Question

A sonometer wire supports a 4 kg load and vibrates in fundamental mode with a tuning fork of frequency 416. Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changed to
(a) 1 kg
(b) 2 kg
(c) 8 kg
(d) 16 kg.

Answer 1

(d) 16 kg

According to the relation of the fundamental frequency of a string
$\nu =\frac{1}{2l}\sqrt{\frac{F}{\mu }}$
where l is the length of the string
F is the tension
μ is the linear mass density of the string
We know that ν₁ = 416 Hz, l₁ = l and l₂ = 2l.
Also, m₁ = 4 kg and m₂ = ?
${\nu }_1=\frac{1}{2l_1}\sqrt{\frac{m_{1}g}{\mu }}\left(1\right)$
${\nu }_2=\frac{1}{2l_2}\sqrt{\frac{m_{2}g}{\mu }}\left(2\right)$
So, in order to maintain the same fundamental mode
${\nu }_1={\nu }_2$
squaring both sides of equations (1) and (2) and then equating
$$\begin{gathered} \frac{1}{4l^2}\frac{4g}{\mu }=\frac{1}{16l^2}\frac{m_{2}g}{\mu } \\ \Rightarrow m_2=16\mathrm{kg} \end{gathered}$$