At the centre of the earth , the value of g becomes

Options

A. zero
B. unity
C. infinity
D. none of these

Answer

To determine the value of gravitational acceleration (g) at the center of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formula for g**:
The acceleration due to gravity (g) at the surface of the Earth is given by the formula:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

**Hint**: Remember that this formula applies at the surface of the Earth, where the entire mass of the Earth is exerting gravitational force.

2. **Considering the Earth as a Uniform Sphere**:
For the purpose of this calculation, we assume the Earth is a perfect sphere with uniform density (\( \rho \)). The mass of the Earth can be expressed in terms of its density and volume:
\[ M = \rho V \]
where \( V \) is the volume of the Earth given by:
\[ V = \frac{4}{3} \pi R^3 \]

**Hint**: The volume of a sphere is crucial for relating mass and density.

3. **Substituting Volume into the Equation for g**:
We can substitute the expression for volume into the equation for mass:
\[ M = \rho \left( \frac{4}{3} \pi R^3 \right) \]
Now substituting this back into the formula for g:
\[ g = \frac{G \left( \rho \frac{4}{3} \pi R^3 \right)}{R^2} \]

**Hint**: Simplifying the equation will help us understand how g changes with radius.

4. **Simplifying the Expression**:
Simplifying the equation gives:
\[ g = \frac{4}{3} G \rho R \]
This shows that g depends on the density of the Earth and the radius.

**Hint**: Notice that this expression still contains R, which we will evaluate for the center of the Earth.

5. **Evaluating g at the Center of the Earth**:
At the center of the Earth, the radius \( R \) becomes 0. Substituting \( R = 0 \) into the equation:
\[ g = \frac{4}{3} G \rho \cdot 0 = 0 \]

**Hint**: Think about why the gravitational force would be zero at the center.

6. **Conclusion**:
Therefore, the value of gravitational acceleration (g) at the center of the Earth is:
\[ g = 0 \]

**Final Answer**: The acceleration due to gravity at the center of the Earth is zero.