Question

Two identical springs (spring constant = $k$ ) are connected to two light rigid rods in parallel combination and then a third spring (identical to the other two) is connected to this combination in series. The combined system is fixed to the ceiling and a block of mass $m$ is attached as shown in the figure. If the mass is slightly displaced and then let go, then the system shall oscillate with a time period of

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

Answer 1

Answer: (B)

Consider the system as shown in figure below.

Here, $k_{2}$ and $k_{3}$ are in parallel. $k_{1}$ is in series with parallel combination of $k_{2}$ and $k_{3}.$
Effective spring constant of the combination is
$k_{e f f}=\frac{\left(k_{2} + k_{3}\right) k_{1}}{k_{1} + k_{2} + k_{3}}$
$=\frac{2 k \times k}{3 k}=\frac{2}{3}k$
$\therefore $ Time period of oscillation
$T=2\pi \sqrt{\frac{m}{k_{e f f}}}$
$=2\pi \sqrt{\frac{m}{\frac{2}{3} k}}$
$=2\pi \sqrt{\frac{3 m}{2 k}}$