Question

How is the mass of the earth calculated?

Answer 1

Hint: In the above question, finding out the mass of the earth is physically not possible. But it could be found out by indirect means. By following Newton’s Law of Universal Gravitation where the relationship of force between two masses is attracted to one another, we can find out the approximate mass of the earth.

Complete step by step answer:
The above question is solved by using Newton’s Law of Universal Gravitation which states that all masses are attracted towards each other in the universe. The formula is
\[F = G\dfrac{{m \times {m_e}}}{{{d^2}}}\] but he didn't have instruments sensitive enough to measure the constant G.
Later on, after many years a torsion balance was used to measure this very tiny constant and the value of G was found where
\[G = 6.67 \times {10^{ - 11}}N{m^2}/kg\]
Imagine there is a mass m sitting on the top and d is the distance from the center
We also have a formula that tells us how much the earth pulls it down.
\[W = mg\]
 And Newton had another formula \[F = G\dfrac{{{m_1}{m_e}}}{{{d^2}}}\]. Both will tell us the force in Newtons, pulling that mass m to the center of the earth. Therefore, we can equate these two equations
\[
  mg = G\dfrac{{m \times {m_e}}}{{{d^2}}} \\
  g = G\dfrac{{{m_e}}}{{{d^2}}} \\
 \]
where \[{m_e}\]is the mass of the earth. Therefore, by substituting the values where
g is acceleration due to gravity and d ids the distance from center and these are constants
and we get,
\[
   \Rightarrow {m_e} = \dfrac{{g{d^2}}}{G} \\
   \Rightarrow {m_e} = \dfrac{{9.8 \times {{(6.38 \times {{10}^6})}^2}}}{{6.67 \times {{10}^{ - 11}}}} \\
   \Rightarrow {m_e} = 5.98 \times {10^{24}}kg \\
 \]
Therefore, mass of the earth is \[5.98 \times {10^{24}}kg\].

Note:
Mass and weight are not the same. An object has mass and this becomes heavy enough to show the weight. The weight tells how hard the gravity is pulling it and it may vary at different places, for example, the moon because the gravity pull is less but the mass remains the same everywhere.