Four massless springs whose force constants are 2 k, 2 k, k and 2 k...

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

Options

1. A$\frac{1}{2 \pi} \sqrt{\frac{k}{4 \mathrm{M}}}$
2. B$\frac{1}{\pi} \sqrt{\frac{k}{\mathrm{M}}}$
3. C$\frac{1}{2 \pi} \sqrt{\frac{k}{7 \mathrm{M}}}$
4. D$\frac{1}{2 \pi} \sqrt{\frac{7 k}{\mathrm{M}}}$

Correct answer

B. $\frac{1}{\pi} \sqrt{\frac{k}{\mathrm{M}}}$

Step-by-step solution

The equivalent system is shown in figure.
The equivalent force constant of which $k_{\mathrm{e}}=k+3 k=4 k\therefore \omega=\sqrt{\frac{k_{e}}{m}}=\sqrt{\frac{4 k}{m}}=2 \sqrt{\frac{k}{m}}$ and $\mathrm{f}=\frac{\omega}{2 \pi}=\frac{2}{2 \pi} \sqrt{\frac{k}{m}}=\frac{1}{\pi} \sqrt{\frac{k}{m}}$

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