Question

A solid hemisphere of radius R is mounted on a solid cylinder of same radius and density as shown. The height of the cylinder is such that resulting center of mass is O. If the radius of the sphere is $2\sqrt{2}m$ and the height of the cylinder is $n$ meter. Find $n$.

Answer 1

Let the point O be the origin.

Mass of cylinder = $\pi R^{2}H\rho$ with center of mass $(0,\frac{-H}{2})$

Mass of hemisphere = $M_h=\frac{2\pi R^3\rho }{3}$

Center of mass of hemisphere calculation

Consider a disc of thickness dy at height y from O

radius of this disc will be given by $r^2=R^2-y^2$

y co-ordinate of center of mass = $\frac{1}{M_h}{\int }_0^{R}y\pi (R^2-y^2)\rho \partial y$

= $\frac{1}{M_h}\left[\frac{\pi \rho R^4}{4}\right]$

= $\frac{3R}{8}$

So co-ordinate of center of mass of hemisphere = $(0,\frac{3R}{8})$

As effective center of mass = $(0,0)$

For the $COM$ of the system to be at O, $m_1\times OS=M_2\times OC$

=>$\pi R^{2}H\rho \ast \frac{H}{2}=\frac{2\pi R^3\rho }{3}\ast \frac{3R}{8}$

=> $H=\frac{R}{\sqrt{2}}=\frac{2\sqrt{2}m}{\sqrt{2}}=2m$