Assume that a tunnel is dug through earth from North pole to south pole and that the earth is a non-rotating, uniform sphere of density `rho`. The gravitational force on a particle of mass m dropped into the tunnel when it reaches a distance r from the centre of earth is

Options

A. `(3/(4pi) mG rho) r`
B. `((4pi)/3 mG rho)r`
C. `((4pi)/3 mG rho) r^(2)`
D. `((4pi)/3 m^(2) G rho)r`

Answer

To solve the problem of finding the gravitational force on a particle of mass \( m \) dropped into a tunnel through the Earth at a distance \( r \) from the center of the Earth, we can follow these steps:

### Step 1: Understand the Setup
We have a tunnel dug straight through the Earth from the North Pole to the South Pole. The Earth is assumed to be a non-rotating, uniform sphere with a density \( \rho \).

### Step 2: Define the Variables
- Let \( R \) be the radius of the Earth.
- Let \( r \) be the distance from the center of the Earth to the particle.
- Let \( m \) be the mass of the particle.

### Step 3: Use Gauss's Law for Gravity
According to Gauss's Law for gravity, the gravitational field \( g \) inside a spherical shell of uniform density is determined only by the mass enclosed within a Gaussian surface.

### Step 4: Calculate the Mass Enclosed
The mass \( M \) enclosed within a sphere of radius \( r \) is given by:
\[ M = \rho \cdot V = \rho \cdot \left(\frac{4}{3} \pi r^3\right) \]
where \( V \) is the volume of the sphere.

### Step 5: Apply Gauss's Law
Using Gauss's Law:
\[ \oint g \cdot dS = 4 \pi G M \]
Since \( g \) is uniform over the surface of the sphere, we can simplify this to:
\[ g \cdot (4 \pi r^2) = 4 \pi G M \]
Substituting \( M \):
\[ g \cdot (4 \pi r^2) = 4 \pi G \left(\rho \cdot \frac{4}{3} \pi r^3\right) \]

### Step 6: Simplify the Equation
Canceling \( 4 \pi \) from both sides:
\[ g r^2 = \frac{4}{3} G \rho r^3 \]
Dividing both sides by \( r^2 \):
\[ g = \frac{4}{3} G \rho r \]

### Step 7: Calculate the Gravitational Force
The gravitational force \( F \) acting on the particle of mass \( m \) is given by:
\[ F = m \cdot g = m \cdot \left(\frac{4}{3} G \rho r\right) \]
Thus, the gravitational force on the particle when it is at a distance \( r \) from the center of the Earth is:
\[ F = \frac{4}{3} G \rho m r \]

### Conclusion
The gravitational force on a particle of mass \( m \) dropped into the tunnel at a distance \( r \) from the center of the Earth is:
\[ F = \frac{4}{3} G \rho m r \]