Question

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

• A. ${\cos }^{-1}\left(\frac{2}{3}\right)$
• B. ${\cos }^{-1}\frac{5}{\sqrt{3}}$
• C. ${\cos }^{-1}\left(\frac{5}{6}\right)$
• D. ${\cos }^{-1}\left[\frac{5}{2\sqrt{3}}\right]$

Answer 1

The correct option is C ${\cos }^{-1}\left(\frac{5}{6}\right)$

At point A skier leaves contact with hemisphere, thus $N=0$

$m{a}_c=mgcos\theta$ where $a_c=\frac{v^2}{R}$

$⟹v^2=gRcos\theta$ .....................(1)

From geometry, $x=R-Rcos\theta =R(1-cos\theta )$

Apply conservation of energy at A and B, $mg(\frac{R}{4}+R)=\frac{1}{2}m{v}^2+mgRcos\theta$

From (1), $mg\left(\frac{5R}{4}\right)=\frac{1}{2}m\left(gRcos\theta \right)+mgRcos\theta$

$⟹cos\theta =\frac{5}{6}$