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

• A

  $2\mathrm{\pi }\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}}$
• Correct answer
  B

  $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}$
• C

  $2\pi \sqrt{\frac{m}{k}}$
• D

  $2\pi \sqrt{\frac{m}{3k}}$

Solution

When the particle of mass $\mathrm{m}$ at O is pushed by $\mathrm{y}$ in the direction of A, the spring A will be compressed by $\mathrm{y}$ while spring B and C will be stretched by $\mathrm{y}\mathrm{\prime }=\mathrm{ycos}45°$. So that the total restoring force on the mass $\mathrm{m}$ along OA.

${\mathrm{F}}_{\mathrm{net}}={\mathrm{F}}_{\mathrm{A}}+{\mathrm{F}}_{\mathrm{B}}\cos 45°+{\mathrm{F}}_{\mathrm{C}}\cos 45°=\mathrm{ky}+2\mathrm{ky}\mathrm{\prime }\cos 45°=\mathrm{ky}+2\mathrm{k}\left(\mathrm{ycos}45°\right)\cos 45°=2\mathrm{ky}$

Also ${\mathrm{F}}_{\mathrm{net}}=\mathrm{k}\mathrm{\prime }\mathrm{y}$

$\Rightarrow \mathrm{k}\mathrm{\prime }\mathrm{y}=2\mathrm{ky}$

$\Rightarrow \mathrm{k}\mathrm{\prime }=2\mathrm{k}$

$\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}\mathrm{\prime }}}=2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}$

Hence, the correct answer is option: 'B'.