Question

In a conical pendulum, the length of the string is $l$ and $\theta$ is the angle of string with vertical. The time period of revolution is

• A. $2\pi \sqrt{\frac{l}{g}}$
• B. $2\pi \sqrt{\frac{l\cos \theta }{g}}$
• C. $2\pi \sqrt{\frac{l}{g\cos \theta }}$
• D. Infinite

Answer 1

The correct option is B $2\pi \sqrt{\frac{l\cos \theta }{g}}$

Equation of motion:

$T\cos \theta =mg$

$⟹T=\frac{mg}{\cos \theta }$

and $T\sin \theta =\frac{m{V}^2}{r}$

$⟹mg\tan \theta =\frac{m{V}^2}{r}$

$⟹V=\sqrt{rg\tan \theta }$

also,

$V=\frac{2\pi r}{t}=\sqrt{rg\tan \theta }$

$\therefore t=2\pi \sqrt{\frac{r}{g\tan \theta }}$

and $r=l\sin \theta$

$\therefore t=2\pi \sqrt{\frac{l\cos \theta }{g}}$