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

• A

  $\frac{1}{2 \pi} \sqrt{\frac{k}{4 M}}$
• Correct answer
  B

  $\frac{1}{2 \pi} \sqrt{\frac{4 k}{M}}$
• C

  $\frac{1}{2 \pi} \sqrt{\frac{k}{7 M}}$
• D

  $\frac{1}{2 \pi} \sqrt{\frac{7 k}{M}}$

Solution

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, $K=\frac{2 k \times 2 k}{2 k+2 k}+(k+2 k)=4 k$ $\therefore$ Frequency $v=\frac{1}{2 \pi} \sqrt{\frac{K}{M}}=\frac{1}{2 \pi} \sqrt{\frac{4 k}{M}}$

Hence, the correct answer is option: 'B'.