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 distanace R/4 from the centre of sphere. The gravitational force on mass `m` is

Options

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

Explanation

To solve the problem, we need to find the gravitational force on the point mass \( m \) placed inside the cavity of a solid sphere with a spherical cavity. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have a solid sphere of mass \( M \) and radius \( R \) with a spherical cavity of radius \( R/2 \). The center of the cavity is located at a distance \( R/2 \) from the center of the sphere. A point mass \( m \) is placed inside the cavity at a distance \( R/4 \) from the center of the sphere.

### Step 2: Gravitational Field Inside the Cavity
According to the shell theorem, the gravitational field inside a spherical cavity is uniform and can be derived from the mass outside the cavity. The total gravitational field \( E \) at a point inside the cavity can be expressed as the sum of the gravitational field due to the entire sphere minus the gravitational field due to the mass that would have occupied the cavity.

### Step 3: Calculate the Gravitational Field Outside the Cavity
The gravitational field \( E_R \) at a distance \( R/2 \) from the center of the sphere (where the cavity's center is located) can be calculated using the formula:
\[ E_R = \frac{GM}{(R/2)^2} = \frac{4GM}{R^2} \]
where \( G \) is the gravitational constant.

### Step 4: Calculate the Gravitational Field Due to the Cavity
The gravitational field \( E_C \) due to the mass that would have occupied the cavity can be calculated as if the cavity were filled with mass. The mass of the cavity (if it were filled) can be calculated as:
\[ M_C = \rho \cdot V_C = \rho \cdot \left(\frac{4}{3} \pi \left(\frac{R}{2}\right)^3\right) = \frac{1}{6} \cdot M \]
where \( \rho \) is the density of the sphere and \( V_C \) is the volume of the cavity.

The gravitational field due to this mass at a distance \( R/4 \) from the center of the sphere is:
\[ E_C = \frac{G \cdot M_C}{(R/4)^2} = \frac{G \cdot \left(\frac{1}{6} M\right)}{(R/4)^2} = \frac{G \cdot \frac{1}{6} M \cdot 16}{R^2} = \frac{8GM}{3R^2} \]

### Step 5: Calculate the Total Gravitational Field
The total gravitational field \( E_T \) at the point mass \( m \) inside the cavity is given by:
\[ E_T = E_R - E_C = \frac{4GM}{R^2} - \frac{8GM}{3R^2} \]
To combine these, find a common denominator:
\[ E_T = \frac{12GM}{3R^2} - \frac{8GM}{3R^2} = \frac{4GM}{3R^2} \]

### Step 6: Calculate the Gravitational Force on Mass \( m \)
The gravitational force \( F \) on the mass \( m \) is given by:
\[ F = m \cdot E_T = m \cdot \frac{4GM}{3R^2} \]

### Final Answer
Thus, the gravitational force on mass \( m \) is:
\[ F = \frac{4GmM}{3R^2} \]