In an experiment with sonometer when a mass of 180g is attached to...

The formula to calculate the fundamental frequency in a stretched string is given by

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

For the first mass, it can be written that

${f}_{1} = \dfrac{1}{2L}\sqrt{\dfrac{{m}_{1}g}{\mu}} . . . \left(2\right)$

And, for the second mass, it can be written that

${f}_{2} = \dfrac{1}{2L}\sqrt{\dfrac{{m}_{2}g}{\mu}} . . . \left(3\right)$

Divide equation (3) by equation (2) and simplify to obtain the required mass.

$\begin{cases}\dfrac{{f}_{2}}{{f}_{1}} & = & \dfrac{\dfrac{1}{2L}\sqrt{\dfrac{{m}_{2}g}{\mu}}}{\dfrac{1}{2L}\sqrt{\dfrac{{m}_{1}g}{\mu}}} \\ & = & \sqrt{\dfrac{{m}_{2}}{{m}_{1}}} \\ & ⇒ & \dfrac{{m}_{1}}{{m}_{2}} = {\left(\dfrac{{f}_{1}}{{f}_{2}}\right)}^{2} \\ & ⇒ & {m}_{2} = {m}_{1}{\left(\dfrac{{f}_{2}}{{f}_{1}}\right)}^{2} . . . \left(4\right)\end{cases}$

Substitute the values of the known parameters into equation (4) to calculate the required mass.

$\begin{cases}{m}_{2} & = & 180g \times {\left(\dfrac{50Hz}{30Hz}\right)}^{2} \\ & = & 500g\end{cases}$