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.

#### Solution

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 ν_{1} = 416 Hz, *l*_{1} = *l* and *l*_{2} = 2*l*.

Also, *m*_{1} = 4 kg and *m*_{2} = ?

\[\nu_1 = \frac{1}{2 l_1}\sqrt{\frac{m_1 g}{\mu}}.................. (1)\]

\[\nu_2 = \frac{1}{2 l_2}\sqrt{\frac{m_2 g}{\mu}} (2)\]

So, in order to maintain the same fundamental mode

\[\nu_1 = \nu_2\]

squaring both sides of equations (1) and (2) and then equating

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

\[ \Rightarrow m_2 = 16 kg\]