Four massless springs whose force constants are \(2 \mathrm{~K}, 2 \mathrm{~K}, \mathrm{~K}\) and \(2 \mathrm{~K}\) respectively are attached to a mass \(\mathrm{M}\) kept on a frictionless plane as shown in figure, If mass \(M\) is displaced in horizontal direction then frequency of oscillating system is

On the right hand side of the block, springs are connected in parallel
\(\therefore\) Their effective spring constant is given by
\(
\begin{aligned}
& \mathrm{K}_1=\mathrm{K}+2 \mathrm{~K} \\
& \mathrm{~K}_1=3 \mathrm{~K}
\end{aligned}
\)
On the left hand side of the block, springs are connected in series.
\(\therefore\) Their effective spring constant is given by,
\(\frac{1}{\mathrm{~K}_2}=\frac{1}{2 \mathrm{~K}}+\frac{1}{2 \mathrm{~K}} \)
\( \therefore \quad \mathrm{K}_2 =\mathrm{K}\)
\(\therefore\) Effective spring constant of the system is given by,
\(\mathrm{K}_{\mathrm{E}}=3 \mathrm{~K}+\mathrm{K}=4 \mathrm{~K} \)
\( \therefore \omega =\sqrt{\frac{\mathrm{K}_{\mathrm{E}}}{\mathrm{M}}}=\sqrt{\frac{4 \mathrm{~K}}{\mathrm{M}}} \)
\( \therefore \mathrm{f} =\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{4 \mathrm{~K}}{\mathrm{M}}}\)