# Elastic Potential Energy Formula

## Elastic Potential Energy

We know that the energy in certain physical equipment such as spring, the energy is stored in the form of potential energy. The elastic potential energy is the energy stored due to the deformation when an external force is applied. The energy is stored until the force is removed and the object bounces back to its original shape (or size), doing work in the process. The deformation of the object may incorporate the process such as compressing, stretching or twisting the object. In this article, we will learn what is elastic potential energy and elastic potential energy formula along with solved problems that will boost our understanding.

### Elastic Potential Energy Formula

Before we start with the elastic potential energy formula, let us have a look at the definition and the meaning of elastic potential energy.

• The elastic potential energy is the total energy obtained as a result of deformation in the shape of an object under consideration.

• As the name itself suggests, it is the elastic potential energy, which implies any object which is capable of regaining its original shape and size after deformation will possess the elastic potential energy.

• The deformation of the objects is only possible as a result of the acquired potential energy which is known as the elastic potential energy. Thus, it can be said that elastic potential energy is the stored energy of a compressible or stretchable object or in other words stored energy of the perfectly elastic materials.

• All the elastic materials such as the twisted rubber, spring, a bouncy ball, get compressed at the moment it strikes the brick wall. All these materials will store the energy in the form of elastic potential energy.

• Any object which is particularly designed to store elastic potential energy must have a high elastic limit, however, all the elastic materials or the objects have a certain limit to the applying the load such that they can sustain. When the elastic materials get deformed beyond the elastic limit, the object can not return to its original shape and size, it will get deformed permanently.

### What is Elastic Potential Energy Formula?

Elastic Potential Energy Formula Derivation:

Let us understand what is potential energy formula. So, the elastic potential energy formula can be derived from the fundamental equation of elastic force. We know that the external force applied to an elastic body is directly proportional to the total displacement in the position. Mathematically, we write:

F = k x…….(1)

Where,

k - The spring constant

x - The displacement in the body as a result of applied external force

It is computed as the work done to stretch the spring which depends on the spring constantk and the displacement stretched. The spring force is a conservative force and conservative forces have potential energies associated with them. Therefore, the total work done in stretching the elastic material will be stored in the form of potential energy and it is known as the elastic potential energy formula. The elastic potential is energy is denoted by the letter U.

Now, the elastic potential energy formula is given by:

U = 1/2 (Force x Displacement)

U = 1/2 (kx) (x)

U = 1/2 k x2 joules………..(2)

Where,

k - The spring constant

x - The displacement in the body as a result of applied external force

Equation (2) is known as the elastic potential energy formula. If we apply the force in a direction opposite to the motion we will end up with negative potential energy, but we should always remember sign convention is just to identify the direction of applied force does mean the energy stored is negative.

### Elastic Potential Energy Examples:

Let us have a look at the solved numerical problems using the elastic potential energy formula.

1. Two balls are connected by a 0.60 m spring that hit a brick wall, compressing spring to 0.30 m.The value of spring constantk is 200 N/m. Then, calculate the elastic potential energy stored due to the compression of the spring.

Sol:

Given,

The total length of the spring = l = 0.60 m

The total compression in the length of spring = l' = 0.30 m

The total displacement as a result of compression = x = 0.60 - 0.30 = 0.30 m

The value of the spring constant = k = 200 N/m

Now our aim is to calculate the elastic potential energy stored due to the compression of the spring. We know that the elastic potential energy is given by the formula:

U = 1/2 k x2 joules

Where,

k - The spring constant

x - The displacement in the body as a result of applied external force

Substituting all the given data in the above equation and simplify it. We get:

U = 1/2 k x2

U = 1/2 (200 N/m) (0.30m)2

U = 9 N-m = 9 joules

Therefore, the elastic potential energy stored due to the compression of the spring is 9 joules.

2. The vertical spring is attached to a load of mass 8kg which is compressed by12 m. Then, calculate the force constant of the spring.

Sol:

Given,

The total mass attached to the vertical spring = m = 8 kg

The total displacement of the spring = x = 12 m

Now our aim is to calculate the spring constant of the vertical spring. We know that force applied can be calculated using the equation:

F = m a

Where,

m - The total mass

a - The total acceleration, here since we are using a vertical spring the only acceleration will be, acceleration due to gravity.

Therefore, we write:

F = mg = 89.8 = 78.4 N

Now, the spring force is given by:

F = k x

k = Fx = 78.4/12 = 6.53 N/m

Therefore, the value of the spring constant is 6.53 N/m.

3. A compressed spring has a potential energy of 60 joules and its spring constant is 200 N/m. Then, estimate the total displacement of the spring due to this stored potential energy.

Sol:

Given,

The total potential energy of the spring = U = 60 joules

The value of the spring constant = k = 200 N/m

Now our aim is to calculate the total displacement of the spring due to the stored elastic potential energy of the spring. We know that the elastic potential energy is given by the formula:

U = 1/2 k x2 joules…….(1)

Where,

k - The spring constant

x - The displacement in the body as a result of applied external force

Rearranging equation (1) for the expression of displacement is given by:

x2 = 2U/k

$\Rightarrow x =\sqrt{\frac{2U}{k}}m$……..(2)

Substituting all the given values in equation (2) and simplifying it. We get:

$\Rightarrow x=\sqrt{\frac{2U}{k}}m=\sqrt{\frac{2\times60}{200}}$

$\Rightarrow x=\sqrt{0.6}m$

x = 0.7746 m 0.8 m

Therefore, the total displacement of the spring due to the stored elastic potential energy of the spring is around 0.8 m.