Question

In the figure shown below, a string of length ${}^{\prime }{l}^{\prime }$ is connected to a bob of mass ${}^{\prime }{m}^{\prime }$ and is moving around a vertical axis through a fixed point. Then which of the following statements is (are) correct?

• A. Angle of inclination of the string is equal to ${\cos }^{-1}\left(\frac{g}{l{\omega }^2}\right)$
• B. Angle of inclination of the string is equal to ${\sin }^{-1}\left(\frac{g}{l{\omega }^2}\right)$
• C. Tension in the string is equal to $T=ml{\omega }^2$
• D. Tension in the string is equal to $T=m\left(l\tan \theta \right){\omega }^2$

Answer 1

Answer: (B), (C)

The correct options are
B Angle of inclination of the string is equal to ${\sin }^{-1}\left(\frac{g}{l{\omega }^2}\right)$
C Tension in the string is equal to $T=ml{\omega }^2$

From FBD of the bob:

$T\sin \theta$ balances the weight of the bob and $T\cos \theta$ provides necessary centripetal force.
$T\sin \theta =mg$ ... (1)
$T\cos \theta =mR{\omega }^2$... (2)
Here, $R=l\cos \theta \Rightarrow T=ml{\omega }^2$ ... (3)
From (1) & (3)
$\sin \theta =\frac{mg}{T}=\frac{mg}{ml{\omega }^2}$
$\Rightarrow \theta ={\sin }^{-1}\left(\frac{g}{l{\omega }^2}\right)$