Question

Two solid hemispheres of uniform densities and masses 'm' and 2m having same radius are joined
from their bases, to form a complete sphere. Find the distance (in centimetres) of the centre of mass
from the centre of the sphere if the radius is 100 cm. (Take 1 = 3.14)​

Answer 1

The centre of mass from the centre of the sphere is 12.5 centimetres on towards 2m hemispheres sides.

Explanation:

1.Here we are consider mass 2 m hemispheres on low side and mass m hemispheres on upper side of heavy mass hemispheres.

2. From equation of centre of gravity. On taking origin at centre of sphere.

  $\bar{Y}= \frac{m_{1}\times y_{1}+m_{2}\times y_{2}}{m_{1}+m_{2}}$    ...1)

  Where

  $\bar{Y}$= position of centre of mass of sphere on Y- axis.

   $m_{1}$ = mass of hemispheres which is top side of heavy hemispheres= m

  $m_{2}$=mass of heavy hemispheres which is bottom side of light hemispheres= 2m

  $y_{1}$ = position of centre of gravity of light hemisphere from origin = $(+\frac{3R}{8})$

$y_{2}$ = position of centre of gravity of heavy hemisphere from origin = $(-\frac{3R}{8})$

  Where R = 100 (cm)

3. Now from equation 1)

  $\bar{Y}= \frac{m\times (+\frac{3R}{8})+2m\times (-\frac{3R}{8})}{m+2m}$

 On solving above equation

 We get

 $\bar{Y}=-\frac{R}{8}$

4. On putting the value of Radius of sphere R =100 (cm)

  $\bar{Y}=$ -12.5(cm)

  Here negative sign indicate that centre of mass lie on negative Y - axis , means that lie on heavy hemispheres.