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)