Q. A particle of mass $m$ is attached to three identical springs $A,\ B$ and $C$ each of force constant $k$ a shown in figure. If the particle of mass $m$ is pushed slightly against the spring $A$ and released then the time period of oscillations is
A) $2\pi\sqrt{\frac{2m}{k}}$
B) $2\pi\sqrt{\frac{m}{2k}}$
C) $2\pi\sqrt{\frac{m}{k}}$
D) $2\pi\sqrt{\frac{m}{3k}}$
Explanation
(b) When the particle of mass $m$ at $O$ is pushed by $y$ in the direction of $A$ the spring $A$ will be compressed by $y$ while spring $B$ and $C$ will be stretched by $y^{'} = y\cos{45{^\circ}}$. So that the total restoring force on the mass $m$ along $OA$
$F_{net} = F_{A} + F_{B}\cos{45{^\circ} + F_{C}\cos{45{^\circ}}}$
$= ky + 2ky^{'}\cos{45{^\circ} = ky + 2k\left( y\cos{45{^\circ}} \right)\cos{45{^\circ} = 2ky}}$
Also $F_{net} = k^{'}y \Rightarrow k^{'}y = 2ky \Rightarrow k^{'} = 2k$
$T = 2\pi\sqrt{\frac{m}{k^{'}}} = 2\pi\sqrt{\frac{m}{2k}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.