What is the equation for elastic potential energy?

The equation for elastic potential energy is \( E = \frac{1}{2} k x^2 \).

Elastic potential energy is the energy stored in an object when it is stretched or compressed. This type of energy is most commonly associated with springs, rubber bands, and other elastic materials. The equation \( E = \frac{1}{2} k x^2 \) helps us calculate the amount of energy stored in such objects.

In this equation, \( E \) represents the elastic potential energy measured in joules (J). The symbol \( k \) stands for the spring constant, which is a measure of the stiffness of the spring or elastic material. The spring constant is measured in newtons per metre (N/m). The variable \( x \) represents the displacement or the amount by which the object is stretched or compressed from its original position, measured in metres (m).

To understand this better, let's break it down with an example. Imagine you have a spring with a spring constant \( k \) of 200 N/m, and you compress it by 0.1 metres. Plugging these values into the equation, you get:

\[ E = \frac{1}{2} \times 200 \times (0.1)^2 \]

\[ E = \frac{1}{2} \times 200 \times 0.01 \]

\[ E = 1 \, \text{J} \]

So, the elastic potential energy stored in the spring is 1 joule.

This equation is derived from Hooke's Law, which states that the force needed to extend or compress a spring by some distance \( x \) is proportional to that distance. The formula for Hooke's Law is \( F = kx \). When you integrate this force over the distance \( x \), you get the elastic potential energy formula.

Understanding this concept is crucial in physics as it helps explain how energy is conserved and transferred in systems involving elastic materials.