Question

A metal rod of Young's modulus \[2\times {{10}^{10}}N/{{m}^{2}}\] undergoes an elastic strain of 0.06%. The energy per unit volume stored in \[J/{{m}^{3}}\] is
(A)3600
(B)7200
(C)10800
(D)14400

Answer 1

Hint: Young’s modulus is given in the question, it means in this problem there we have to use the concept of extension of the length of the material when a force acts on it. Also, the elastic strain is given and that too in percentage that we can convert into decimals.

Complete step by step answer:
From the above data, we see that
Young’s modulus, Y= \[2\times {{10}^{10}}N/{{m}^{2}}\]
Strain, S= 0.06% =0.0006
We need to find the energy per unit volume stored.
From the formula,
\[\begin{align}
  & E=\tfrac{1}{2}\times stress\times strain \\
 & E=\tfrac{1}{2}\times (strain\times Y)\times strain \\
 & E=\tfrac{1}{2}\times Y\times {{S}^{2}} \\
\end{align}\]
Substituting the values in the above equation we get
\[\begin{align}
  & E=\tfrac{1}{2}\times 2\times {{10}^{10}}\times {{(0.0006)}^{2}} \\
 & E=3600 \\
\end{align}\]
Thus, energy per unit volume stored is 3600 \[J/{{m}^{3}}\]

So, the correct answer is “Option A”.

Additional Information:
Stress is the force applied to a material, divided by the material's cross-sectional area. The strain is the displacement of material that results from applied stress. The stress and strain can be normal, shear or mixture depending upon the situation.

Note:
We should keep in mind that Young modulus is a mechanical property that measures the stiffness of solid material and is different for different materials. If we are given a problem in which two rods of different materials are attached then we have to use two sets of young’s moduli for each of them. Also, the strain is a ratio and so it has no units. The stress is measured in units of the pressure.