The Earth’s gravitational
attraction keeps the Moon going around the Earth, rather than
the Moon going straight off into space. By looking at how fast
the Moon orbits (travels around) the Earth and how far away it is
from the Earth we can find out the mass of the Earth.
We know that because the Earth has mass it has a force of
gravity that pulls objects towards the Earth. The size of this
force of gravity depends on the mass of the Earth and the other
object and the distance between the center of the two. We also
know that if something is dropped, it falls back to Earth because
of the Earth’s gravity. As the object falls, it falls faster
and faster (accelerates) until it hits the ground.
Objects near the surface of the Earth accelerate at a rate of
9.8 metres/second2 (9.8 m/s2). If you go
way up high on a mountain the acceleration due to gravity is a
little less than 9.8 m/s2. If you go higher still the
acceleration is even less and so on.
All objects accelerate to the Earth at exactly the same rate
if they are the same distance from the Earth's center. Even if
one object has a bigger mass than the other, it will still fall
to the Earth at the same rate. The Moon is accelerating towards
the Earth at a rate that is correct for its distance from the
Earth's center. If another object, with the mass of this
computer, were to be placed next to the Moon, it would circle the
Earth at exactly the same speed as the Moon.
The mass of the Earth can be found by looking at the distance
between the center of the Earth and the Moon, and the time it
takes the Moon to orbit the Earth.
CALCULATING THE MASS OF THE EARTH
The equation to use is:
M = 4 x pi2 x R3 / G x T2
Where:
M is the mass of the Earth (in kilograms)
R is the distance between the center of the Earth and the Moon
(in metres)
T is the time it takes the Moon to orbit the Earth (in
seconds)
G is the universal constant of gravitation (6.7 x 10-11
Nm2/kg2)
pi is 3.14
The average distance between the Earth and the Moon is 385 000
km (3.85 x 108 m).
The Moon takes 27.3 days (2358720 seconds) to orbit the Earth
once.
Therefore:
M = 4 x pi2 x R3 / G x T2
Step 1: M = 4 x 3.142 x (3.85 x 108 m)3
/ (6.7 x 10-11 Nm2/kg2) x
(2358720 s)2
Step 2: M = 4 x 9.8596 x (3.85 x 108)3 /
(6.7 x 10-11) x (5.56 x 1012)
Step 3: M = 2.25 x 1027 / 372.52
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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