Which graph correctley presents the variation of acceleration due to gravity with the distance form the centre of the earth?

Options

A.
B.
C.
D.

Answer

To determine the correct graph that represents the variation of acceleration due to gravity (g) with the distance from the center of the Earth (r), we can analyze the behavior of g in two distinct regions: outside the Earth and inside the Earth.

### Step-by-Step Solution:

1. **Understanding the Concept**:
- The acceleration due to gravity (g) varies with distance from the center of the Earth. We need to analyze how g changes as we move from the center of the Earth to the surface and then beyond the surface.

2. **Case 1: Outside the Earth** (r > R, where R is the radius of the Earth):
- The formula for gravitational acceleration outside the Earth is given by:
\[ g = \frac{G M_e}{r^2} \]
where \( G \) is the gravitational constant and \( M_e \) is the mass of the Earth.
- As the distance (r) increases, g decreases with the square of the distance, indicating that g is inversely proportional to \( r^2 \).

3. **Case 2: Inside the Earth** (r < R):
- The formula for gravitational acceleration inside the Earth is:
\[ g = \frac{G M_e}{R^3} r \]
Here, g is directly proportional to the distance (r) from the center of the Earth.
- This means that as we move from the center of the Earth towards the surface, g increases linearly with r.

4. **Graphical Representation**:
- At the center of the Earth (r = 0), g = 0.
- As we move towards the surface, g increases linearly until it reaches its maximum value at the surface (r = R).
- Beyond the surface, g decreases following an inverse square law.

5. **Identifying the Correct Graph**:
- The graph should start from zero at the center, increase linearly to a maximum at the surface, and then decrease following a curve that represents the inverse square relationship as we move further away from the Earth.

6. **Conclusion**:
- After analyzing the options provided, the correct graph that represents this variation is the one that shows a linear increase from 0 to maximum at the surface and then a decrease following the inverse square law.

### Final Answer:
The fourth option is the correct graph representing the variation of acceleration due to gravity with the distance from the center of the Earth.