Question

A sono-meter wire supports a 4 kg load and vibrates in fundamental node with a tuning fork of frequency 416 Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental node, the load should be changed to

• A. 1 kg
• B. 2 kg
• C. 8 kg
• D. 16 kg

Answer 1

The correct option is D

16 kg

so initially say load is $m_1$

$T=m_{1}g$ v(velocity) $=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{m_{1}g}{\mu }}$

Fundamental frequency $f=\frac{v}{2L}$

$f=\sqrt{\frac{T}{\mu }}\left(\frac{1}{2L}\right)$

$=\sqrt{\frac{m_{1}g}{\mu }}\left(\frac{1}{2L}\right)$

On doubling the length $f^{\prime }=\sqrt{\frac{m_{2}g}{\mu }}\times \frac{1}{\left(4L\right)}$ and changing load to $m_2$

Question says $f=f^{\prime }$

$\sqrt{\frac{m_{1}g}{\mu }}\times \frac{1}{2L}=\sqrt{\frac{m_{2}g}{\mu }}\times \frac{1}{4L}$

$\frac{m_{1}g}{\mu }=\frac{1}{4}\frac{m_{2}g}{\mu }$

$4m_1=m_2$

Load must be increased 4 times initial load was 4 kg (given). So now it should be 16 kg!