Question

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. $2\pi \sqrt{\frac{2 m}{k}}$
• B. $2\pi \sqrt{\frac{m}{2 k}}$
• C. $2\pi \sqrt{\frac{m}{k}}$
• D. $2\pi \sqrt{\frac{m}{3 k}}$

Answer 1

Answer: (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′=ycos45^\circ $ . So that the total restoring force on the mass $m$ along OA.
$F_{net}=F_{A}+F_{B}cos45^\circ +F_{C}cos45^\circ =ky+2ky′cos45^\circ =ky+2k\left(ycos 45 ^\circ \right)cos45^\circ =2ky$
Also $F_{net}=k′y$ $\Rightarrow k′y=2ky$
$\Rightarrow k′=2k$
$T=2\pi \sqrt{\frac{m}{k ′}}=2\pi \sqrt{\frac{m}{2 k}}$