Question

As shown in figure, $A$ rod $AB$ $\left(l_{AB} = 2 \,m\right)$ is hinged at point $A$ and its other end $B$ attached to a platform on which a block of mass $m$ is kept. The platform is not frictionless. Rod rotates about point $A$ maintaining angle $\theta =30^{o}$ with the vertical in such a way that platform remains horizontal and revolves on the horizontal circular path. What is the maximum angular velocity in $rads^{- 1}$ of rod so that the block does not slip on the platform? $\left(g = 10 \,m s^{- 2} , \mu = 0 . 1\right)$

Answer 1

$N=mg, \, \mu N=mr\omega ^{2}$
$r=2sin\theta $
$\mu mg=m\left(2 sin \theta \right)\left(\omega \right)^{2}$
$\Rightarrow \, \, \omega = \, \sqrt{\frac{\mu \theta }{2 sin \theta }}= \, \sqrt{\frac{0.1 \times 10}{2 \times sin ⁡ 20}}=1 \, rads^{- 1}$