Acceleration due to gravity at any point inside the earth

Options

A. varies inversely as the square of the distance from the centre
B. is independent of the distance from the centre
C. is directly proportional to the distance from the centre
D. is inversely proportional to the distance from centre

Answer

To find the acceleration due to gravity at any point inside the Earth, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity
The acceleration due to gravity (g) at a distance r from the center of the Earth is given by the formula:
\[ g = \frac{G M}{r^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( r \) is the distance from the center of the Earth.

### Step 2: Consider the scenario inside the Earth
When we are inside the Earth, the effective mass that contributes to the gravitational force is only the mass that is at a distance less than r from the center. According to the shell theorem, the gravitational force inside a uniform spherical shell of mass is zero. Therefore, only the mass of the sphere of radius r contributes to the gravitational force.

### Step 3: Modify the formula for inside the Earth
For a point inside the Earth at a distance r from the center, the formula for acceleration due to gravity becomes:
\[ g' = \frac{G M'}{r^2} \]
where \( M' \) is the mass of the sphere of radius r.

### Step 4: Relate mass M' to the volume and density
Assuming the Earth has a uniform density \( \rho \), the mass \( M' \) can be expressed as:
\[ M' = \rho \cdot V = \rho \cdot \frac{4}{3} \pi r^3 \]
Substituting this back into the equation for g':
\[ g' = \frac{G \cdot \rho \cdot \frac{4}{3} \pi r^3}{r^2} \]
This simplifies to:
\[ g' = \frac{4}{3} \pi G \rho r \]

### Step 5: Conclusion
From the derived formula, we can see that the acceleration due to gravity inside the Earth is directly proportional to the distance r from the center. Thus, as you move towards the center of the Earth, the acceleration due to gravity decreases linearly with r.

### Final Answer
The acceleration due to gravity at any point inside the Earth is given by:
\[ g' = \frac{4}{3} \pi G \rho r \]
This indicates that \( g' \) varies linearly with the distance from the center of the Earth.

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