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Have you ever wondered how the mass of Earth is measured? Obviously you can't put Earth on a scale. What you can do is measure the orbits of objects going around it, such as the Moon. The Moon orbits Earth once every 27.3215 days. The distance to the Moon can be calculated from the time it takes a laser beam to travel to the Moon's surface and back (2.514 s on average). Use the data given in this problem to determine the mass of Earth. Here is some other possibly useful information:

1. Speed of laser beam: 2.998x108 m/s
2. Radius of Earth: 6.38 106m
3. Radius of the Moon: 1.74x106m

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Answer:

[tex]5.68458\times 10^{24}\ kg[/tex]

Explanation:

M = Mass of Earth

t = Time for round trip of laser = 2.514 s

T = Orbital time = 27.3215 days

Distance between Earth and Moon is given by

[tex]2R=ct\\\Rightarrow R=\dfrac{ct}{2}\\\Rightarrow R=\dfrac{2.998\times 10^8\times 2.514}{2}\\\Rightarrow R=376848600\ m[/tex]

The centripetal accelaration will balance the gravitational force

[tex]\dfrac{GMm}{R^2}=m\dfrac{v^2}{R}\\\Rightarrow v=\sqrt{\dfrac{GM}{R}}[/tex]

Orbital time is given by

[tex]T=\dfrac{2\pi R}{v}\\\Rightarrow T=\dfrac{2\pi R}{\sqrt{\dfrac{GM}{R}}}\\\Rightarrow M=\dfrac{4\pi^2 R^3}{GT^2}\\\Rightarrow M=\dfrac{4\pi^2\times 376848600^3}{6.67\times 10^{-11}\times (27.3215\times 24\times 3600)^2}\\\Rightarrow M=5.68458\times 10^{24}\ kg[/tex]

The mass of Earth is [tex]5.68458\times 10^{24}\ kg[/tex]

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