Average density of earth is

Options

A. is directly proportional to g
B. is inversely proportional to g
C. is complex function of g
D. does not depend on g

Answer

To determine the average density of the Earth in relation to the acceleration due to gravity (g), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of g**:
- The acceleration due to gravity (g) at the surface of a planet 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.

**Hint**: Remember that \( g \) is influenced by both the mass of the Earth and its radius.

2. **Express Mass in Terms of Density**:
- The mass \( M \) of the Earth can be expressed in terms of its average density \( \rho \) and volume \( V \):
\[ M = \rho V = \rho \left(\frac{4}{3} \pi R^3\right) \]
where \( \rho \) is the average density of the Earth.

**Hint**: The volume of a sphere is calculated using the formula \( V = \frac{4}{3} \pi R^3 \).

3. **Substitute Mass into the g Formula**:
- Substitute the expression for mass \( M \) into the formula for \( g \):
\[ g = \frac{G \left(\rho \frac{4}{3} \pi R^3\right)}{R^2} \]
- Simplifying this gives:
\[ g = \frac{4}{3} \pi G \rho R \]

**Hint**: Notice how \( g \) is now expressed in terms of \( \rho \) and \( R \).

4. **Analyze the Relationship**:
- From the equation \( g = \frac{4}{3} \pi G \rho R \), we see that \( g \) is directly proportional to the average density \( \rho \) of the Earth when \( R \) is constant.
- However, the average density \( \rho \) itself does not depend on the value of \( g \).

**Hint**: The density is a property of the material and does not change with the gravitational acceleration.

5. **Conclusion**:
- Based on the analysis, we conclude that while \( g \) can change with density, the average density of the Earth does not depend on the value of \( g \). Thus, the correct answer is that the average density of the Earth does not depend on \( g \).

**Final Answer**: The average density of the Earth does not depend on \( g \) (Option 4).