Potential Energy; Stored Energy

49

Stored Energy

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of Contents

50

Where does Kinetic Energy Come From?

Imagine a roller coaster car that is at the top of the first hill and is
stopped.

Does the car stay
stopped at the top
of the hill for the
entire ride?

What happens?

51

Where does Kinetic Energy Come From?

Once the car leans over the
edge, gravity pulls it down.
The ride is taking advantage
of the gravitational attraction
between the car and Earth to
give the car kinetic energy
and make it go faster as it
falls.

The kinetic energy the car is
receiving is coming from
another type of energy called
potential energy.

52

Where does Kinetic Energy Come From?

Potential energy is energy stored in an object due to the object's
position. The roller coaster car on the previous slide had stored
energy due to its height above the ground.

There are two forms of potential energy that we will be looking at in
this unit:

Gravitational Potential Energy

and

Elastic Potential Energy

53

Gravitational Potential Energy

The potential energy due to
elevated positions is called
gravitational potential energy.

Gravitational potential energy is
stored energy and it can be used at
a later time to cause an object to
move.

Once the person steps off the
diving board, the gravitational
potential energy is converted into
kinetic energy and the person falls
(moves!)

54

Gravitational Potential Energy

Work is required to elevate objects against Earth's
gravity.

For example, work is done on the truck to elevate it
off the ground. The amount of work done on the
truck is equal to the truck's gravitational potential
energy at this new height.

55

Gravitational Potential Energy

Gravitational potential energy is determined by three factors: mass,
gravitational acceleration, and height. All three factors are directly
proportional to energy.

Mass: The heavier the object is, the _______
gravitational potential energy the object has.

Gravitational Acceleration: The larger the 'g', the _________
gravitational potential energy the object has. Since gravity on Earth
is considered a constant, this will not change.

Height: The higher the object is off the ground, the _________
gravitational potential energy the object has.

more

more

more

GREATER

GREATER

GREATER

*on Earth g = 9.8 m/s2

56

h = 2 m

h = 2 m

m = 2 kg

m = 1 kg

In this picture, the mass of a tennis ball was doubled when it was at
the same height off of the ground.

How Does Mass Affect Gravitational
Potential Energy?

How does the gravitational
potential energy compare
for the two objects?

57

h = 2 m

h = 2 m

m = 2 kg

m = 1 kg

Mass: doubled

Gravitational Acceleration:
stayed the same, no change

Height: stayed the same, no change

Since the only thing that changed was the mass, which doubled,
the gravitational potential energy also doubled.

How Does Mass Affect Gravitational
Potential Energy?

59

h = 4 m

h = 2 m

Mass: stayed the same, no change

Gravitational Acceleration: stayed the
same, no change

Height: doubled

Since the only thing that changed was the height which doubled,
the gravitational potential energy also doubled.

How Does Height Affect Gravitational
Potential Energy?

58

h = 4 m

h = 2 m

In this picture, a tennis ball is lifted to a height that is twice as high.

How Does Height Affect Gravitational
Potential Energy?

How would the gravitational
potential energy compare at the
higher height?

Multiple Choice

21 A bowling ball, which has a mass that is 30 times larger than a softball, is lifted to the same height as the softball. How does the gravitational potential energy of the bowling ball compare to the softball?

• they are the same

• thirty times smaller

• ten times as large

• thirty times as large

60

21 A bowling ball, which has a mass that is 30 times larger
than a softball, is lifted to the same height as the softball.
How does the gravitational potential energy of the
bowling ball compare to the softball?

A they are the same

B thirty times smaller

C ten times as large

D thirty times as large

D

Answer

*since height is the same, we
only need to look at mass which

is 30 x larger and directly proportional
so GPE is also 30 x larger

Multiple Choice

22 Two balloons are floating in the sky. If one balloon is floating at a height of 30 m and the other identical balloon, is floating at a height of 45 m, how much larger is the gravitational potential energy of the higher balloon compared to the lower one?

• half as large

• they are the same

• they are the same

• twice as large

61

22 Two balloons are floating in the sky. If one balloon is
floating at a height of 30 m and the other identical
balloon, is floating at a height of 45 m, how much larger is
the gravitational potential energy of the higher balloon
compared to the lower one?

A half as large

B they are the same

C 1.5 times larger

D twice as large

Answer

C

1.5 x larger

since the balloons are identical, mass
is the same, so we only need to look

at height. 45 m is 15 m higher than

30 m which is half of it. This makes
it 1.5 x larger

62

Calculating Gravitational Potential Energy

Gravitational potential energy can be solved by using the
equation:

variable

units

Name

Gravitational
Potential Energy
m
m

Gravity

GPE = mgh

Let's fill in the table below.

63

Example ­ Calculating Gravitational
Potential Energy

A basketball with a mass of 0.5 kg, is held at a height of 2 m
above the ground. How much gravitational potential energy
does the basketball possess?

GPE = mgh

GPE = (0.5 kg)(9.8 m/s2)(2 m)

GPE = 9.8 J

Click on the box to see the solution.

Multiple Choice

23 A 50 kg diver is standing on top of a 10 m platform. How much gravitational potential energy does he have?

• 4,000J

• 4,900J

• 900J

64

23 A 50 kg diver is standing on top of a 10 m platform. How
much gravitational potential energy does he have?

Answer

Multiple Choice

24 A 3,000 kg hot air balloon is hovering at a height of 100 m above Earth's surface. How much gravitational potential energy does it possess?

• 3,940,000J

• 2,000,000J

• 2,940,000J

65

24 A 3,000 kg hot air balloon is hovering at a height of 100
m above Earth's surface. How much gravitational
potential energy does it possess?

Answer

GPE= mgh
= 3000 kg(9.8 m/s2)(100 m)
= 294,000,000 J

66

GPE= mgh

GPE = mgh

GPE = mgh

GPE = mgh

GPE = mgh

GPE = mgh

Thinking Mathematically

If any of these decrease,

then the GPE decreases

by the same factor.

We know that GPE is directly

proportional to mass, to

gravity, and to height. This

means that as any of these

increase, the GPE increases

by the same factor.

73

Elasticity: The more elastic a material is, the more elastic potential
energy the object has.

Distance of stretch (or compression): The larger
the distance the elastic material is stretched (or
compressed) the more elastic potential energy
it has.

Elastic Potential Energy

Multiple Choice

25 A ball is at a height of 30 m. It is then moved to a height of 60m. By what factor does the GPE increase?

• 2

• 1

• 3

• 4

67

25 A ball is at a height of 30 m. It is then moved to a height
of 60m. By what factor does the GPE increase?

Answer

2

Multiple Choice

26 A 3 kg object and a 9 kg object are elevated from the same height. Which has more GPE?

• 3 Kg

• 9Kg

68

26 A 3 kg object and a 9 kg object are elevated from the
same height. Which has more GPE?

A 3 kg object

B 9 kg object

Answer

Multiple Choice

27 A 3 kg object and a 9 kg object are dropped from the same height. How much less is the GPE of the 3 kg object than the 9 kg object?

• 1/2 less

• 1/3 less

• 1/4 less

69

27 A 3 kg object and a 9 kg object are dropped from the
same height. How much less is the GPE of the 3 kg
object than the 9 kg object?

Answer

1/3

Multiple Choice

28 An object is 5 m above the ground. The object triples its mass and doubles its height. By what factor does the object's GPE change?

• 2

• 3

• 5

• 6

70

28 An object is 5 m above the ground. The object triples its
mass and doubles its height. By what factor does the
object's GPE change?

Answer

6

71

Elastic Potential Energy

Another type of stored energy is
called elastic potential energy.

Looking at the picture to the right,
can you come up with an idea about
what elastic potential energy is?

72

Elastic potential energy is determined by two factors: the elasticity
of the material and how far it is stretched or compressed.

Elastic Potential Energy

Think about what you know
about rubber bands.

Do you think elasticity and
distance stretched are directly
proportional or indirectly
proportional to the energy?

Talk about this at your table.

74

What is the Difference Between Stretching
and Compression in a Spring?

Think about a slinky sitting
on a desk. A spring has no
potential energy stored in it
if it is neither stretched nor
compressed. This relaxed
state is shown in figure (a).

Stretching a spring is
caused when the spring is
pulled increasing the length
of the spring compared to
the relaxed length, as shown
in figure (b).

(a)

(b)

(c)

75

What is the Difference Between Stretching
and Compression in a Spring?

Compressing a spring is caused when the spring is squeezed. This
causes a decrease in the length of the spring compared to the relaxed
length, as shown in figure (c).

The stretched and compressed spring below store the same elastic
potential energy because both springs are displaced the same
distance, x.

(a)

(b)

(c)

relaxed

stretched

compressed

no EPE stored

76

How Does Elastic Potential Energy Depend
Upon Compression and Stretching?

Both pictures to the right show a spring, which
is an elastic material.

In the top picture the spring is stretched from
its relaxed state.

In the bottom picture, the spring is compressed
from its relaxed state.

For each case, is elastic potential energy stored in the
spring?

Multiple Choice

29 A child jumps on a trampoline. When will the trampoline have more elastic potential energy?

• When the child is standing on the trampoline

• When the child is in the air

• The trampoline will always have the same elastic potential energy

• When the child lands on the trampoline after jumping

77

29 A child jumps on a trampoline. When will the trampoline
have more elastic potential energy?

A When the child is standing on the trampoline

B When the child is in the air

C When the child lands on the trampoline after
jumping

D The trampoline will always have the same elastic
potential energy

Answer

C

*their weight
will cause the
trampoline to stretch the most at this moment

78

Calculating Elastic Potential Energy

Elastic potential energy can be solved by using the equation:

EPE = kx2
1
2

EPE = Elastic Potential Energy (J)

k = spring constant (N/m)

x = distance of stretch or compression (m)

Elastic Potential Energy

spring constant

distance stretched
or compressed

79

Spring Constant

The energy and distance variables in this equation are likely familiar.

1
2
EPE = kx2

But what is the spring constant

(k)? Look at the two springs to

the right. Which do you think

would be easier to stretch?

Every spring has a different

degree of stretchiness and

that is what the spring

constant represents.

80

Spring Constant

1
2
EPE = kx2

Breaking down the units for

spring constant also

explains what the variable

represents.

Can you explain what

Newtons per Meter (N/m)

means?

81

Example ­ Calculating Elastic Potential
Energy

A spring that has a spring constant of 10 N/m, is stretched a
distance of 1 m from its relaxed length. How much elastic
potential energy is stored in the spring?

EPE = kx2

EPE = ( )(10 N/m)(1 m)2

EPE = ( )(10 N/m)(1 m2)

EPE = (5 N*m)

EPE = 5 J

1
2
1
2

1
2

Click on the box to see the solution.

Teacher Notes

Multiple Choice

30 A child bouncing on a pogo stick compresses the spring by 0.25 m. If the spring constant of the spring on the bottom of the pogo stick is 200 N/m, what is the elastic potential energy stored in the spring when it is compressed?

• 6.25 J

• 625 J

• 62.5 J

82

30 A child bouncing on a pogo stick compresses the spring
by 0.25 m. If the spring constant of the spring on the
bottom of the pogo stick is 200 N/m, what is the elastic
potential energy stored in the spring when it is
compressed?

Answer

EPE= 1/2 kx2
= 1/2 (200 N/m) (0.25 m)2
= 1/2 (200 N/m)(0.0625 m2)
= 100 N/m (0.0625 m2)
EPE = 6.25 J

Multiple Choice

31 A rubber band with a spring constant of 40 N/m is pulled back 0.5 m. How much elastic potential energy is stored in the elastic band?

• 5 J

• 0.5 J

• 50 J

• 500 J

83

31 A rubber band with a spring constant of 40 N/m is pulled
back 0.5 m. How much elastic potential energy is stored
in the elastic band?

Answer

EPE= 1/2 kx2
= 1/2 (40 N/m) (0.5 m)2
= 20 N/m (0.25 m2)

EPE = 5 J

Multiple Choice

32 Which of the following would you expect to have the smallest spring constant?

• a garage door spring

• a slinky

• a spring in a pen

• a trampoline spring

84

32 Which of the following would you expect to have the
smallest spring constant?

A a garage door spring

B a slinky

C a spring in a pen

D a trampoline spring

Answer

C

85

Thinking Mathematically

EPE = kx2
1
2

1
2
KE = mv2

Notice that the equation for EPE is similar to the equation for KE.

Remember that in the equation for KE, energy was directly

proportional to the mass and it was also directly proportional to

the square of the velocity.

What do you think the relationship is between EPE and the
spring constant, k?

What do you think is the relationship between EPE and the
distance, x, the spring is stretched or compressed?

86

Thinking Mathematically

EPE = kx2
1
2

EPE is _________________________ to the spring constant.

EPE is _________________________ to the square of the

distance the spring is compressed or stretched.

directly proportional

directly proportional

DIRECTLY proportional

DIRECTLY proportional

Multiple Choice

33 If the spring constant, k, is tripled, by what factor does the EPE increase?

• 1

• 2

• 3

• 6

87

33 If the spring constant, k, is tripled, by what factor does
the EPE increase?

Answer

3

Multiple Choice

34 If the spring constant, k, is halved, by what factor does the EPE decrease

• 1

• 2

• 1/2

• 1/4

88

34 If the spring constant, k, is halved, by what factor does
the EPE decrease?

Answer

Multiple Choice

35 If the distance a spring is stretched is increased by a factor of 6, by what factor is the EPE increased?

• 8

• 12

• 24

• 36

89

35 If the distance a spring is stretched is increased by a
factor of 6, by what factor is the EPE increased?

Answer

36