Two planets A and B have the same averge density . Their radii `R_A and R_B` are such that `R_A : R_B = 3 : 1`. If `g_A` and `g_B` are the acceleration due to gravity at the surfaces of the planets, the `g_A : g_B` equals

Options

A. `3 : 1`
B. `1 : 3`
C. `1 : 9`
D. `sqrt(3) : 1`

Answer

To solve the problem, we need to determine the ratio of the acceleration due to gravity at the surfaces of two planets, A and B, given that they have the same average density and their radii are in the ratio of 3:1.

### Step-by-Step Solution:

1. **Understanding the Formula for Acceleration due to Gravity**:
The acceleration due to gravity \( g \) at 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. **Expressing Mass in Terms of Density**:
The mass \( M \) of a planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho \cdot V \]
The volume \( V \) of a sphere (which we assume the planets are) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the mass can be rewritten as:
\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

3. **Substituting Mass into the Gravity Formula**:
Substituting the expression for mass into the formula for \( g \):
\[ g = \frac{G \cdot \left(\rho \cdot \frac{4}{3} \pi R^3\right)}{R^2} \]
Simplifying this gives:
\[ g = \frac{4}{3} \pi G \rho R \]
This shows that the acceleration due to gravity \( g \) is directly proportional to the radius \( R \) of the planet when the density \( \rho \) is constant.

4. **Finding the Ratio of Accelerations**:
Let \( R_A \) and \( R_B \) be the radii of planets A and B, respectively. Given that:
\[ R_A : R_B = 3 : 1 \]
We can express this as:
\[ R_A = 3R \quad \text{and} \quad R_B = R \]
Now, using the proportionality of \( g \) to \( R \):
\[ \frac{g_A}{g_B} = \frac{R_A}{R_B} = \frac{3R}{R} = 3 \]

5. **Final Result**:
Therefore, the ratio of the acceleration due to gravity at the surfaces of planets A and B is:
\[ g_A : g_B = 3 : 1 \]

### Conclusion:
The final answer is:
\[ g_A : g_B = 3 : 1 \]