Question

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

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

Answer 1

Answer: (B)

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

Answer 2

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, $\quad\quad K=\frac{2k\times2k}{2\kappa+2k}+\left(k+2k\right)=4k$ $\therefore\quad Frequency ? = \frac{1}{2\pi}\sqrt{\frac{K}{M}}=\frac{1}{2\pi}\sqrt{\frac{4k}{M}}$