Potential Energy
An object can store energy as the result of its position. For
example, the heavy ball of a demolition machine is storing energy when it is
held at an elevated position. This stored energy of position is referred to as
potential energy. Similarly, a drawn bow is able to store energy as the result
of its position. When assuming its usual position (i.e.,
when not drawn), there is no energy stored in the bow. Yet when its position is
altered from its usual equilibrium position, the bow is able to store energy by
virtue of its position. This stored energy of position is referred to as
potential energy. Potential energy is the
stored energy of position possessed by an object.
Gravitational Potential Energy
The two examples above illustrate the two forms of potential
energy to be discussed in this course - gravitational potential energy and elastic
potential energy.
Gravitational potential energy is the energy stored in an object as the result
of its vertical position or height. The energy is stored as the result of the
gravitational attraction of the Earth for the object. The gravitational
potential energy of the massive ball of a demolition machine is dependent on
two variables - the mass of the ball and the height to which it is raised.
There is a direct relation between gravitational potential energy and the mass
of an object. More massive objects have greater gravitational potential energy.
There is also a direct relation between gravitational potential energy and the
height of an object. The higher that an object is elevated, the greater the
gravitational potential energy. These relationships are expressed by the
following equation:
PEgrav = mass • g • height
PEgrav = m *• g • h
In the above equation, m represents
the mass of the object, h represents the height of the object and grepresents the gravitational field strength (9.8 N/kg on Earth) - sometimes
referred to as the acceleration of gravity.
To determine the gravitational potential energy of an object,
a zero height position must 
first be arbitrarily assigned. Typically, the ground is considered to be
a position of zero height. But this is merely an arbitrarily assigned position
that most people agree upon. Since many of our labs are done on tabletops, it is often customary to assign the tabletop to be the zero height position. Again this is
merely arbitrary. If the tabletop is the
zero position, then the potential energy of an object is based upon its height
relative to the tabletop. For example, a
pendulum bob swinging to and from above the tabletop has
a potential energy that can be measured based on its height above the tabletop. By measuring the mass of the bob and the height
of the bob above the tabletop, the potential
energy of the bob can be determined.
Since the gravitational potential energy of an object is
directly proportional to its height above the zero position, a doubling of the
height will result in a doubling of the
gravitational potential energy. A tripling of the
height will result in a tripling of the
gravitational potential energy.
Use this principle to determine the blanks in the following diagram.
Knowing that the potential energy at the top of the tall platform is 50 J, what
is the potential energy at the other positions shown on the stair steps and the
incline?
A: PE = 40 J (since the same mass is elevated to
4/5-ths height of the top stair)
B: PE = 30 J (since the same mass is elevated to
3/5-ths height of the top stair)
C: PE = 20 J (since the same mass is elevated to
2/5-ths height of the top stair)
D: PE = 10 J (since the same mass is elevated to 1/5-ths
height of the top stair)
E and F: PE = 0 J (since the same mass
is at the same zero height position as shown for the bottom stair).
Elastic Potential Energy
The second form of potential energy that we will discuss is elastic
potential energy. Elastic potential energy is the
energy stored in elastic materials as the result of their stretching or
compressing. Elastic potential energy can be stored in rubber bands,
bungee chords, trampolines, springs, an arrow
drawn into a bow, etc. The amount of elastic potential energy stored in such a
device is related to the amount of stretch of the device - the more stretch,
the more stored energy.
Springs are a special instance of a device that can store
elastic potential energy due to either compression or stretching. A force is
required to compress a spring; the more compression there is, the more force
that is required to compress it further. For certain springs, the amount of
force is directly proportional to the amount of stretch or compression (x); the
constant of proportionality is known as the spring constant (k).
Fspring = k • x
Such springs are said to follow Hooke's Law. If a spring is
not stretched or compressed, then there is no elastic potential energy stored
in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally
assumes when there is no force applied to it. In terms of potential energy, the
equilibrium position could be called the zero-potential energy position. There
is a special equation for springs that relates the amount of elastic potential
energy to the amount of stretch (or compression) and the spring constant. The
equation is
PEspring = 0.5 • k • x2
where k = spring constant
x = amount of compression
(relative to equilibrium position)
To summarize, potential energy is the energy that is stored
in an object due to its position relative to some zero position. An object
possesses gravitational potential energy if it is positioned at a height above
(or below) the zero height. An object possesses elastic potential energy if it
is at a position on an elastic medium other than the equilibrium position.
Check Your Understanding
Check your understanding of the concept of potential energy
by answering the following questions. When finished, click the button to view
the answers.
1. A cart is loaded with a brick and pulled at constant speed along an
inclined plane to the height of a seat-top. If the mass of the loaded cart is
3.0 kg and the height of the seat top is 0.45 meters, then what is the
potential energy of the loaded cart at the height of the seat-top?
PE
= m*g*h
PE = (3 kg ) * (9.8 m/s/s) *
(0.45 m)
PE = 13.2 J
2. If a force of 14.7 N is used to drag the loaded cart (from
previous question) along the incline for a distance of 0.90 meters, then how
much work is done on the loaded cart?
W = F * d * cos Theta
W = 14.7 N * 0.9 m * cos (0 degrees)
W = 13.2 J