Question

32. Two particles tied to different strings are whirled in a
horizontal circle as shown in figure. The ratio of lengths of the
strings so that they complete their circular path with equal
time period is:
(B)underroot3/2
(C) 1
(D) None of these
(A)underroot 2/3​

Answer 1

Answer:

Option (A) $\sqrt{\frac{2}{3}}$

Explanation:

Each of the arrangement is a conical pendulum.

In general way, if the angle is θ and the length of the string is L then the Time Period T is given by

$\boxed{T=\sqrt{\frac{L\cos\theta}{g}}}$

Since time periods are equal

Therefore

$\sqrt{\frac{L_1\cos\theta_1}{g}}=\sqrt{\frac{L_2\cos\theta_2}{g}}$

Squaring on both sides

$\frac{L_1\cos\theta_1}{g}=\frac{L_2\cos\theta_2}{g}$

$\implies L_1\cos\theta_1=L_2\cos\theta_2$

$\implies L_1\cos30^\circ=L_2\cos45^\circ$

$\implies L_1\times\frac{\sqrt{3}}{2}=L_2\times\frac{1}{\sqrt{2}}$

$\implies L_1\times\frac{\sqrt{3}}{\sqrt{2}}=L_2$

$\implies \frac{L_1}{L_2}=\sqrt{\frac{2}{3}}$

Hope this helps.