Question

A particle is freely sliding down from the top of smooth convex hemisphere of radius $30cm$. The particle leaves the surface at a vertical distance, from highest point, equal to:

• A. $5cm$
• B. $10cm$
• C. $15cm$
• D. $20cm$

Answer 1

Answer: (B)

$10cm$

Let the particle leave the hemisphere at an angle $\theta$.
$\Rightarrow mgR(1-\cos \theta )=\frac{1}{2}m{\nu }^2\left(fromconservationofenergy\right)$

At that point $mg\cos \theta =\frac{m{\nu }^2}{R}\left(Forceequation\right)$

$\therefore mgR(1-\cos \theta )=\frac{1}{2}mgR\cos \theta$
$2-2\cos \theta =\cos \theta$
$\Rightarrow \cos \theta =\frac{2}{3}$
$\therefore$ point is at distance $\frac{2}{3}\times 30cm$ = 20 above centre or 10 cm below top point.