Question

Which of the following is true for elastic potential energy density?
A) Energy density $ = \dfrac{1}{2} \times \text{Stress} \times \text{Strain}$
B) Energy density $ = {\left( {\text{Strain}} \right)^2} \times \text{Volume}$
C) Energy density $ = \left( {\text{Strain}} \right) \times \text{Volume}$
D) Energy density $ = \left( {\text{Stress}} \right) \times \text{Volume}$

Answer 1

Hint: The energy that is possessed by the body in its position is called potential energy. Energy is related to the virtue of the position of the body with reference to the zero position. When a material is stretched or compressed, the energy that is stored in those kinds of materials is called elastic potential energy. To answer the given question, consider the formula of the elastic potential energy.

Complete answer:
As discussed in the hint, potential energy is the energy that is possessed by the body of the materials. It is always a stored form of energy. There are two types of potential energy. One is elastic potential energy and another one is spring potential energy.
Have you ever seen someone who is jumping on the trampoline? When they jump on the trampoline the trampoline stretches downwards and the person jumping on that is thrown upwards. This is the example of the elastic potential energy. The elastic potential energy is nothing but the energy that is stored on the elastic materials. The materials that undergo stretch or compress are called elastic materials. The amount of energy that is stressed or stretched will always be proportional to the stored amount of energy of the material.
To answer the given question the formula of this elastic energy will help. The elastic potential energy is given by,
$ \Rightarrow \dfrac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume}$
The energy density of the elastic potential energy is given by,
$ \Rightarrow \dfrac{{\text{Energy}}}{{\text{Volume}}}$
Substitute the values of energy in the density formula.
 $ \Rightarrow \dfrac{{ \dfrac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume}}}{{\text{Volume}}}$
Cancel out the common term.
$ \Rightarrow \dfrac{1}{2} \times \text{Stress} \times \text{Strain}$
Therefore, the elastic potential energy density is $\dfrac{1}{2} \times \text{Stress} \times \text{Strain}$ .
Hence option $\left( A \right)$ is the correct answer.

Note: When the applied force is zero, the equilibrium of the position of the material, the potential energy at such positions is zero. When a spring is stretched or stressed, the force that we experience will always be equal to the force that we apply in the opposite direction. As soon as the spring is relieved from the stretch or stress, it always regains its original position. This energy is known as spring potential energy.