An astronaut on a strange planet finds that acceleration due to gravity is twice as that on. the surface of earth. Which of the following could explain this?

Options

A. Both the mass and radius of the planet are half of that north
B. Radius of the planet Is half as that of earth, but the mass Is same as that of earth
C. Both the mass and radius of the planet are twice as that of earth
D. Mass of the planet Is half as that of earth, but radius Is same as that of earth

Answer

To solve the problem of determining which option explains why the acceleration due to gravity on a strange planet is twice that of Earth, we will analyze each option using the formula for gravitational acceleration.

### Step-by-Step Solution:

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

2. **Identify the Given Information**:
The problem states that the acceleration due to gravity on the strange planet is twice that on Earth:
\[ g' = 2g \]
where \( g \) is the acceleration due to gravity on Earth.

3. **Analyze Each Option**:

- **Option 1**: Both mass and radius of the planet are half that of Earth.
- If \( M' = \frac{M}{2} \) and \( R' = \frac{R}{2} \):
\[ g' = \frac{G \cdot \frac{M}{2}}{\left(\frac{R}{2}\right)^2} = \frac{G \cdot \frac{M}{2}}{\frac{R^2}{4}} = \frac{2GM}{R^2} = 2g \]
- This option satisfies the condition \( g' = 2g \).

- **Option 2**: Radius of the planet is half that of Earth, but the mass is the same as that of Earth.
- If \( M' = M \) and \( R' = \frac{R}{2} \):
\[ g' = \frac{G \cdot M}{\left(\frac{R}{2}\right)^2} = \frac{G \cdot M}{\frac{R^2}{4}} = 4 \frac{GM}{R^2} = 4g \]
- This option does not satisfy the condition.

- **Option 3**: Both mass and radius are twice that of Earth.
- If \( M' = 2M \) and \( R' = 2R \):
\[ g' = \frac{G \cdot 2M}{(2R)^2} = \frac{G \cdot 2M}{4R^2} = \frac{1}{2} \frac{GM}{R^2} = \frac{1}{2}g \]
- This option does not satisfy the condition.

- **Option 4**: Mass of the planet is half that of Earth, but radius is the same as that of Earth.
- If \( M' = \frac{M}{2} \) and \( R' = R \):
\[ g' = \frac{G \cdot \frac{M}{2}}{R^2} = \frac{1}{2} \frac{GM}{R^2} = \frac{1}{2}g \]
- This option does not satisfy the condition.

4. **Conclusion**:
The only option that satisfies the condition that the acceleration due to gravity on the strange planet is twice that on Earth is **Option 1**.

### Final Answer:
**Option 1**: Both mass and radius of the planet are half that of Earth.