Question

Two blocks each of mass m are connected as shown in figure. Pulleys are smooth and light. The coefficient of static friction at all surfaces is ${\mu }_s$. If inclination $\theta$ is increased gradually, '$\theta$' at which the blocks begin to slide is given by $\theta =ta{n}^{-1}\left(\frac{10{\mu }_s}{P}\right)$. Find the value of $P$.

Answer 1

Since the friction force between the inclined plane and B is more than that between A and B, A will slide down, while B will slide up the plane.


$mg\sin \theta -T-{\mu }_{s}mg\cos \theta =0$
$T=mg\sin \theta -{\mu }_{s}mg\cos \theta$
$2T=mg\sin \theta +3{\mu }_{s}mg\cos \theta$
$2(mg\sin \theta -{\mu }_{s}mg\cos \theta )=mg\sin \theta +3{\mu }_{s}mg\cos \theta$
$mg\sin \theta =5{\mu }_{s}mg\cos \theta$
$\tan \theta =5{\mu }_s$
$\theta ={\tan }^{-1}\left(5{\mu }_s\right)$