Question

1. Two solid hemispheres of radii R
and R/2 with centers o and o
respectively as shown in figure. The
density of bigger hemisphere is p
and that of smaller hemisphere is
2p. Taking center of bigger
hemisphere is at origin and the
distance between centers of two
hemispheres oo is R/10 find co-
ordinates of center of mass of the
system.​

Answer 1

Given :

Two solid hemispheres of radii R and 0.5R.

Density of hemispheres are ρ and 2ρ.

Distance between centers = R/10.

To find :

Coordinates of center of mass of system.

Solution :

Radius of first body = R

Volume of first body (hemisphere) :

$= \frac{2}{3} \pi {R}^{3}$

Radius of second body = 0.5R

Volume of second body (hemisphere) :

$= \frac{2}{3} \pi ( {0.5R})^{3} = \frac{\pi \: {R}^{3} }{12}$

Mass of a body:

$m = \rho \times V \:$

Density of first body = ρ

so

Mass of first body :

$= \rho \times (\frac{2}{3} \pi {R}^{3})$

Density of second body = 2ρ

si

Mass of second body :

$= 2\rho \times (\frac{ \pi {R}^{3}}{12}) \\ \: = \rho \times (\frac{ \pi {R}^{3}}{6})$

With respect to first body,

Distance of first body with respect to first body= 0 m.

and

Distance of second body with respect to first body= 0.1R.

Formula of Center of mass :

$r = \frac {m_1r_1 + m_2r_2}{m_1 + m_2}$

Putting all the values of distances and masses.

Center of mass of system :

$r = \frac { ( \rho \times (\frac{2}{3} \pi {R}^{3}) )\times 0 + (\rho \times (\frac{ \pi {R}^{3}}{6}) )\times 0.1R}{ \rho \times (\frac{2}{3} \pi {R}^{3})+ (\rho \times (\frac{ \pi {R}^{3}}{6}) )}$

$r = \frac { (\rho \times (\frac{ \pi {R}^{3}}{6})\times 0.1R}{ \rho \times (\frac{ \pi {R}^{3}}{6})(4 + 1) }$

On solving this, we get :

$r = \frac{0.1R}{5} = \frac{R}{50} = 0.02R$

so,

The center of mass of system = 0.02 R.

Answer 2

Answer:

Radius of first body = R

Volume of first body (hemisphere) :

Radius of second body = 0.5R

Volume of second body (hemisphere) :

Mass of a body:

Density of first body = ρ

so

Mass of first body :

Density of second body = 2ρ

si

Mass of second body :

With respect to first body,

Distance of first body with respect to first body= 0 m.

and

Distance of second body with respect to first body= 0.1R.

Formula of Center of mass :

Putting all the values of distances and masses.

Center of mass of system :

On solving this, we get :

so,

The center of mass of system = 0.02 R.