A copper wire is held at the two ends by rigid supports. At $30^{0}$ the wire is just taut with negligible tension. The speed of transverse waves in this wire at $10^{0} C$ is $10 \times D m / s .$ Find the value of D(Take, $Y_{c u} = 1.3 \times 10^{11} P a,$ $α_{c u} 1.7 \times 10^{- 5} /^{0} C, ρ_{c u} = 9000 k g / m^{3}$ )

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answer is 7.

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Detailed Solution

Thermal stress $= \frac{F}{A} = y α Δ θ,$ where $F$ is tension and $A$ is area or cross-section.
We known, $v = \sqrt{\frac{F}{ρ A}} = \sqrt{\frac{T h e r m a l s t r e s s}{D e n s i t y}} = \sqrt{\frac{Y α Δ θ}{ρ}}$
$= \sqrt{\frac{1.3 \times 10^{11} \times 1.7 \times 10^{- 5} \times (30 - 10)}{9000}} = 70 m s^{- 1}$
$∴ D = 7$