Two pendulums begin to swing simultaneously. The first pendulum makes 9 full oscillations when the other makes 7. The ratio of lengths of the two pendulum is:

Dear Student,

Let in time t seconds the first pendulum will make 9 full oscillation and the other makes 7 full oscillations.

Time period of the first pendulum, T₁ = $\frac{t}{9}$ s    .....(i)
Time period of the second pendulum, T₂ = $\frac{t}{7}$ s   .....(ii)
Let l₁ and l₂ are the lengths of the first and second pendulum, respectively.

From (i) and (ii), we have
$$\begin{gathered} \frac{T_1}{T_2}=\frac{2\mathrm{\pi }\sqrt{{\displaystyle \frac{l_1}{g}}}}{2\mathrm{\pi }\sqrt{\frac{l_2}{g}}}=\sqrt{\frac{l_1}{l_2}} \\ \Rightarrow \frac{T_1}{T_2}=\sqrt{\frac{l_1}{l_2}} \\ \Rightarrow \frac{{\displaystyle \frac{t}{9}}}{{\displaystyle \frac{t}{7}}}=\sqrt{\frac{l_1}{l_2}} \\ \Rightarrow \sqrt{\frac{l_1}{l_2}}=\frac{7}{9} \\ \Rightarrow \frac{l_1}{l_2}={\left(\frac{7}{9}\right)}^2=\frac{49}{81} \end{gathered}$$