If the ratio of the masses of two planets is 8 : 3 and the ratio of their diameters is 2 : 3, then what will be the ratio of their acceleration due to gravity ?

Explanation

To find the ratio of the acceleration due to gravity (g) on two planets based on their masses and diameters, we can use the formula for gravitational acceleration:

\[ g = \frac{GM}{R^2} \]

where \( G \) is the universal 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:
\[ \frac{m_1}{m_2} = \frac{8}{3} \]
- The ratio of the diameters of the two planets is given as:
\[ \frac{d_1}{d_2} = \frac{2}{3} \]
- Since the diameter is twice the radius, we can express the ratio of the radii as:
\[ \frac{r_1}{r_2} = \frac{d_1}{d_2} = \frac{2}{3} \]

2. **Express the Ratio of Radii:**
- From the diameter ratio, we have:
\[ r_1 = \frac{2}{3} r_2 \]

3. **Write the Formula for Acceleration Due to Gravity:**
- For the first planet:
\[ g_1 = \frac{G m_1}{r_1^2} \]
- For the second planet:
\[ g_2 = \frac{G m_2}{r_2^2} \]

4. **Find the Ratio of Acceleration Due to Gravity:**
- The ratio \( \frac{g_1}{g_2} \) can be expressed as:
\[ \frac{g_1}{g_2} = \frac{G m_1 / r_1^2}{G m_2 / r_2^2} = \frac{m_1}{m_2} \cdot \frac{r_2^2}{r_1^2} \]
- The \( G \) cancels out.

5. **Substituting the Values:**
- Substitute the ratios:
\[ \frac{g_1}{g_2} = \frac{8/3}{1} \cdot \frac{r_2^2}{r_1^2} \]
- Since \( r_1 = \frac{2}{3} r_2 \), we have:
\[ r_1^2 = \left(\frac{2}{3} r_2\right)^2 = \frac{4}{9} r_2^2 \]
- Therefore:
\[ \frac{g_1}{g_2} = \frac{8}{3} \cdot \frac{r_2^2}{\frac{4}{9} r_2^2} = \frac{8}{3} \cdot \frac{9}{4} \]

6. **Calculating the Final Ratio:**
- Simplifying:
\[ \frac{g_1}{g_2} = \frac{8 \cdot 9}{3 \cdot 4} = \frac{72}{12} = 6 \]
- Thus, the ratio of the acceleration due to gravity is:
\[ \frac{g_1}{g_2} = 6 : 1 \]

### Final Answer:
The ratio of the acceleration due to gravity on the two planets is \( 6 : 1 \).