In an experiment with sonometer when a mass of $$180$$ g is attached to the string, it vibrates with fundamental frequency of $$30$$ Hz. When a mass $$m$$ is attached, the string vibrates with fundamental frequency of $$50$$ Hz. The value of $$m$$ is _____ g.

The fundamental frequency of a sonometer wire is:

$$f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$$

where $$T = Mg$$ is the tension. Since $$L$$ and $$\mu$$ are the same:

$$f \propto \sqrt{T} \propto \sqrt{M}$$

$$\frac{f_1}{f_2} = \sqrt{\frac{M_1}{M_2}}$$

$$\frac{30}{50} = \sqrt{\frac{180}{m}}$$

$$\frac{9}{25} = \frac{180}{m}$$

$$m = \frac{180 \times 25}{9} = 500 \text{ g}$$

The value of $$m$$ is $$\mathbf{500}$$ g.