A small mass is hung from the ceilling of a train compartment. The train moves up an incline of $30 ^{\circ}$ with horizontal. Acceleration of train up the incline is $a=\large\frac{g}{2}.$ The angle with which the string supporting the mass makes with normal to ceiling in equilibrium

$(a)\;30 ^{\circ} \quad (b)\;\tan ^{-1} \bigg(\frac{2}{\sqrt 3}\bigg) \quad (c)\;\tan^{-1}\bigg(\frac{\sqrt 3}{2}\bigg) \quad (d)\;\tan ^{-1} 2$

Let $\theta$ be the angle string makes,

Equating force along the incline and perpendicular to the incline
$T\cos \theta=mg \cos 30^{\circ}$-----(1)
$T \sin \theta=mg \sin 30=m g/2$-----(2)
Solving (1) and (2)
$\theta= \tan ^{-1}\bigg(\large\frac{2}{\sqrt 3}\bigg)$
Hence b is the correct answer.

edited Feb 10, 2014 by meena.p