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

Options

• 1 kg

• 2 kg

• 8 kg

• 16 kg.

Answer 1

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 = ? 
\[\nu_1 = \frac{1}{2 l_1}\sqrt{\frac{m_1 g}{\mu}}..................  (1)\]
\[\nu_2  = \frac{1}{2 l_2}\sqrt{\frac{m_2 g}{\mu}}                (2)\]
So, in order to maintain the same fundamental mode
\[\nu_1  =  \nu_2\]
squaring both sides of equations (1) and (2) and then equating

\[\frac{1}{4 l^2}\frac{4g}{\mu}   =   \frac{1}{16 l^2}\frac{m_2 g}{\mu}\] 

\[ \Rightarrow  m_2  = 16  kg\]