bridge to the river below. Papa Bear throws his rock
upward at an angle of 30° above the horizontal. Mama Bear throws hers
horizontally. Baby Bear throws the rock at an angle of 30° below the
horizontal. Assuming that all three bears throw with the same speed, which
rock will be traveling fastest when it hits the water?” Three students meet
after the exam and discuss their answers.
Emma: “Baby Bear’s rock will be going the fastest because it starts with a
downward component of velocity.”
Hector: “But Papa Bear’s rock will stay in the air the longest, so it will have
more time to speed up. I think his rock will be traveling the fastest.”
M’Lynn: “Papa Bear’s rock does stay in the air longer, but part of that time
it is moving upward and slowing down. I think Mama Bear’s rock will be
traveling fastest when it hits the water because it is in the air longer than
Baby Bear’s and it is speeding up all of the time.”
With which student (if any) do you agree?
Flawed Reasoning
Conservation of Mechanical Energy


Answer:
All three students are wrong. They are making an easy
problem much too difficult by ignoring the power of the
energy approach to problem solving.
Because each of the three rocks started with the same
kinetic energy (same speed) and the same gravitational
potential energy (same height), they must all end up with
the same final kinetic energy before hitting the water. All
three rocks must therefore hit the water with the same
speed.

Note that the three rocks will not hit the water at the same time,
with the same velocity, or at the same distance from the bridge.
However, the energy method will not give us this information.
Friction






The blue path is shorter than the red
path
The work required is less on the blue
path than on the red path
Friction depends on the path and so
is a nonconservative force
Energy changes to thermal… not
mechanical= nonconservative
WNC=fd
So energy at an instant of time after
motion might be thermal because
friction did work
Bernoulli’s Effect

If the fluid is not compressible, the fluid must be
moving faster in the narrow region.



This is because the same amount of fluid must pass by
every point in the pipe, or it would pile up. Therefore, the
fluid must flow faster in the narrow regions.
This might lead one to conclude incorrectly that the pressure
would be higher in this region.
Swiss mathematician and physicist Daniel Bernoulli
stated the correct result as a principle.
The pressure in a fluid decreases as its velocity
increases.
Bernoulli’s Effect

We can understand Bernoulli’s principle by
“watching” a small cube of fluid flow through the
pipe.




The cube must gain kinetic energy as it speeds up
entering the narrow region.
Because there is no change in its gravitational potential
energy, there must be a net force on the cube that does
work on it.
Therefore, the force on the front of the cube must be less
than on the back. That is, the pressure must decrease as
the cube moves into the narrow region.
As the cube of fluid exits the narrow region, it slows
down. Therefore, the pressure must increase again.
Everyday Examples
Bernoulli’s Effect


There are many examples of Bernoulli’s effect in our
everyday activities.
Smoke goes up a chimney partly because hot air
rises but also because of the Bernoulli effect.


The wind blowing across the top of the chimney reduces the
pressure and allows the smoke to be pushed up.
This effect is also responsible for houses losing roofs
during tornadoes (or attacks by big bad wolves).

When a tornado reduces the pressure on the top of the roof,
the air inside the house lifts the roof off.
Everyday Examples
Bernoulli’s Effect

A fluid moving past an object is equivalent to the
object moving in the fluid, so the Bernoulli effect
should occur in these situations.



A tarpaulin over the back of a truck lifts up as the truck
travels down the road due to the reduced pressure on the
outside surface of the tarpaulin produced by the truck
moving through the air.
This same effect causes your car to be sucked toward a
truck as it passes you going in the opposite direction.
The upper surfaces of airplane wings are curved so that the
air has to travel a farther distance to get to the back edge of
the wing.
Therefore, the air on top of the wing must travel faster than
that on the underside and the pressure on the top of the
wing is less, providing lift to keep the airplane in the air.
Work Done by Varying Forces

The work done by a
variable force acting
on an object that
undergoes a
displacement is equal
to the area under the
graph of F versus x
Other Forms of Energy



Roller coasters and pendulums do it; but we can identify
other places where kinetic energy is temporarily stored.
For example, a moving ball can compress a spring and
lose its kinetic energy (next slide, Figure 7-12).
While the spring is compressed, it stores energy much
like a pendulum bob at the top of its arc. If we latch the
spring while it is in the compressed state, we can store
its energy indefinitely as elastic potential energy.
Releasing it at some future date will transform the
spring’s elastic potential energy back into the ball’s
kinetic energy.
Figure 7-12
Other Forms of Energy
Potential Energy Stored in a
Spring


Involves the spring constant (or force
constant), k
Hooke’s Law gives the force

F=-kx



F is the restoring force
F is in the opposite direction of x
k depends on how the spring was formed, the
material it is made from, thickness of the wire,
etc.
Potential Energy in a Spring

Elastic Potential Energy


related to the work required to compress a
spring from its equilibrium position to some
final, arbitrary, position x
U= ½ kx2
Conceptual Question
Other Forms of Energy

Question: When a ball is hung from a vertical
spring, it stretches the spring. As it drops, it
loses gravitational potential energy, but this
does not all show up as kinetic energy.
What happens to the gravitational potential
energy?
Answer: The gravitational potential energy is
converted to kinetic energy and to elastic
potential energy of the spring. At the bottom,
it is all elastic potential energy.
Bungee Jumping
Other Forms of Energy


Bungee jumping is a thrilling example of energy
storage and release.
A nylon rope is securely fastened to the ankles
of the jumper, who then dives off a high platform.




As the jumper falls, gravitational potential
energy is converted to kinetic energy.
As the rope stretches, both the kinetic energy and
some additional gravitational potential energy are
converted to elastic potential energy.
When the rope reaches its maximum stretch, the jumper bounces
back up into the air because much of the elastic potential energy is
converted back into kinetic and gravitational potential energy.
After several bounces the bungee jumper is lowered to the ground.

Notice that if there were no loss of mechanical energy, the bungee
jumper would bounce forever!
Power

In previous chapters we discussed how various quantities
change with time.





For example, speed is the change of position with time,
and acceleration is the change of velocity with time.
The change of energy with time is called power.
Power P is equal to the amount of energy converted from one
form to another (W) divided by the time (Δt ) during which this
conversion takes place:
W
P
 Fv
t per second, a metric unit
Power is measured in units of joules
known as a watt (W). One watt of power would raise a 1-kg
mass (with a weight of 10 newtons) a height of 0.1 meter each
second.
J kg  m2
W
s

s2
Some Different Units of Measurement
Power

The English unit for electric power is the
watt, but a different English unit is used
for mechanical power.


One horsepower is equal to 746 watts.
Can define units of work or energy in terms
of units of power:


kilowatt hours (kWh) are often used in electric
bills
1kWh=3.6 x 106 J
Human Power



A human can generate 1500 watts
(2 horsepower) for very short periods of
time, such as in weightlifting.
The maximum average human power for an
8-hour day is more like 75 watts (0.1
horsepower).
Each person in a room generates thermal
energy equivalent to that of a 75-watt
lightbulb. That’s one of the reasons why
crowded rooms warm up!
Working It Out
Power



A compact car traveling at 27 m/s (60 mph) on a level
highway experiences a frictional force of about 300 N
due to the air resistance and the friction of the tires with
the road.
Therefore, the car must obtain enough energy by
burning gasoline to compensate for the work done by the
frictional forces each second:
This means that the power needed is 8100 W, or 8.1 kW.

This is equivalent to a little less than 11 horsepower.
Conceptual Question
Power

Question: How much energy is
required to leave a 75-watt yard light
on for 8 hours?
Answer:
ΔE = PΔt
= (75 watts)(8 hours)
= 600 watt-hours
= 0.6 kWh.