Question

A small body of mass $m$ moves downward from the hemisphere of radius $r$ if it leaves the hemisphere at a point that is at a distance $h$ below the vertex then:

• A. $h=r$
• B. $h=\frac{r}{3}$
• C. $h=\frac{r}{2}$
• D. $h=\frac{2r}{3}$

Answer 1

The correct option is D $h=\frac{2r}{3}$

Given that,

Mass = m

Height = h

Radius = r

Now,

$F_c=mg\cos \theta -N$

When the body leaves the semi sphere

Then, N = 0

Now,

$F_c=mg\cos \theta ....\left(I\right)$

Now,

$F_c=\frac{m{v}^2}{r}....\left(II\right)$

From equation (I) and (II)

$mg\cos \theta =\frac{m{v}^2}{r}$

$v^2=gr\cos \theta$

$cos\theta =\frac{h}{r}$

Now, from conservation of energy

$U_i+K_i=U_f+K_f$

$mgr+0=mgh+\frac{1}{2}m{v}^2$

$gr=gh+\frac{1}{2}{v}^2$

$2gr-2gh=gr\cos \theta$

$2r=r\times \frac{h}{r}+2h$

$h=\frac{2}{3}r$

Hence, the height is $\frac{2}{3}r$