Question

The length of a sonometer wire is $0.75\text{m}$ and density is $9\times {10}^3{\text{kg/m}}^3$. It can bear a stress of $8.1\times {10}^8{\text{N/m}}^2$ without exceeding elastic limit. The fundamental frequency that can be produced in the wire is

• A. $200\text{Hz}$
• B. $150\text{Hz}$
• C. $175\text{Hz}$
• D. $225\text{Hz}$

Answer 1

The correct option is A $200\text{Hz}$

Given,
$l=0.75\text{m},\rho =9\times {10}^3{\text{kg/m}}^3$
$\sigma =8.1\times {10}^8{\text{N/m}}^2$
Let the cross-sectional area be $A$.
Tension, $T=$ Stress $\times$ Area
$\Rightarrow T=8.1\times {10}^{8}A$
Linear mass density, $\mu =\rho A=9\times {10}^{3}A$
Now, speed of the wave,
$v=\sqrt{\frac{T}{\mu }}$
$\Rightarrow v=\sqrt{\frac{8.1\times {10}^{8}A}{9\times {10}^{3}A}}$
$\Rightarrow v=300\text{m/s}$
So, fundamental frequency,
$f_0=\frac{v}{2l}=\frac{300}{2\times 0.75}$
$\Rightarrow f_0=200\text{Hz}$