Question

A small body of mass $m$ slides without friction from the top of a hemispherical cup of radius $r$ as shown in the following figure. If it leaves the surface of the cup at a vertical distance $h$ below the highest point, 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 B $h=\frac{r}{3}$


$mgcos\theta -N=f_c$
when the mass $m$ detaches from hemispere,
$N=0$
$mgcos\theta =f_c=\frac{m{v}^2}{r}$
$v^2=grcos\theta ...\left(i\right)$
from conservation of mechanical energy
$mgr+0=mg(r-h)+\frac{1}{2}m{v}^2$
$mgr=m\left(g\right(r-h)+\frac{1}{2}grcos\theta )$
$gr=g\left(\right(r-h)+\frac{1}{2}r\times \frac{r-h}{r})$
$r=(r-h)+\frac{1}{2}(r-h)$
$r=\frac{3(r-h)}{2}$
$h=\frac{r}{3}$