First slide

Elastic Modulie

Question

The ratio of diameters of two wires, made of same material is 2:1. The ratio of their lengths is 1:2. On applying the same tensile force to the wires, the ratio of elastic potential energies stored in the wires will be

Easy

A

1:2
B

1:4
C

1:16
D

1:8

Solution

stress $σ = \frac{F}{π r^{2}}$
strain energy per unit volume $u = \frac{σ^{2}}{2 Y}$
$∴$ Total elastic potential energy stored in the wire, $U = u \times π r^{2} l = \frac{F^{2} l}{2 π Y r^{2}}$
$∴ \frac{U_{1}}{U_{2}} = (\frac{l_{1}}{l_{2}}) {(\frac{r_{2}}{r_{1}})}^{2} = (\frac{1}{2}) {(\frac{1}{2})}^{2} = \frac{1}{8}$

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