Question

Two wires of the same material and same length but diameters in the ratio $2: 1$ are stretched by the same force. The potential energy per unit volume of the second wire will be _______ times that of first wire.

Answer 1

Potential energy per unit volume of the wire is
$u =\frac{1}{2} \times \frac{(\text { Stress })^{2}}{\text { Young's modulus }}=\frac{1}{2} \frac{ S ^{2}}{ Y }$
Also, stress, $S=\frac{\text { Force }}{\text { Area }}$
Given: $F _{1}= F _{2}$
$\therefore \frac{ S _{1}}{ S _{2}}=\frac{ A _{2}}{ A _{1}}$
In this case, $Y_{1}=Y_{2}$
$\therefore \frac{u_{1}}{u_{2}}=\left(\frac{S_{1}}{S_{2}}\right)^{2}=\left(\frac{A_{2}}{A_{1}}\right)^{2}$
$=\left[\left(\frac{d_{2}}{d_{1}}\right)^{2}\right]^{2}=\frac{1}{16} $
$\therefore u_{2}=16 u_{1}$