`g_(e)` and `g_(p)` denote the acceleration due to gravity on the surface of the earth and another planet whose mass and radius are twice as that of earth. Then

Options

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

Answer

To solve the problem, we need to find the relationship between the acceleration due to gravity on the surface of the Earth (denoted as \( g_e \)) and the acceleration due to gravity on the surface of another planet (denoted as \( g_p \)) whose mass and radius are both twice that of the Earth.

### Step-by-Step Solution:

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

2. **Calculate \( g_e \) for Earth**:
For Earth, we denote its mass as \( M_e \) and its radius as \( R_e \). Thus, the acceleration due to gravity on Earth is:
\[ g_e = \frac{G M_e}{R_e^2} \]

3. **Determine the Mass and Radius of the New Planet**:
The problem states that the mass and radius of the new planet are both twice that of Earth:
\[ M_p = 2 M_e \]
\[ R_p = 2 R_e \]

4. **Calculate \( g_p \) for the New Planet**:
Now, we can find the acceleration due to gravity on the surface of the new planet:
\[ g_p = \frac{G M_p}{R_p^2} \]
Substituting the values of \( M_p \) and \( R_p \):
\[ g_p = \frac{G (2 M_e)}{(2 R_e)^2} \]
Simplifying the denominator:
\[ g_p = \frac{G (2 M_e)}{4 R_e^2} \]

5. **Relate \( g_p \) to \( g_e \)**:
Now we can express \( g_p \) in terms of \( g_e \):
\[ g_p = \frac{2 G M_e}{4 R_e^2} = \frac{1}{2} \cdot \frac{G M_e}{R_e^2} = \frac{1}{2} g_e \]

6. **Final Relationship**:
Therefore, the relationship between the acceleration due to gravity on the new planet and that on Earth is:
\[ g_p = \frac{1}{2} g_e \]

### Conclusion:
The acceleration due to gravity on the surface of the new planet is half that of the acceleration due to gravity on the surface of the Earth.