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.

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. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Potential energy per unit volume of the wire is

u = \frac{1}{2} \times \frac{( Stress ) ^{2}}{Young’s modulus} = \frac{1}{2} \frac{S ^{2}}{Y}

Also, stress,

S = \frac{Force}{Area}

Given:

F_{1} = F_{2}

∴ \frac{S _{1}}{S _{2}} = \frac{A _{2}}{A _{1}}

In this case,

Y_{1} = Y_{2}

∴ \frac{u _{1}}{u _{2}} = (\frac{S _{1}}{S _{2}})^{2} = (\frac{A _{2}}{A _{1}})^{2}

= [(\frac{d _{2}}{d _{1}})^{2}]^{2} = \frac{1}{16}

∴ u_{2} = 16 u_{1}

Q. 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.

Q. 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.