Two planets have the same average density but their radii are `R_(1)` and `R_(2)`. If acceleration due to gravity on these planets be `g_(1)` and `g_(2)` respectively, then

Options

A. `(g_(1))/(g_(2))=(R_(1))/(R_(2))`
B. `(g_(1))/(g_(2))=(R_(2))/(R_(1))`
C. `(g_(1))/(g_(2))=(R_(1)^(2))/(R_(2)^(2))`
D. `(g_(1))/(g_(2))=(R_(1)^(3))/(R_(2)^(3))`

Explanation

To solve the problem, we need to relate the acceleration due to gravity on two planets with the same average density but different radii. Let's denote the average density of the planets as ρ, and their radii as R₁ and R₂. The acceleration due to gravity on the planets will be denoted as g₁ and g₂ respectively.

### 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:
\[ g = \frac{G \cdot M}{R^2} \]
where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.

2. **Finding the mass of the planet**:
The mass (M) of a planet can be expressed in terms of its volume and density:
\[ M = \text{Volume} \times \text{Density} = \frac{4}{3} \pi R^3 \cdot \rho \]
where ρ is the average density of the planet.

3. **Substituting mass into the gravity formula**:
Substituting the expression for mass into the formula for g, we get:
\[ g = \frac{G \cdot \left(\frac{4}{3} \pi R^3 \cdot \rho\right)}{R^2} \]
This simplifies to:
\[ g = \frac{4}{3} \pi G \rho R \]

4. **Calculating g₁ and g₂**:
For the first planet with radius R₁:
\[ g_1 = \frac{4}{3} \pi G \rho R_1 \]
For the second planet with radius R₂:
\[ g_2 = \frac{4}{3} \pi G \rho R_2 \]

5. **Finding the ratio of g₁ to g₂**:
Now, we can find the ratio of the accelerations due to gravity on the two planets:
\[ \frac{g_1}{g_2} = \frac{\frac{4}{3} \pi G \rho R_1}{\frac{4}{3} \pi G \rho R_2} \]
The terms \(\frac{4}{3} \pi G \rho\) cancel out, leading to:
\[ \frac{g_1}{g_2} = \frac{R_1}{R_2} \]

6. **Conclusion**:
Thus, we conclude that the ratio of the accelerations due to gravity on the two planets is directly proportional to the ratio of their radii:
\[ g_1 : g_2 = R_1 : R_2 \]

### Final Answer:
The relation between the accelerations due to gravity on the two planets is:
\[ g_1 : g_2 = R_1 : R_2 \]