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

  $2\pi \sqrt{\frac{m}{3k}}$
• Correct answer
  B

  $2\pi \sqrt{\frac{3m}{2k}}$
• C

  $2\pi \sqrt{\frac{2m}{3k}}$
• D

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

Solution

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

$k_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\left(k_2+k_3\right)k_1}{k_1+k_2+k_3}$

$=\frac{2k\times k}{3k}=\frac{2}{3}k$

$\therefore$ Time period of oscillation

$T=2\pi \sqrt{\frac{m}{k_{\mathrm{e}\mathrm{f}\mathrm{f}}}}$

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

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

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