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

Options

A. `(GMm)/R^3x`
B. `(GMm)/R^2x`
C. `(GMm)/R^3x^2`
D. zero

Explanation

To find the gravitational force acting 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 1: Understand the Setup
Consider a sphere representing the Earth with radius \( R \). A tunnel is dug along its diameter, and we have a particle of mass \( m \) located at a distance \( x \) from the center of the Earth.

### Step 2: Use the Concept of Gravitational Force
According to the shell theorem, the gravitational force acting on a mass inside a spherical shell is zero. Therefore, only the mass of the Earth that is at a radius less than \( x \) contributes to the gravitational force acting on the mass \( m \).

### Step 3: Calculate the Mass of the Sphere Inside Radius \( x \)
The mass \( M \) of the Earth that is inside the radius \( x \) can be calculated using the formula for the volume of a sphere and the density \( \rho \) of the Earth:
\[ M = \text{Volume} \times \text{Density} = \frac{4}{3} \pi x^3 \rho \]

### Step 4: Write the Gravitational Force Equation
The gravitational force \( F \) acting on the mass \( m \) at distance \( x \) from the center is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m \cdot M}{x^2} \]
Substituting the expression for \( M \):
\[ F = \frac{G \cdot m \cdot \left(\frac{4}{3} \pi x^3 \rho\right)}{x^2} \]

### Step 5: Simplify the Expression
Now, we can simplify the expression:
\[ F = \frac{G \cdot m \cdot \frac{4}{3} \pi x^3 \rho}{x^2} = \frac{4}{3} \pi G \rho m \cdot x \]

### Step 6: Final Result
Thus, the force acting on the particle of mass \( m \) at a distance \( x \) from the center of the Earth is:
\[ F = \frac{4}{3} \pi G \rho m \cdot x \]