A mass \(M\) is split into two parts, \(m\) and \((M–m),\) which are then separated by a certain distance. What ratio of \(m/M\) maximizes the gravitational force between the two parts?
1. \(1/3\)
2. \(1/2\)
3. \(1/4\)
4. \(1/5\)
The figure shows two concentric shells of masses \(m,~\) and \(m.~\) At which point a particle of mass \(m\) shall experience zero gravitational force because of them?

1. \(A\)
2. \(C\)
3. \(D\)
4. \(B\)
| 1. | \( \dfrac{2}{9}{F} \) | 2. | \(\dfrac{16}{9} F\) |
| 3. | \(\dfrac{8}{9} F\) | 4. | \(F\) |
| 1. | occurs only when the objects have very different masses |
| 2. | is greater on the more massive of the two objects |
| 3. | is not an attractive force |
| 4. | increases in magnitude as the two objects approach each other |

| Column-I | Column-II | ||
| \(\mathrm{(A)}\) | Force on any particle (units of \(Gm^2/a^2\)) |
\(\mathrm{(I)}\) | \(3\) |
| \(\mathrm{(B)}\) | Potential energy of the system (units of \(-Gm^2/a\)) |
\(\mathrm{(II)}\) | \(\sqrt3\) |
| \(\mathrm{(C)}\) | Gravitational potential due to any particle at the centre \((O)\) (units of \(-Gm/a\)) |
\(\mathrm{(III)}\) | \(\dfrac43\) |
| \(\mathrm{(D)}\) | Gravitational field at the mid-point of a side (units of \(Gm/a^2\)) |
\(\mathrm{(IV)}\) | \(\dfrac23\) |
| 1. | \(\mathrm{A-I, B-II, C-IV, D-III}\) |
| 2. | \(\mathrm{A-III, B-I, C-I, D-II}\) |
| 3. | \(\mathrm{A-II, B-I, C-II, D-III}\) |
| 4. | \(\mathrm{A-I, B-III, C-IV, D-II}\) |
| 1. | \({\dfrac{2Gm^2}{a^2}}\) | 2. | \({\dfrac{Gm^2}{a^2}}\) |
| 3. | \({\dfrac{\sqrt3}{2}\dfrac{Gm^2}{a^2}}\) | 4. | \({\dfrac{\sqrt3Gm^2}{a^2}}\) |
| Statement I: | The force of gravitation between two particles acts on the respective particles only when there is no other medium between them. |
| Statement II: | The gravitational force between two uniform spheres is inversely proportional to the square of the distance between their centres. |
| 1. | Statement I is incorrect and Statement II is correct. |
| 2. | Both Statement I and Statement II are correct. |
| 3. | Both Statement I and Statement II are incorrect. |
| 4. | Statement I is correct and Statement II is incorrect. |