Calculate the mass of the earth from the following data. Radius of the earth, 6371km, `g = 9.8 ms^(-2)`

Explanation

To calculate the mass of the Earth using the given data, we can use the formula for gravitational acceleration:

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

Where:
- \( g \) is the acceleration due to gravity (9.8 m/s²),
- \( G \) is the universal gravitational constant (\( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 1: Rearranging the formula
We need to rearrange the formula to solve for \( M \):

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

### Step 2: Convert the radius of the Earth to meters
The radius of the Earth is given as 6371 km. We need to convert this to meters:

\[ R = 6371 \, \text{km} = 6371 \times 10^3 \, \text{m} = 6.371 \times 10^6 \, \text{m} \]

### Step 3: Substitute the values into the formula
Now we can substitute the values of \( g \), \( R \), and \( G \) into the rearranged formula:

\[ M = \frac{9.8 \, \text{m/s}^2 \cdot (6.371 \times 10^6 \, \text{m})^2}{6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2} \]

### Step 4: Calculate \( R^2 \)
First, calculate \( R^2 \):

\[ R^2 = (6.371 \times 10^6 \, \text{m})^2 = 4.059 \times 10^{13} \, \text{m}^2 \]

### Step 5: Substitute \( R^2 \) back into the formula
Now substitute \( R^2 \) back into the equation for \( M \):

\[ M = \frac{9.8 \cdot 4.059 \times 10^{13}}{6.67 \times 10^{-11}} \]

### Step 6: Calculate the numerator
Calculate the numerator:

\[ 9.8 \cdot 4.059 \times 10^{13} = 3.98 \times 10^{14} \]

### Step 7: Calculate the mass of the Earth
Now, divide the numerator by \( G \):

\[ M = \frac{3.98 \times 10^{14}}{6.67 \times 10^{-11}} \approx 5.97 \times 10^{24} \, \text{kg} \]

### Final Answer
The mass of the Earth is approximately:

\[ M \approx 5.97 \times 10^{24} \, \text{kg} \]

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