The values of the acceleration due to gravity on two planets are `g_(1) - g_(2)`, then the two planets must have the same

Options

A. raduis
B. mass
C. `(("mass")/("radius"))^(2)`
D. `("mass")/(("radius")^(2))`

Explanation

To solve the problem, we need to analyze the relationship between the acceleration due to gravity on two different planets, denoted as \( g_1 \) and \( g_2 \). The formula for the acceleration due to gravity on a planet is given by:

\[ g = \frac{G \cdot M}{R^2} \]

where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step-by-Step Solution:

1. **Write the equations for \( g_1 \) and \( g_2 \)**:
- For the first planet:
\[ g_1 = \frac{G \cdot M_1}{R_1^2} \]
- For the second planet:
\[ g_2 = \frac{G \cdot M_2}{R_2^2} \]

2. **Set the two equations equal to each other**:
Since we know \( g_1 \) and \( g_2 \) are given, we can equate them:
\[ g_1 = g_2 \]
This leads to:
\[ \frac{G \cdot M_1}{R_1^2} = \frac{G \cdot M_2}{R_2^2} \]

3. **Cancel the universal gravitational constant \( G \)**:
Since \( G \) is a constant and appears on both sides of the equation, we can cancel it out:
\[ \frac{M_1}{R_1^2} = \frac{M_2}{R_2^2} \]

4. **Rearranging the equation**:
This can be rearranged to show the relationship between the mass and radius of the two planets:
\[ \frac{M_1}{M_2} = \frac{R_1^2}{R_2^2} \]

5. **Conclusion**:
From the above relationship, we can conclude that if the values of acceleration due to gravity \( g_1 \) and \( g_2 \) are equal, then the two planets must have the same ratio of mass to the square of their radius:
\[ \frac{M_1}{R_1^2} = \frac{M_2}{R_2^2} \]

### Final Answer:
The two planets must have the same mass-to-radius squared ratio.