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 $4.9\times {10}^{-4}m$. 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 is 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 is 1st overtone is $70Hz$
D The stress in the wire is $4.41\times {10}^{7}N/m^2$

Speed of wave in wire $V=\sqrt{\frac{T}{\rho A}}=\sqrt{\frac{Y\Delta l}{l}A\times \frac{1}{\rho A}}=\sqrt{\frac{Y\Delta l}{\rho l}}$

For lowest frequency; the standing wave must be in fundamental mode.

$\therefore f=\frac{V}{\lambda }=\frac{V}{2l}$

$\therefore TimePeriod=\frac{2l}{V}=2l\sqrt{\frac{\rho l}{Y\Delta l}}=\frac{1}{35}second$

$\therefore (f=35Hz)$

and, frequency of first overtone:


$f=\frac{V}{l}=70Hz$
Stress in the wire:
$Y=\frac{stress}{strain}\Rightarrow stress=Y\times \frac{\Delta l}{l}$
$stress=9\times {10}^{10}\times 4.9\times {10}^{-4}stress=44.1\times {10}^{6}Pa$