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

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,

& 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{aligned}$$ Substituting the values in the above equation we get $$\begin{aligned} & E=\tfrac{1}{2}\times 2\times {{10}^{10}}\times {{(0.0006)}^{2}} \\\ & E=3600 \\\ \end{aligned}$$ Thus, energy per unit volume stored is 3600 $$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.