The ratio between masses of two planets is 3 : 5 and the ratio between their radii is 5 : 3. The ratio between their acceleration due to gravity will be

Options

A. `(9)/(25)`
B. `(25)/(9)`
C. `(125)/(27)`
D. `(27)/(125)`

Answer

To find the ratio between the acceleration due to gravity of two planets, we can use the formula for gravitational acceleration, which is given by:

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

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

### Step-by-Step Solution:

1. **Identify the given ratios:**
- The ratio of the masses of the two planets is given as:
\[ \frac{M_1}{M_2} = \frac{3}{5} \]
- The ratio of the radii of the two planets is given as:
\[ \frac{R_1}{R_2} = \frac{5}{3} \]

2. **Express the ratio of the radii in terms of squares:**
- Since we need \( R^2 \) in the formula for gravitational acceleration, we will square the radius ratio:
\[ \frac{R_1^2}{R_2^2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \]

3. **Set up the ratio of the accelerations due to gravity:**
- The ratio of the accelerations due to gravity \( g_1 \) and \( g_2 \) can be expressed as:
\[ \frac{g_1}{g_2} = \frac{G \cdot M_1}{R_1^2} \div \frac{G \cdot M_2}{R_2^2} = \frac{M_1}{M_2} \cdot \frac{R_2^2}{R_1^2} \]

4. **Substitute the known values:**
- Substitute the ratios we have:
\[ \frac{g_1}{g_2} = \frac{3}{5} \cdot \frac{R_2^2}{R_1^2} = \frac{3}{5} \cdot \frac{9}{25} \]

5. **Calculate the final ratio:**
- Now, calculate the product:
\[ \frac{g_1}{g_2} = \frac{3 \cdot 9}{5 \cdot 25} = \frac{27}{125} \]

### Final Result:
Thus, the ratio between the acceleration due to gravity of the two planets is:
\[ \frac{g_1}{g_2} = \frac{27}{125} \]