Assuming the mass of Earth to be ten times the mass of Mars, its radius to be twice the radius of Mars and the acceleration due to gravity on the surface of Earth is `10 m//s^(2)` . Then the accelration due to gravity on the surface of Mars is given by

Options

A. `0.4 ms^(-2)`
B. `2 ms^(-2)`
C. `4 ms^(-2)`
D. `5 ms^(-2)`

Answer

To find the acceleration due to gravity on the surface of Mars (g_m), we can use the formula for gravitational acceleration:

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

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

### Step 1: Define the known quantities
Given:
- Mass of Earth \( M_e = 10 \times M_m \) (mass of Mars)
- Radius of Earth \( R_e = 2 \times R_m \) (radius of Mars)
- Acceleration due to gravity on Earth \( g_e = 10 \, \text{m/s}^2 \)

### Step 2: Write the formula for acceleration due to gravity on Mars
Using the formula for gravitational acceleration, we can express the acceleration due to gravity on Mars as:

\[ g_m = \frac{G M_m}{R_m^2} \]

### Step 3: Relate \( g_m \) to \( g_e \)
We can relate \( g_m \) and \( g_e \) using the ratio of their masses and radii:

\[ \frac{g_m}{g_e} = \frac{M_m}{M_e} \cdot \left(\frac{R_e}{R_m}\right)^2 \]

Substituting the known relationships:

\[ \frac{g_m}{10} = \frac{M_m}{10 M_m} \cdot \left(\frac{2 R_m}{R_m}\right)^2 \]

### Step 4: Simplify the equation
This simplifies to:

\[ \frac{g_m}{10} = \frac{1}{10} \cdot (2)^2 \]

\[ \frac{g_m}{10} = \frac{1}{10} \cdot 4 \]

\[ \frac{g_m}{10} = \frac{4}{10} \]

### Step 5: Solve for \( g_m \)
Now, multiplying both sides by 10 gives:

\[ g_m = 4 \, \text{m/s}^2 \]

### Conclusion
The acceleration due to gravity on the surface of Mars is \( g_m = 4 \, \text{m/s}^2 \).

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