A solid sphere of mass M and radius R has a spherical cavity of radius R/2 such that the centre of cavity is at a distance R/2 from the centre of the sphere. A point mass `m` is placed inside the cavity at a distance R/4 from the centre of sphere. The gravitational force on mass `m` is

Options

A. `(11GMm)/(R^(2)`
B. `(14GMm)/(R^(2)`
C. `(GMm)/(2R^(2)`
D. `(GMm)/(4R^(2)`

Answer

To solve the problem of finding the gravitational force on a point mass \( m \) placed inside a spherical cavity of a solid sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a solid sphere of mass \( M \) and radius \( R \).
- There is a spherical cavity of radius \( \frac{R}{2} \) whose center is at a distance \( \frac{R}{2} \) from the center of the solid sphere.
- A point mass \( m \) is placed inside the cavity at a distance \( \frac{R}{4} \) from the center of the solid sphere.

2. **Determine the Mass of the Cavity**:
- The volume of the solid sphere is given by \( V = \frac{4}{3} \pi R^3 \).
- The volume of the cavity is \( V_{cavity} = \frac{4}{3} \pi \left(\frac{R}{2}\right)^3 = \frac{4}{3} \pi \frac{R^3}{8} = \frac{1}{6} \pi R^3 \).
- The mass of the cavity (which is the mass that is missing) can be calculated as:
\[ M_{cavity} = \frac{M}{8} \]
- This is because the density of the sphere remains constant, and the cavity occupies \( \frac{1}{8} \) of the volume of the sphere.

3. **Calculate the Gravitational Field Inside the Sphere**:
- The gravitational field \( g \) inside a solid sphere at a distance \( r \) from the center is given by:
\[ g = \frac{G M_{inside}}{r^2} \]
- Where \( M_{inside} \) is the mass of the sphere that is at a distance \( r \) from the center.
- For our case, we need to consider the gravitational field due to the remaining mass of the sphere after removing the cavity.

4. **Calculate the Gravitational Field at the Point Mass \( m \)**:
- The distance of the point mass \( m \) from the center of the sphere is \( \frac{R}{4} \).
- The gravitational field due to the remaining mass of the sphere at this point is:
\[ g_{remaining} = \frac{G (M - M_{cavity})}{(\frac{R}{4})^2} = \frac{G (M - \frac{M}{8})}{\frac{R^2}{16}} = \frac{G \left(\frac{7M}{8}\right)}{\frac{R^2}{16}} = \frac{14G M}{R^2} \]

5. **Consider the Effect of the Cavity**:
- The gravitational field due to the cavity can be treated as a negative contribution since it is missing mass. The gravitational field due to the cavity at the point mass \( m \) is directed towards the center of the cavity.
- The distance from the center of the cavity to the point mass \( m \) is \( \frac{R}{4} - \frac{R}{2} = -\frac{R}{4} \).
- The gravitational field due to the cavity is:
\[ g_{cavity} = -\frac{G M_{cavity}}{(\frac{R}{4})^2} = -\frac{G \left(\frac{M}{8}\right)}{\frac{R^2}{16}} = -\frac{2G M}{R^2} \]

6. **Combine the Gravitational Fields**:
- The total gravitational field \( g_{total} \) at the point mass \( m \) is:
\[ g_{total} = g_{remaining} + g_{cavity} = \frac{14G M}{R^2} - \frac{2G M}{R^2} = \frac{12G M}{R^2} \]

7. **Calculate the Gravitational Force on Mass \( m \)**:
- The gravitational force \( F \) on the mass \( m \) is given by:
\[ F = m \cdot g_{total} = m \cdot \frac{12G M}{R^2} \]

### Final Answer:
The gravitational force on mass \( m \) is:
\[ F = \frac{12G M m}{R^2} \]