Q: A particle is placed at rest inside a hollow hemisphere of radius R. The coefficient of friction between the particle and the hemisphere is μ = 1/√3 . The maximum height up to which the particle can remain stationary is
(a) $\displaystyle \frac{R}{2}$
(b) $ \displaystyle (1-\frac{\sqrt3}{2}) R $
(c) $ \displaystyle (\frac{\sqrt3}{2}) R $
(d) $ \displaystyle (\frac{3 R}{8}) $
Ans: (b)
Solution: mgcosθ = N …(i)
mgsinθ = f …(ii)
On dividing (i) by (ii)
cotθ = N/f = N/μN = 1/μ
tanθ = μ = 1/√3
θ = 30°
h = R – Rcosθ
h = R(1-cos30)
h = R(1-√3/2)