JEE Main 2013 — Oscillations Question with Solution

Let $T_1$ and $T_2$ be the time period of the two pendulums ${\mathrm{T}}_1=2\pi \sqrt{\frac{1}{\mathrm{g}}}$ and ${\mathrm{T}}_2=2\pi \sqrt{\frac{4}{\mathrm{g}}}$ As ${\ell }_1<{\ell }_2$ therefore ${\mathrm{T}}_1<{\mathrm{T}}_2$ Let at $\mathrm{t}=0$ they start swinging together. Since their time periods are different, the swinging will not be in unison always. Only when number of completed oscillations differ by an integer, the two pendulums will again begin to swing together Let longer length pendulum complete $\mathrm{n}$ oscillation and shorter length pendulum complete $(n+1)$ oscillation. For unison swinging $$\begin{array}{rl}& (n+1)T_1=n_2\\ & (n+1)\times 2\pi \sqrt{\frac{1}{g}}=(n)\times 2\pi \sqrt{\frac{4}{g}}\\ & \Rightarrow n=1\\ & \therefore n+1=1+1=2\end{array}$$