A solid sphere of mass M and a ring of mass m have their centres lying on the x-axis separated by a distance $2\sqrt{2}\mathrm{r}$ where r is the radius of the ring as shown in Fig. The gravitational force exerted by the sphere on the ring is

detailed solution

Correct option is A

The entire mass of the sphere can be assumed to be concentrated at its centre of mass O. Consider a small element of the ring at A. Let dm be the mass of the element (see Fig) The gravitational force exerted by M on dm isdF=GMdm3r2=GMdm9r2This force can be resolved into x and y components as shown. Now consider another element at B. We find that  components (namely dFcos⁡θ) add up but y-components (namely of dFsin⁡θ) cancel each other. Hence the x components of forces on all elements add up and y-components cancel. Hence, the total force exerted by the sphere on the ring isF=∮dFcos⁡θ=∮GMdm9r2×223=22GM27r2∮dm=22GMm27r2So the correct choice is (1).