Question

A bead $A$ can slide freely along a smooth rod bent in the form of a half circle of Radius $R$. The system is set in rotation with a constant angular velocity $\omega$ about a vertical axis $O{O}^{\prime }$. Find the angle $\theta$ corresponding to steady position of the bead.

• A. ${\cos }^{-1}\left(\frac{R{\omega }^2}{g}\right)$
• B. ${\cos }^{-1}\left(\frac{g}{R{\omega }^2}\right)$
• C. ${\sin }^{-1}\left(\frac{g}{R{\omega }^2}\right)$
• D. ${\sin }^{-1}\left(\frac{R{\omega }^2}{g}\right)$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{g}{R{\omega }^2}\right)$

The bead is in steady position if the net force along the tangent is zero. i,e,
$mg\sin \theta -m{w}^{2}r\cos \theta =0$
or $g\sin \theta =w^2\left(R\sin \theta \right)\cos \theta$ where $r=R\sin \theta$
or $\cos \theta =\frac{g}{w^{2}R}$
or $\theta ={\cos }^{-1}\left(\frac{g}{w^{2}R}\right)$