Question

A wire of density $9\times {10}^{3}kg/m^3$ is stretched between two clamps 1 m apart and is stretched to an extension of $4.9\times {10}^{-4}metre.$ Young's modulus of material is $9\times {10}^{10}N/m^2$. Then

• A. The lowest frequency of standing wave is 35 Hz
• B. The frequency of 1st overtone is 70 Hz
• C. The frequency of 1st overtone is 105 Hz
• D. The stress in the wire is $4.41\times {10}^{7}N/m^2$

Answer 1

Answer: (A)

The lowest frequency of standing wave is 35 Hz

For wire if M = mass, r = density, A = Area of cross section V = volume, l = length, Dl = change

in length Then mass per unit length $m=\frac{M}{l}=\frac{Al\rho }{l}=A\rho$ And Young? s modules of elasticity

$y=\frac{T/A}{\Delta l/l}ÞT=\frac{Y\Delta lA}{l}.$ Hence lowest frequency of vibration $n=\frac{1}{2l}\sqrt{\frac{T}{m}}$

$\frac{1}{2l}\sqrt{\frac{y\left(\frac{\Delta l}{l}\right)A}{A\rho }}=\frac{1}{2l}\sqrt{\frac{y\Delta l}{l\rho }}Þn=\frac{1}{2\times 1}\sqrt{\frac{9\times {10}^{10}\times 4.9\times {10}^{-4}}{1\times 9\times {10}^3}}=35Hz$