# Elastic Potential Energy and Spring Potential Energy

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## Elastic Potential Energy

Elastic energy can be defined as the mechanical potential energy reserved in the configuration of a material or physical system. It is exposed to elastic deformation by work performed upon it. Elastic energy generated when objects are temporarily stretched, compressed, or generally deformed in any manner. Elasticity theory mainly develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used for calculating positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as sound energy and kinetic energy, when the object is allowed to reform by its elasticity.

$U = \frac{1}{2}$  $k \Delta x^{2}$

### What Causes Elastic Energy?

A force acting on an object temporarily changes its shape, such as when you stretch an elastic band or squish a squishy ball with your hand.

### Spring Potential Energy

Since the potential energy's change of an object between two positions is equal to the work that must be done to move the object from one point to another, the calculation of potential energy is identical to calculating the work. Since the force requires stretching a spring changes with distance, the calculation of the work involves an integral.

$W = \int_{0}^{x} kxdx = k \frac{x^{2}}{2}$

### The Potential Energy of A Spring

When we compress or extend a stretched spring, we feel a force equal to that applied by us in the opposite direction. So the reason for this happening is when a spring deviates from its mean position, it tends to restore its equilibrium by exerting a force equal and opposite to the external force. But the question remains in which way can this force be helpful to us? We all must have seen the uses of spring force in bicycle carriers and launching devices. The energy gained by disturbing the equilibrium of the spring is used as its potential energy and converted to other forms.

### Hooke’s Law

The force that requires stretching an elastic object like a metal spring is always directly proportional to the spring extension for small-scale distances. The force applied back by the spring is known as Hooke's Law.

$\overline{F}_{s} = - k \overrightarrow{x}$

Where Fs  is the force exerted by the spring, x is the displacement relative to the unstretched length of the spring, and k is the spring constant.

The spring force can be called a restoring force because the force exerted by the spring is always in the opposite direction to the displacement, this is the reason behind a negative sign in the Hooke's law equation. Pulling down on a spring stretches the spring downward, which results in the spring exerting an upward force.

### We Have Listed a few Uses of Elastic Energy Below:

A spring is used to reserve elastic potential energy in many mechanical devices like the shock absorbers present in cars. Elastic energy can be used in many ways since the spring can remain in its compressed or stretched state for extended periods without dissipating energy. Balloons, rubber bands, bungees, and trampolines use elastic energy for the stretch. We can find uses of elastic energy in squishy balls, a bow and arrow, and coiled springs. Catapults and slingshots are also uses of elastic energy.

### Solved Examples

Question 1: What happens when a spring is stretched too far?

Answer: If a force is applied to spring to exceed its elastic limit, then it will no longer return to its original shape.

Question 2: How to analyse a spring force versus displacement graph?

Answer: The area under the force in the spring versus displacement curve is done in the spring. The diagram below shows a plot of force on the spring versus displacement where displacement is 0 when the spring is unstretched. The work is done on a spring store elastic potential energy Us in the spring until the spring returns to its original length. Therefore, the Us is equal to the work done and also to the area under the curve.