Question

A skier plans to ski a smooth fixed hemisphere of radius R. He starts from rest from a curved smooth surface of height (R/4). The angle $\theta$ at which he leaves the hemisphere is

• A. $co{s}^{-1}\left(2/3\right)$
• B. $co{s}^{-1}\left(5/\sqrt{3}\right)$
• C. $co{s}^{-1}\left(5/6\right)$
• D. $co{s}^{-1}\left(5/2/\sqrt{3}\right)$

Answer 1

The correct option is C $co{s}^{-1}\left(5/6\right)$

$\Delta h=\frac{R}{4}+R(1-cos\theta )\frac{1}{2}m{v}^2=mg\Delta h=\frac{mgR}{4}\{1+4(1-cos\theta \left)\right\}$
$\therefore \frac{m{v}^2}{R}=\frac{mg}{2}(5-4cos\theta )mgcos\theta -N=\frac{m{v}^2}{R}$
He will leave hemisphere when $N$ = 0.

$mgcos\theta =\frac{mg}{2}(5-4cos\theta )cos\theta =5/6$