Question

A particle of mass $m$ is attached to three identical springs $A,B$ and $C$ each of force constant $k$ as shown in figure. If the particle of mass $m$ in is pushed slightly against the spring $A$ and released, then the time period of oscillation is

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

Answer 1

The correct option is B $2\pi \sqrt{\frac{m}{2k}}$

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 springs $B$ and $C$ will be stretched by $y^{\prime }=y\cos {45}^{\circ }$, so that the total restoring force on the mass $m$ along $OA$,
Resultant force $=F_A+F_B\cos {45}^{\circ }+F_C\cos {45}^{\circ }$
$=ky+2k{y}^{\prime }\cos {45}^{\circ }$
$=ky+2k\left(y\cos {45}^{\circ }\right)\cos {45}^{\circ }=k^{\prime }y$
where $k^{\prime }=2k$
$\therefore T=2\pi \sqrt{\frac{m}{k^{\prime }}}=2\pi \sqrt{\frac{m}{2k}}$
Hence, the correct answer is option (b).

Why this question? Caution: Component followed by component- not to miss on.