A copper wire is held at the two ends by rigid supports. At $30^{0} C$ , the wire just taut, with negligible tension. Find the speed of transverse waves in this wire at $10^{0} C$ (Young modulus of copper $= 1.3 \times 10^{11} N / m^{2}$ )
Co-efficient of linear expansion of copper $= 1.7 \times 10^{- 5} /^{0} c$
Density of copper $= 9 \times 10^{3} k g / m^{3}$

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$70 m / \sec$

$60 m / \sec$

$40 m / \sec$

$100 m / \sec$

answer is A.

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Detailed Solution

$Y = \frac{s t r e s s}{s t r a i n} \Rightarrow Y = \frac{F / A}{Δ l / l} \Rightarrow Y = \frac{F l}{A Δ l} \Rightarrow F = \frac{Y A Δ l}{l} A l s o, c h a n g e i n l e n g t h d u e t o t e m p e r a t u r e = Δ l = α l ∆ θ F = Y A α ∆ θ S p e e d o f w a v e o n s t r i n g : V = \sqrt{\frac{F}{μ}} w h e r e μ = \frac{m}{l} = A ρ ∴ V = \sqrt{\frac{Y α ∆ θ}{ρ}} = \sqrt{\frac{1.3 \times 10^{11} \times 1.7 \times 10^{- 5} \times 20}{9 \times 10^{3}}} V = 70 m / \sec .$