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Four massless springs whose force constants are $2k, 2k, k$ and $2k$ respectively are attached to a mass $M$ kept on a frictionless plane (as shown in figure). If the mass $M$ is displaced in the horizontal direction, then the frequency of oscillation of the system is
$\frac{1}{{2\pi }}\sqrt {\frac{k}{{4M}}} $
$\frac{1}{{2\pi }}\sqrt {\frac{{4k}}{M}} $
$\frac{1}{{2\pi }}\sqrt {\frac{k}{{7M}}} $
$\frac{1}{{2\pi }}\sqrt {\frac{{7k}}{M}} $
Solution
(b) The two springs on left side having spring constant of $2k$ each are in series, equivalent constant is $\frac{1}{{\left( {\frac{1}{{2k}} + \frac{1}{{2k}}} \right)}} = k$. The two springs on right hand side of mass $M$ are in parallel. Their effective spring constant is $(k + 2k) = 3k$.
Equivalent spring constants of value $k$ and $3k$ are in parallel and their net value of spring constant of all the four springs is $k + 3k = 4k$
$\therefore $ Frequency of mass is $n = \frac{1}{{2\pi }}\sqrt {\frac{{4k}}{M}} $
