A mars satellite moving in an orbit of radius `9.4xx10^3 km` take `27540 s` to complete one revolution. Calculate the mass of mars.

Explanation

To calculate the mass of Mars based on the given information about a satellite orbiting it, we can follow these steps:

### Step 1: Convert the radius from kilometers to meters
The radius of the orbit is given as \( 9.4 \times 10^3 \) km. We need to convert this to meters for our calculations.

\[ r = 9.4 \times 10^3 \text{ km} = 9.4 \times 10^3 \times 10^3 \text{ m} = 9.4 \times 10^6 \text{ m} \]

### Step 2: Write down the formula for the mass of Mars
The formula to calculate the mass of a planet based on the orbital radius and period of a satellite is derived from Kepler's third law:

\[ T^2 = \frac{4\pi^2 r^3}{G M} \]

Rearranging this formula to solve for the mass \( M \) of Mars gives:

\[ M = \frac{4\pi^2 r^3}{G T^2} \]

### Step 3: Substitute the known values into the formula
We know:
- \( r = 9.4 \times 10^6 \) m
- \( T = 27540 \) s
- \( G = 6.67 \times 10^{-11} \, \text{m}^3/\text{kg s}^2 \)

Now substitute these values into the formula:

\[ M = \frac{4\pi^2 (9.4 \times 10^6)^3}{(6.67 \times 10^{-11}) (27540)^2} \]

### Step 4: Calculate \( r^3 \)
First, calculate \( r^3 \):

\[ r^3 = (9.4 \times 10^6)^3 = 8.27 \times 10^{20} \text{ m}^3 \]

### Step 5: Calculate \( T^2 \)
Next, calculate \( T^2 \):

\[ T^2 = (27540)^2 = 758,916,000 \text{ s}^2 \]

### Step 6: Substitute \( r^3 \) and \( T^2 \) into the mass formula
Now substitute \( r^3 \) and \( T^2 \) back into the mass formula:

\[ M = \frac{4\pi^2 (8.27 \times 10^{20})}{(6.67 \times 10^{-11}) (758916000)} \]

### Step 7: Calculate the numerator and denominator
Calculate the numerator:

\[ 4\pi^2 \approx 39.478 \]

So,

\[ \text{Numerator} = 39.478 \times 8.27 \times 10^{20} \approx 3.26 \times 10^{22} \]

Now calculate the denominator:

\[ \text{Denominator} = (6.67 \times 10^{-11}) \times (758916000) \approx 5.06 \times 10^{-2} \]

### Step 8: Calculate the mass of Mars
Now, divide the numerator by the denominator:

\[ M = \frac{3.26 \times 10^{22}}{5.06 \times 10^{-2}} \approx 6.43 \times 10^{23} \text{ kg} \]

### Final Result
Thus, the mass of Mars is approximately:

\[ M \approx 6.43 \times 10^{23} \text{ kg} \]