Q. 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
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) $\frac{1}{2\pi}\sqrt{\frac{7k}{M}}$
Explanation
(b) The two spring 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$\ n = \frac{1}{2\pi}\sqrt{\frac{4k}{M}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.