$$ F = \frac{Gm_1m_2}{r^2}$$
where:
- F is the gravitational force between two objects
- G is the gravitational constant (approximately 6.674 × 10^-11 N m^2 kg^-2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
To calculate the mass of the Earth (m1), we need to know the gravitational force (F), the mass of an object on the Earth's surface (m2), and the radius of the Earth (r).
By measuring the acceleration due to gravity (g) at the Earth's surface, which is approximately 9.8 m/s², we can calculate the gravitational force (F) acting on an object with mass m2 using the formula:
$$ F = m2g $$
Next, we need to find the distance (r) between the center of the Earth and the object. This distance is equal to the radius of the Earth, which is approximately 6.371 × 10^6 meters.
Now, substituting these values into the gravitational force equation, we can solve for the mass of the Earth (m1):
$$ m1 = \frac{F r^2}{Gm_2} $$
$$ m1 = \frac{(m_2g) (r^2)}{G}$$
By plugging in the values for g, r, and the mass of the object on the Earth's surface (m2), we can calculate the mass of the Earth.
For example, if we assume the object on the Earth's surface has a mass of 1 kilogram (m2 = 1 kg), then the mass of the Earth (m1) would be:
$$ m1 = \frac{(1 kg)(9.8 m/s^2) (6.371 × 10^6 m)^2}{(6.674 × 10^-11 N m^2 kg^-2)}$$
$$ m1 \approx 5.972 × 10^24 kg $$
This calculation gives an approximate value for the mass of the Earth, which is close to the accepted value of 5.972 × 10^24 kilograms.
