Question

Two solid hemispheres of radii R and R/2 with centers respectively as shown in figure. The density of bigger hemisphere is and that of smaller hemisphere is .Taking center of bigger hemisphere is at origin and the distance between centers of two hemispheres 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 hemisphere of radius R =  d and

Density of hemisphere of radius 0.5R = 2d.

Distance between its centers = R/10.

To find :

The Coordinates of center of mass of system.

Solution :

Consider the body of Radius R

• Mass = density x Volume

• Volume of first body (hemisphere) :
• V = 2πR³/3
• M1 = 2πdR³/3

Consider the body of Radius 0.5R

• Mass = density x Volume

• Volume of second body (hemisphere) :
• V = 2π(0.5R)³/3
• V =  πR³/12
• M2 = 2πdR³/12

Let the first body be at origin.

Then the position of second body = 0.1R

• Formula of Center of mass : $\frac{M1R+ M2R2 }{M1 + M2 }$

Solving ,

Coordinates of center of mass ,

• d =  $\frac{\frac{2\pi d R^{3}}{3} * 0 + \frac{\pi d R^{3}}{6} * 0.1R}{\frac{2\pi d R^{3}}{3} + \frac{\pi d R^{3}}{6}}$  = 0.1R / ( 4 + 1 ) = 0.1R/5

The co-ordinates of center of mass of the system is at 0.02 R.