Physics - SHM Question with Solution | TestHub

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

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

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

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