A tunnel is dug along a diameter of the earth. Find the force on a particle of mass m placed in the tunnel at a distance x from the centre.

Explanation

To find the gravitational force on a particle of mass \( m \) placed in a tunnel dug along the diameter of the Earth at a distance \( x \) from the center, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Consider a tunnel dug through the Earth along its diameter.
- A particle of mass \( m \) is placed at a distance \( x \) from the center of the Earth.

2. **Using Gauss's Law**:
- According to Gauss's law for gravity, the gravitational force acting on a mass inside a spherical shell of uniform density is zero.
- Therefore, only the mass of the Earth that is at a radius less than \( x \) contributes to the gravitational force on the mass \( m \).

3. **Calculating the Mass Inside the Radius \( x \)**:
- The volume \( V \) of a sphere of radius \( x \) is given by:
\[ V = \frac{4}{3} \pi x^3 \]
- If \( \rho \) is the uniform density of the Earth, the mass \( m' \) of the sphere of radius \( x \) is:
\[ m' = \rho V = \rho \left(\frac{4}{3} \pi x^3\right) \]

4. **Relating Density to Total Mass of Earth**:
- The total mass \( M_E \) of the Earth can be expressed in terms of its radius \( R \) and density \( \rho \):
\[ M_E = \rho \left(\frac{4}{3} \pi R^3\right) \]
- Therefore, we can express \( \rho \) as:
\[ \rho = \frac{3M_E}{4 \pi R^3} \]

5. **Substituting for Mass \( m' \)**:
- Substitute \( \rho \) into the equation for \( m' \):
\[ m' = \left(\frac{3M_E}{4 \pi R^3}\right) \left(\frac{4}{3} \pi x^3\right) = M_E \frac{x^3}{R^3} \]

6. **Calculating the Gravitational Force**:
- The gravitational force \( F \) on the mass \( m \) at distance \( x \) from the center is given by Newton's law of gravitation:
\[ F = G \frac{m m'}{x^2} \]
- Substitute \( m' \) into this equation:
\[ F = G \frac{m \left(M_E \frac{x^3}{R^3}\right)}{x^2} \]
- Simplifying this gives:
\[ F = G \frac{m M_E}{R^3} x \]

7. **Final Expression for the Force**:
- Thus, the force on the particle of mass \( m \) at a distance \( x \) from the center of the Earth is:
\[ F = \frac{G M_E}{R^3} m x \]

### Final Answer:
The gravitational force on a particle of mass \( m \) placed in the tunnel at a distance \( x \) from the center of the Earth is:
\[ F = \frac{G M_E}{R^3} m x \]