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

Answer: (B)

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

Answer 2

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}}