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 the figure. If the mass M is displaced in the horizontal direction, then the frequency of the system is

$\frac{1}{2 π} \sqrt{\frac{k}{4 M}}$

$\frac{1}{2 π} \sqrt{\frac{4 k}{M}}$

$\frac{1}{2 π} \sqrt{\frac{k}{7 M}}$

$\frac{1}{2 π} \sqrt{\frac{7 k}{M}}$

Solution

$\frac{1}{2 π} \sqrt{\frac{4 k}{M}}$

Two springs on the L.H.S. of mass M are in series and two springs on the R.H.S. of mass M are in parallel. These combinations of springs will be considered in parallel to mass M. Thus effective spring constant,

K = $\frac{2 k \times 2 k}{2 k + 2 k}$ + ( k + 2k)

= 4k

∴ Frequency ν = $\frac{1}{2 π} \sqrt{\frac{K}{M}}$

ν = $\frac{1}{2 π} \sqrt{\frac{4 k}{M}}$

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