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Calculation: Mass of earth
We calculate Earth’s mass as follows:
Everything pulls on everything else in a simple way that involves only mass
and distance. Newton said every body attracts every other body with a force that,
for any two bodies, is directly proportional to the product of the masses of the
bodies and inversely proportional to the square of the distance separating them.
The equation, expressing Newton’s statement, is
(1) F = G (m1)
(m2) / d²
where G is the gravitational constant.
Newton also said that the acceleration of an object is directly proportional
to the net force acting on the object, is in the direction of the net force, and
is inversely proportional to the mass of the object.
The equation, expressing this statement, is
(2) a = F / m.
We substitute Earth’s mass (me) and Earth’s radius (r) in
Equation (1) to get the force that Earth attracts a body of mass (m) on
the surface of Earth.
(3) F = G (me)(m) / r²
We substitute the acceleration (g) due to gravity in Equation (2) to
get the force that the object of mass (m) resists the acceleration due to
gravity
(4) F = mg
To calculate Earth’s mass, we equate the forces given by Equations (3) and (4) and solve for
(me). We get
(5) me = g r² / G
where the constants are given by G = 6.67300 x 10-11m³/(kg
s²),
g = 9.8 m/s²,
and r = 6.378 x 106m.
Plugging the values of the constants in Equation (5) we get the mass of
Earth:
me = 5.9742 x 1024 kilograms.
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