Question

A particle of mass m is attached to three identical massless springs of spring constant k as shown in the figure. The time period of vertical oscillation of the particle is .

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

Answer 1

The Correct Option is 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 is along OA $\hspace20mm F_{net} = F_A + F_B cos \, 45^\circ + F_C cos \, 45^\circ$ $\hspace25mm = ky + 2ky' \, cos \, 45^\circ$ $\hspace25mm = ky + 2k (y \, cos \, 45^\circ) cos \, 45^\circ$ $\hspace25mm = 2ky$ Also,$\hspace10mm F_{net} =k' y \, \, \Rightarrow k' \, y = 2 ky$ $\Rightarrow \hspace20mm k' = 2 k$ $\hspace20mm T = 2 \pi \sqrt{\frac{m}{k'}} = 2 \pi \sqrt{\frac{m}{2k}}$