The scientists named Isaac Newton, Galileo, Henry Cavendish, and Eratosthenes contributed to a way of finding out the mass of the Earth using the universal law of gravitation, acceleration due to the force of gravity, and the radius of the Earth.
This calculation is done using Newton's universal law of gravitation, which formulates the attractive force (gravitational force) that two masses place on each other:
F is the gravitational force between two masses (in Newtons)
m and M are the two masses (in kilograms)
R is the distance between the center of the two masses (in metres)
G is the universal constant of gravitation
G was calculated by Henry Cavendish in 1798, and was determined to be:
6.7 x 10-11 Nm2/kg2
Also needed is Newton's second law of motion: F = m x a
F is the force applied to an object
m is the mass of the object
a is its acceleration due to the force.
Galileo determined that the acceleration due to the force of gravity of Earth was a constant equal to 9.8 metres/second2 (9.8 m/s2) near the surface of the Earth.
Lastly, you need to know that the radius of the Earth is 6,400,000 metres (6.4 x 106m). This was first calculated by the Greek scientist Eratosthenes thousands of years ago (by comparing shadows in wells during the summer solstice about 230 B.C.).
CALCULATING THE MASS OF THE EARTH
Assume that Earth is one of the masses (M) and a 1 kg sphere is the other mass (m) near the surface of the Earth.
F = G x m x M / R2
Where:
F is the gravitational force
G is the gravitational constant
M is the mass of the Earth
m is the mass of the sphere close to the Earth’s surface
R is the radius of the Earth
The force between the Earth and the sphere can be measured by dropping the 1 kg sphere to Earth.
F = m x a
F = 1 kg x 9.8 m/s2
F = 9.8 kg x m/s2
We already know that:
R = 6.4 x 106 m
G = 6.7 x 10-11 Nm2/kg2
Therefore:
F = G x m x M / R2
Step 1: 9.8 kg x m/s2 = 6.7 x 10-11 Nm2/kg2 x 1 x M / (6.4 x 106 m)2
Step 2: M = 9.8 kg x m/s2 x (6.4 x 106 m)2 / 6.7 x 10-11 Nm2/kg2 x 1
Step 3: M = 9.8 x (6.4 x 106)2 / 6.7 x 10-11
Step 4: M = 9.8 x (4.096 x 1013) / 6.7 x 10-11
Step 5: M = 4.01408 x 1014 / 6.7 x 10-11
Answer: M = 6.0 x 1024 kg
Therefore the mass of the Earth is 6.0 x 1024 kg or 6.0 x 1021 tonnes.
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