The value of acceleration due to gravity 'g' on the surface of a planet with radius double that of the earth and same mean density as that of the earth is (`g_(e)` =acceleration due to gravity on the surface of the earth )

Options

A. `g_(p)=2g_(e)`
B. `g_(p)=g_(e)//2`
C. `g_(p)=g_(e)//4`
D. `g_(p)=4g_(e)`

Explanation

To find the value of acceleration due to gravity 'g' on the surface of a planet with a radius double that of the Earth and the same mean density as that of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Gravitational Acceleration**:
The gravitational acceleration \( g \) at the surface of a planet is given by the formula:
\[ g = \frac{G \cdot M}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Express Mass in Terms of Density and Volume**:
The mass \( M \) of the planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the mass becomes:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

3. **Substitute Mass into the Gravitational Acceleration Formula**:
Substituting the expression for mass into the formula for gravitational acceleration gives:
\[ g = \frac{G \cdot \left(\rho \cdot \frac{4}{3} \pi R^3\right)}{R^2} \]
Simplifying this, we get:
\[ g = \frac{4}{3} \pi G \rho R \]

4. **Calculate for the Earth**:
For the Earth, we denote the gravitational acceleration as \( g_e \):
\[ g_e = \frac{4}{3} \pi G \rho R_e \]
where \( R_e \) is the radius of the Earth.

5. **Calculate for the New Planet**:
For the new planet, which has a radius \( R_p = 2R_e \) and the same density \( \rho \):
\[ g_p = \frac{4}{3} \pi G \rho R_p = \frac{4}{3} \pi G \rho (2R_e) \]
This simplifies to:
\[ g_p = 2 \cdot \frac{4}{3} \pi G \rho R_e = 2g_e \]

6. **Final Result**:
Therefore, the acceleration due to gravity on the surface of the planet is:
\[ g_p = 2g_e \]

### Conclusion:
The value of acceleration due to gravity \( g \) on the surface of the planet is \( 2g_e \), where \( g_e \) is the acceleration due to gravity on the surface of the Earth.