Assuming the earth to be a sphere of uniform density, the acceleration 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 inversely proportional to its distance from the centre.

Answer

To solve the problem of determining the acceleration due to gravity assuming the Earth is a sphere of uniform density, we can follow these steps:

### Step 1: Understand the Concept of Gravitational Acceleration
The acceleration due to gravity (g) at a distance R from the center of the Earth can be calculated using the formula:
\[ g = \frac{G \cdot M}{R^2} \]
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: Determine the Mass of the Earth
Since we are assuming the Earth has a uniform density (ρ), we can express the mass (M) of the Earth in terms of its density and volume:
\[ M = \rho \cdot V \]
The volume (V) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass of the Earth can be expressed as:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

### Step 3: Substitute Mass into the Gravitational Formula
Now, substituting the expression for mass (M) into the gravitational acceleration formula:
\[ g = \frac{G \cdot \left(\rho \cdot \frac{4}{3} \pi R^3\right)}{R^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho R \]

### Step 4: Analyze the Results
From the derived formula, we can see that the acceleration due to gravity (g) is directly proportional to the radius (R) of the Earth when the density (ρ) is constant. This means:
- Inside the Earth, the acceleration due to gravity increases linearly with distance from the center (up to the surface).
- Outside the Earth, the acceleration due to gravity decreases with the square of the distance from the center.

### Step 5: Conclusion
Thus, the acceleration due to gravity is:
- Inversely proportional to the square of the distance from the center of the Earth when outside the Earth.
- Directly proportional to the distance from the center when inside the Earth.

### Final Answer
The correct option is that the acceleration due to gravity is inversely proportional to the square of the distance from the center of the Earth when outside.

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