The earth is an approximate sphere. If the interior contained matter which is not of the same density every where, then on the surface of the earth, the acceleration due to gravity

Options

A. willl be directed towards the centre but not the same everywhere.
B. will have the same value everywhere but not directed towards the centres.
C. will be same everywhere in magnitude directed towards the centres.
D. cannot be zero at any point.

Answer

To solve the problem, we need to analyze the implications of having a non-uniform density inside the Earth on the acceleration due to gravity at its surface.

### Step-by-Step Solution:

1. **Understanding the Concept of Acceleration Due to Gravity**:
The acceleration due to gravity (g) at a point on the surface of the Earth is determined by the mass of the Earth and the distance from its center. It is given by the formula:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Considering Non-Uniform Density**:
If the density of the Earth is not uniform, it means that the distribution of mass is uneven. This can affect the location of the center of mass of the Earth. The center of mass may not coincide with the geometric center of the Earth.

3. **Effect on Direction of Acceleration**:
The direction of the acceleration due to gravity is always towards the center of mass of the Earth. If the center of mass is not at the geometric center, then the direction of gravity will not be perfectly vertical everywhere on the surface.

4. **Magnitude of Acceleration**:
The magnitude of the acceleration due to gravity depends on the total mass of the Earth and how that mass is distributed. With non-uniform density, we cannot assume that \( g \) will be the same at all points on the surface. Different regions may experience different gravitational forces.

5. **Conclusion**:
Therefore, with a non-uniform density inside the Earth:
- The acceleration due to gravity will vary in magnitude across the surface.
- The direction of the acceleration due to gravity will be directed towards the center of mass, which may not be at the center of the Earth.

### Final Answer:
The acceleration due to gravity on the surface of the Earth will not be the same everywhere and will be directed towards the center of mass, which may not be at the geometric center of the Earth.