Density of earth is `5.488 xx 10^(3) kgm^(-3)` . Assume earth to be a hemogeneous sphere, find the value of g on the surface of the earth. Use the known values of R and G.

Explanation

To find the value of \( g \) on the surface of the Earth using the given density, we can follow these steps:

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

### Step 2: Relate mass to density and volume
The mass \( M \) can be expressed in terms of density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, we can write:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

### Step 3: Substitute mass into the formula for \( g \)
Substituting the expression for \( M \) into the formula for \( g \):
\[ g = \frac{G \left( \rho \cdot \frac{4}{3} \pi R^3 \right)}{R^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho R \]

### Step 4: Substitute known values
Now, we can substitute the known values:
- Density of Earth, \( \rho = 5.488 \times 10^3 \, \text{kg/m}^3 \)
- Radius of Earth, \( R = 6371 \times 10^3 \, \text{m} \)
- Universal gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)

Substituting these values into the equation:
\[ g = \frac{4}{3} \pi (6.67 \times 10^{-11}) (5.488 \times 10^3) (6371 \times 10^3) \]

### Step 5: Calculate \( g \)
Now we can calculate \( g \):
1. Calculate \( 4/3 \pi \):
\[ \frac{4}{3} \pi \approx 4.18879 \]
2. Calculate the product:
\[ g \approx 4.18879 \times (6.67 \times 10^{-11}) \times (5.488 \times 10^3) \times (6371 \times 10^3) \]

3. Performing the multiplication:
\[ g \approx 9.8 \, \text{m/s}^2 \]

### Final Answer
Thus, the value of \( g \) on the surface of the Earth is approximately:
\[ g \approx 9.8 \, \text{m/s}^2 \]