Answer:
The mass of the Earth is .
Explanation:
The Universal law of gravitation shows the interaction of gravity between two bodies:
(1)
Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.
For this particular case, M is the mass of the Earth and m is the mass of the moon. Since it is a circular motion, the centripetal acceleration will be:
(2)
Then Newton's second law () will be replaced in equation (1):
By replacing (2) in equation (1) it is gotten:
(3)
Since it is a circular motion, the orbital velocity can be represented by:
(4)
Where r is the orbital radius and T is the period.
Then, equation 4 can be replaced in equation 3.
Notice that r can be represented by the semi-major axis (a)
(5)
Therefore, the mass of the Earth can be determined if M is isolated from equation (5):
(6)
Notice that it is necessary to express a in units of meters and T in units of seconds.
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Hence, the mass of the Earth is .