Question

A particle is released on a vertical smooth semicircular track from point $X$ so that $OX$ makes angle $\theta$ from the vertical (see figure). The normal reaction of the track on the particle vanishes at point $Y$ where $OY$ makes angle $\phi$ with the horizontal. Then :

• A. $sin\,\phi = cos \,\theta$
• B. $sin\,\phi =\frac{1}{2} cos \,\theta$
• C. $sin\,\phi =\frac{2}{3} cos \,\theta$
• D. $sin\,\phi =\frac{3}{4} cos \,\theta$

Answer 1

Answer: (C)

$\frac{ mv ^{2}}{ r }= mg \,\sin \,\phi\,\,\,\,\,\dots(i)$
$mg \,r \,\cos \theta=\frac{1}{2} mv ^{2}+rgsin\, \phi$
$\frac{ v ^{2}}{ rg }=2 \,\cos\, \theta-2 \,\sin \,\phi \,\,\,\,\,\,\dots(ii)$
$\sin \,\phi=2\, \cos \,\theta-2 \,\sin \,\phi$
$3\, \sin \,\phi=2\, \cos \,\theta$
$\sin \,\phi=\frac{2}{3} \,\cos \,\phi$