Question

A particle initially at rest starts moving from point A on the surface of a fixed smooth hemisphere of radius r as shown. The particle looses its contact with hemisphere at point B. C is centre of the hemisphere. The equation relating $\alpha$ and $\beta$ is:

• A. $3sin\alpha =2cos\beta$
• B. $2sin\alpha =3cos\beta$
• C. $3sin\beta =2cos\alpha$
• D. $2sin\beta =3cos\alpha$

Answer 1

The correct option is C $3sin\beta =2cos\alpha$

Initial height is $r(1-\cos \alpha )$

Final height is $r(1-\sin \beta )$

Net vertical displacement is $r(1-\sin \beta )-r(1-\cos \alpha )=r(\cos \alpha -\sin \beta )$

Apply equation of kinematic

$v^2-u^2=2as$

$v^2=2gr(\cos \alpha -\sin \beta )$

Normal reaction,

$N=mg\sin \beta -\frac{m{v}^2}{r}$

$0=mg\sin \beta -\frac{m2gr(\cos \alpha -\sin \beta )}{r}$

$3\sin \beta =2\cos \alpha$

Hence, the relation is $3\sin \beta =2\cos \alpha$