First slide

Elastic behaviour of solids

Question

A uniform cylindrical wire is subjected to a longitudinal tensile stress of $5 \times 10^{7} N / m^{2}$ . Young's modulus of the material of the wire is $2 \times 10^{11} N / m^{2}$ . The volume change in the wire is 0.02%.The fractional change in the radius is

Moderate

A

$0.25 \times 10^{- 4}$
B

$0.5 \times 10^{- 4}$
C

$1.0 \times 10^{- 4}$
D

$1.5 \times 10^{- 4}$

Solution

$Y = \frac{stress}{ΔL / L} or \frac{ΔL}{L} = \frac{stress}{Y} = \frac{5 \times 10^{7}}{2 \times 10^{11}} = 2 .5 \times 10^{- 4}$
$\begin{array}{l} Now, V = {πr}^{2} L \\ \frac{ΔV}{V} = \frac{πΔ (r^{2} L)}{{πr}^{2} L} \\ or \frac{ΔV}{V} = \frac{r^{2} ΔL + L \times 2 rΔr}{r^{2} L} or \frac{ΔV}{V} = \frac{ΔL}{L} + 2 \frac{Δr}{r} \\ or 2 \frac{Δr}{r} = \frac{ΔV}{V} - \frac{ΔL}{L} or 2 \frac{Δr}{r} = \frac{0 .02}{100} - 2 .5 \times 10^{- 4} \\ or \frac{Δr}{r} = 1 \times 10^{- 4} - \frac{2 .5}{2} \times 10^{- 4} = - 0 .25 \times 10^{- 4} \end{array}$

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