Question

A hollow hemisphere and a solid hemisphere of equal radius and mass are placed as shown, the coordinates of their mutual centre of mass are

Answer 1

Let the mass be $m$.

The $Y_1$ and $Y_2$ co-ordinates will be zero cause the center of masses of both the objects are on $X$- axis.

Now according to the formula, the distance of center of mass of solid hemisphere is $\frac{R}{2}$

Thus,

$X_1=\frac{3R}{8}$

Now according to the formula, the distance of center of mass of hollow hemisphere is $\frac{3R}{8}$

Thus,

$X_2=\frac{-R}{2}$

(Minus sigh indicates that the center of mass of this hemisphere is on the left side)

Now according to the formula,

$\frac{m\left(X_1\right)-m\left(X_2\right)}{m+m}$

$\frac{m\left(3R/8\right)-m\left(R/2\right)}{2m}$

$=\frac{m(3R/8-R/2)}{2m}$

$\frac{(3R/8-4R/8)}{2}$

$=\frac{(-R/8)}{2}=\frac{-R}{16}$

So, the co-ordinates are $(\frac{-R}{16},0)$