Question

A small sphere of radius R held against the inner surface of a smooth spherical shell of radius 6R as shown in figure. The masses of the shell and small spheres are 4M and M respectively. This arrangement is placed on a smooth horizontal table. The small sphere is now released. The change in x-coordinate of the centre of the shell when the smaller sphere reaches the other extreme position is:

(a) R

(b) 2R

(c) 3R

(d) 4R

Answer 1

Correct option is (b) 2R

There is no external force on the system of the small and big spheres in the horizontal direction. So the center of mass C (horizontal coordinate of COM) remains at the same position.

X coordinate of COM:

\(C = \frac{-5R \times M + 0\times 4M}{M + 4M}=- R\) 

Thus the center of mass is a distance R away from the center of big sphere.

When the small sphere goes to the other acute extreme, the center of mass C remains at the same position horizontally. COM will still be at a distance R from the center of big sphere. But on the other side.

Then the center of big sphere moves by 2R to the other side of COM. 

The change in the position of large sphere = R−(−R) = 2R