Question

A copper wire is held at the two ends by rigid supports. At ${30}^{\circ }C$ the wire is just taut, with negligible tension. The speed of a transverse wave in this wire at ${10}^{\circ }C$ is $x\text{m/s}$. The value of $x$ is [Give your answer as nearest integer value]

[Given Young modulus of copper$=1.3\times {10}^{11}{\text{N/m}}^2$,
Co-efficient of linear expansion of copper $=1.7\times {10}^{-5}{}^{\circ }{C}^{-1}$,
Density of copper $=9\times {10}^3{\text{kg/m}}^3$]

Answer 1

Step 1,

As the temperature of copper wire decreases, it will get to contract. Thus there will be outward tension developed in the wire at the two ends as shown in the figure below.

Step 2,

So, the amount of contract,

$\Delta L=\alpha L\Delta T$

From the data given in the question,

$\Rightarrow \Delta L=1.7\times {10}^{-5}\times L\times (30-10)$

$\Rightarrow \Delta L=3.4\times {10}^{-4}L...........\left(1\right)$

From Young's modulus formula,

$Y=\frac{FL}{A\Delta L}$

$\Rightarrow 1.3\times {10}^{11}=\frac{TL}{A\times 3.4\times {10}^{-4}L}$

$\Rightarrow T=4.42\times {10}^{7}A..........\left(2\right)$

Now, speed of transverse wave,

$v=\sqrt{\frac{T}{\mu }}$

$\Rightarrow v=\sqrt{\frac{T}{\rho A}}[\mu =\frac{m}{L}and\rho =\frac{m}{LA}]$

$\Rightarrow v=\sqrt{\frac{4.42\times {10}^7\times A}{9\times {10}^3\times A}}$

After solving

v = 70.1 m/s

Hence the value of x is 70.1 m/s