The mass of Jupiter is 314 times that of earth and the diameter of Jupiter is `11.35` times that of earth . If 'g' has a value of `9.8 ms^(-2)` on the earth , what is its value on Jupiter ?

Answer

To find the value of gravitational acceleration (g) on Jupiter, we can use the relationship between gravitational acceleration, mass, and radius of the planet. The formula for gravitational acceleration is given by:

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

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

### Step-by-Step Solution:

1. **Identify the Known Values**:
- Mass of Jupiter \( M_J = 314 \times M_E \) (where \( M_E \) is the mass of Earth).
- Diameter of Jupiter \( D_J = 11.35 \times D_E \) (where \( D_E \) is the diameter of Earth).
- Therefore, the radius of Jupiter \( R_J = \frac{D_J}{2} = \frac{11.35 \times D_E}{2} = 11.35 \times R_E \) (where \( R_E \) is the radius of Earth).
- The value of \( g \) on Earth \( g_E = 9.8 \, \text{m/s}^2 \).

2. **Express Gravitational Acceleration on Jupiter**:
- The gravitational acceleration on Jupiter \( g_J \) can be expressed as:
\[ g_J = \frac{G \cdot M_J}{R_J^2} \]

3. **Substitute the Values**:
- Substitute \( M_J \) and \( R_J \) into the equation:
\[ g_J = \frac{G \cdot (314 \cdot M_E)}{(11.35 \cdot R_E)^2} \]

4. **Simplify the Equation**:
- The equation can be simplified as follows:
\[ g_J = \frac{G \cdot 314 \cdot M_E}{11.35^2 \cdot R_E^2} \]
- Recognizing that \( \frac{G \cdot M_E}{R_E^2} = g_E \):
\[ g_J = \frac{314}{11.35^2} \cdot g_E \]

5. **Calculate \( g_J \)**:
- Now substituting \( g_E = 9.8 \, \text{m/s}^2 \):
\[ g_J = \frac{314}{11.35^2} \cdot 9.8 \]

6. **Calculate \( 11.35^2 \)**:
- Calculate \( 11.35^2 = 128.4225 \).

7. **Final Calculation**:
- Now, substituting this value back into the equation:
\[ g_J = \frac{314}{128.4225} \cdot 9.8 \]
- Calculate \( \frac{314}{128.4225} \approx 2.448 \).
- Therefore, \( g_J \approx 2.448 \cdot 9.8 \approx 23.88 \, \text{m/s}^2 \).

### Conclusion:
The value of gravitational acceleration on Jupiter \( g_J \) is approximately \( 23.88 \, \text{m/s}^2 \).