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 sphere are $4M$ and $M$ respectively. This arrangement is placed on a smooth horizontal table. The smaller sphere is now released. The $x-coordinate$ 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

Answer: (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:

$C=\frac{-5R\times M+O\times 4M}{M+4M}=-R$

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 $=R-(-R)=2R$