A sleeve `A` can slide freely along a smooth rod bent in the shape of a half circle of radius `R` .The system is set in rotaion with a constant angular velocity `omega` about a vertical axis `OO'` .Find the angle `theta` corresponding to the steady position of the sleeve

Explanation

`r = R sin theta , N cos theta = mg` …(i)
`N sin theta = m omega ^(2)r`
`rArr N = m omega ^(2)R`….(ii)
From Eqs(i) and (ii) we get
`cos theta = (g)/(omega^(2)R) rArr theta = cos^(-1)((g)/(omega ^(2) R))`