Q 11-13-095JEE MainJEE Main 2022 (26 Jul, Shift 2)Medium
As per the given figures, two springs of spring constants $k$ and $2k$ are connected to mass $m$. If the period of oscillation in figure (a) is $3\ \text{s}$, then the period of oscillation in figure (b) will be $\sqrt x\ \text{s}$. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 2
(a) Series: $k_a = \dfrac{k\cdot2k}{3k} = \dfrac{2k}{3}$, so $T_a = 2\pi\sqrt{\dfrac{3m}{2k}} = 3\ \text{s}$.
(b) Parallel: $k_b = 3k$, so $T_b = 2\pi\sqrt{\dfrac{m}{3k}}$.
$$\frac{T_b}{T_a} = \sqrt{\frac{m/3k}{3m/2k}} = \sqrt{\frac29}\ \Rightarrow\ T_b = 3\times\frac{\sqrt2}{3} = \sqrt2\ \text{s}$$
So $x = 2$.
Solution by Sreeraj P, M.Sc Physics