Question

If x, longitudinal strain is produced in a wire of young's modulus y then energy stored in the material of the wire per unit volume is.......
(A) $$y{x}^2$$
(B) $$2y{x}^2$$
(C) $${\displaystyle \frac{y^{2}x}{2}}$$
(D) $${\displaystyle \frac{y{x}^2}{2}}$$

Answer 1

Hint: Strain and stress comes into play when on a wire a force is applied to one of the ends and then there is change in length and area on account of application of that force. Stress is defined as the force acting per unit area and strain is defined as the ratio of change in length to the original length. The amount of stress and strain is determined by the Young’s modulus, as Young's modulus of elasticity depends upon the nature of material of the substance.

Complete step by step answer:
Given strain= x
Young modulus= y
Energy stored= U
We know there exist a relationship between the stress, the strain and the potential energy stored inside the wire and it is given by:
U= $${\displaystyle \frac{1}{2}}$$ $$\times$$stress $$\times$$strain---(1)
But the relationship between stress, strain and Young modulus is given by,
Stress= young modulus $$\times$$strain, thus eq (1) gets modified as,
$$U={\displaystyle \frac{1}{2}}\times xy\times x$$
$$\Rightarrow U={\displaystyle \frac{y{x}^2}{2}}$$

So, the correct answer is “Option D”.

Additional Information:
Young’s modulus is the property of the material which measures the elasticity of the material. Stress is the force applied to a material, divided by the material's cross-sectional area.

Note:
stress is measured in units of pressure viz pascals and since strain is the ratio of change in length to the original length, it is a dimensionless quantity. The energy gets stored on account of the position and thus is a form of potential energy. The strain is the displacement of material that results from applied stress.