Two planets of radii `R_(1)` and `R_(2` have masses `m_(1)` and `M_(2)` such that `(M_(1))/(M_(2))=(1)/(g)`. The weight of an object on these planets is `w_(1)` and `w_(2)` such that `(w_(1))/(w_(2))=(4)/(9)`. The ratio `R_(1)/R_(2)`

Options

A. 2
B. `(1)/(2)`
C. 4
D. `(1)/(4)`

Answer

To solve the problem, we need to find the ratio \( \frac{R_1}{R_2} \) given the conditions about the masses and weights of objects on two different planets.

### Step-by-Step Solution:

1. **Understanding Weight on Planets**:
The weight \( w \) of an object on a planet is given by the formula:
\[ w = mg \]
where \( g \) is the acceleration due to gravity on that planet.

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

3. **Weight on Each Planet**:
For the first planet with mass \( m_1 \) and radius \( R_1 \):
\[ w_1 = mg = m \cdot \frac{Gm_1}{R_1^2} \]
For the second planet with mass \( m_2 \) and radius \( R_2 \):
\[ w_2 = mg = m \cdot \frac{Gm_2}{R_2^2} \]

4. **Ratio of Weights**:
Given that \( \frac{w_1}{w_2} = \frac{4}{9} \), we can express this as:
\[ \frac{w_1}{w_2} = \frac{m \cdot \frac{Gm_1}{R_1^2}}{m \cdot \frac{Gm_2}{R_2^2}} = \frac{m_1}{m_2} \cdot \frac{R_2^2}{R_1^2} \]

5. **Substituting Given Ratios**:
We know that \( \frac{m_1}{m_2} = \frac{1}{g} \) (where \( g = 9 \) from the problem statement):
\[ \frac{m_1}{m_2} = \frac{1}{9} \]
Therefore, substituting into the weight ratio:
\[ \frac{4}{9} = \frac{1}{9} \cdot \frac{R_2^2}{R_1^2} \]

6. **Simplifying the Equation**:
Multiply both sides by 9:
\[ 4 = \frac{R_2^2}{R_1^2} \]
Rearranging gives:
\[ \frac{R_1^2}{R_2^2} = \frac{1}{4} \]

7. **Taking Square Roots**:
Taking the square root of both sides:
\[ \frac{R_1}{R_2} = \frac{1}{2} \]

### Final Answer:
Thus, the ratio \( \frac{R_1}{R_2} \) is:
\[ \frac{R_1}{R_2} = \frac{1}{2} \]