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, 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 1: Define the variables
Let:
- \( M_m \) = mass of Mars
- \( R_m \) = radius of Mars
- \( M_e \) = mass of Earth = \( 10 M_m \) (given)
- \( R_e \) = radius of Earth = \( 2 R_m \) (given)
- \( g_e \) = acceleration due to gravity on Earth = \( 10 \, \text{m/s}^2 \)

### Step 2: Write the equation for gravity on Earth
Using the formula for gravitational acceleration on Earth:

\[ g_e = \frac{G M_e}{R_e^2} \]

Substituting the values we have:

\[ 10 = \frac{G (10 M_m)}{(2 R_m)^2} \]

### Step 3: Simplify the equation
Now simplify the equation:

\[ 10 = \frac{G (10 M_m)}{4 R_m^2} \]

This can be rearranged to:

\[ 10 = \frac{10 G M_m}{4 R_m^2} \]

Dividing both sides by 10:

\[ 1 = \frac{G M_m}{4 R_m^2} \]

### Step 4: Write the equation for gravity on Mars
Now, we can write the equation for gravitational acceleration on Mars:

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

### Step 5: Relate \( g_m \) to the previous equation
From the previous step, we know that:

\[ \frac{G M_m}{4 R_m^2} = 1 \]

Thus, we can express \( G M_m \) in terms of \( g_m \):

\[ G M_m = 4 R_m^2 \]

Substituting this into the equation for \( g_m \):

\[ g_m = \frac{4 R_m^2}{R_m^2} \]

### Step 6: Simplify to find \( g_m \)
This simplifies to:

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

### Conclusion
The acceleration due to gravity on the surface of Mars is:

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