# Potential Energy

## What is Potential Energy?

As we know, an object can store energy as the result of its position. In the case of a bow and an arrow, when the bow is drawn, it stores some amount of energy, which is responsible for the kinetic energy it gains, when released. Similarly, in the case of a spring, when it is displaced from its equilibrium position, it gains some amount of energy which we observe in the form of stress we feel in our hand upon stretching it. We can define potential energy as a form of energy that results from the alteration of its position or state.

### What are the different types of potential energy?

There are two main types of potential energy and they are:

• Gravitational Potential Energy
• Elastic Potential Energy

## Gravitational Potential Energy

Gravitational potential energy of an object is defined as the energy possessed by an object rose to a certain height against gravity. We shall formulate gravitational energy with the following example.

• Consider an object of mass = m.
• Placed at a height h from the ground, as shown in the figure.

Now, as we know, the force required to raise the object is equal to m×g of the object.

As the object is raised against the force of gravity, some amount of work (W) is done on it.

Work done on the object = force × displacement.

So,

 W = m×g×h = mgh

Above is the potential energy formula.

As per the law of conservation of energy, since the work done on the object is equal to m×g×h, the energy gained by the object = m×g×h, which in this case is the potential energy E.

E of an object raised to a height h above the ground = m×g×h

It is important to note that, the gravitational energy does not depend upon the distance travelled by the object, but the displacement i.e., the difference between the initial and the final height of the object. Hence, the path along which the object has reached the height is not taken into consideration. In the example shown above, the gravitational potential energy for both the blocks A and B will be the same.

## Elastic Potential Energy

Elastic potential energy is the energy stored in objects that can be compressed or stretched such as rubber bands, trampoline and bungee cords. The more an object can stretch, the more elastic potential energy it has. Many objects are specifically designed to store elastic potential energy such as the following:

• A twisted rubber band that powers a toy plane
• An archer’s stretched bow
• A bent diver’s board just before a diver dives in
• Coil spring of a wind-up clock

An object that stores elastic potential energy will typically have a high elastic limit, however all elastic objects have a threshold to the load they can sustain. When deformed beyond the elastic limit, the object will no longer return to its original shape.

An elastic potential energy can be calculated using the following formula:
$U=\frac{1}{2}kx^2$

where, U is the eleastic potential energy, k is the spring force constant and x is the string stretch length in m.

## Potential Energy Examples

Following are a few potential energy examples:

### Potential Energy Practice Question:

Q1: What will be the gravitational potential energy possessed by a ball of mass 1 kg when it is raised to a height of 6 m above the ground. (g = 9.8 m s–2)
Solution:
Here, mass of the object (m) = 1 kg,
Displacement (height) (h) = 10 m,
Acceleration due to gravity (g) = 9.8 m s–2.
Hence, Potential energy (p) = m×g×h = 1 kg × 9.8 m s–2 × 10 m = 98 J.