Assume the earth to be a sphere of uniform density the accleration due to gravity

Options

A. at a point outside the earth is inversely proportional to the square of its distance from the centre
B. at a point outside the earth is inversely proportional to its distance from the centre
C. at a point inside is zero
D. at a point inside is proportional to its distance from the centre.

Explanation

To find the acceleration due to gravity \( g \) at a distance \( r \) from the center of the Earth, assuming the Earth is a sphere of uniform density, we can follow these steps:

### Step 1: Understand the Formula for Gravitational Force
The gravitational force \( F \) between two masses \( m \) (mass of an object) and \( M \) (mass of the Earth) at a distance \( r \) from the center of the Earth is given by Newton's law of gravitation:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
where \( G \) is the gravitational constant.

### Step 2: Relate Gravitational Force to Acceleration
The acceleration due to gravity \( g \) is defined as the gravitational force per unit mass:
\[ g = \frac{F}{m} = \frac{G \cdot M}{r^2} \]

### Step 3: Consider Different Cases
1. **Outside the Earth** (\( r > R \)):
- Here, \( g \) is given by:
\[ g = \frac{G \cdot M}{r^2} \]
where \( R \) is the radius of the Earth.

2. **On the Surface of the Earth** (\( r = R \)):
- The acceleration due to gravity at the surface is:
\[ g = \frac{G \cdot M}{R^2} \]

3. **Inside the Earth** (\( r < R \)):
- For a uniform sphere, the gravitational force inside the Earth varies linearly with distance from the center. The formula for \( g \) inside the Earth is:
\[ g = \frac{G \cdot M(r)}{r^2} \]
where \( M(r) \) is the mass enclosed within radius \( r \). Since the density \( \rho \) is uniform, we have:
\[ M(r) = \rho \cdot \frac{4}{3} \pi r^3 \]
Thus,
\[ g = \frac{G \cdot \rho \cdot \frac{4}{3} \pi r^3}{r^2} = \frac{4}{3} \pi G \rho r \]
This shows that \( g \) is directly proportional to \( r \) inside the Earth.

### Step 4: Conclusion
- For \( r > R \): \( g = \frac{G \cdot M}{r^2} \)
- For \( r = R \): \( g = \frac{G \cdot M}{R^2} \)
- For \( r < R \): \( g = \frac{4}{3} \pi G \rho r \)