Question

A wire of density $9\times {10}^{3}kg/m^3$ is stretched between two clamps $1m$ 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 $35Hz$
• B. the frequency of 1st overtone is $70Hz$
• C. the frequency of 1st overtone is $105Hz$
• D. the stress in the wire is $4.41\times {10}^{7}N/m^2$

Answer 1

Answer: (A), (B), (D)

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

$\frac{T}{A}=\gamma \frac{\Delta L}{L}$ (by hook's law)
$⟹\frac{T}{A}=\frac{9\times {10}^{10}\times 4.9\times {10}^{-4}}{1}$
$\therefore$ Stress in the wire =$4.41\times {10}^{7}N/m^2$
Frequency of the wave in fundamental mode is given by$f=\frac{1}{2l}\sqrt{\frac{T}{\rho A}}=\frac{1}{2l}\sqrt{\frac{Y\Delta \ell }{\ell }A\times \frac{1}{\rho A}}=\frac{1}{2l}\sqrt{\frac{Y\Delta \ell }{\ell \rho }}$
$\therefore f=35Hz$
and, frequency of first overtone $=\frac{V}{\ell }=70Hz$