Gravitational force of Earth and Moon

Question

The Earth has a mass of 5.97*10²⁴ kg and the Mood has a mass of 7.36*10²² kg. The average distance between the center of the Earth and the center of the moon is 3.84*10⁸ m.

a) What is the gravitational force acting on the Moon due to the Earth?

b)What is the gravitational force acting on the Earth due to the Moon?

c) How far away would the Moon need to be for the magnitude of the gravitational force acting on it due to the Earth be the same as the magnitude of the gravitational force of a 72kg student sitting at their desk on the surface of he earth?

Answer 1

Newton's Law of Universal Gravitation states that the magnitude of the gravitational force between ANY two objects that have mass is given by the relationship...

|F_G| = Gm₁m₂/r² where G = 6.67x10^-11 and r is the distance between the centers of mass of the two objects.

a) We just have to substitute the masses and distance given into the above equation and solve. Don't forget to include the direction, which is "toward the Earth".

b) According to Newton's Third Law of Motion, the force of the Moon on the Earth and the force of the Earth on the Moon are equal in magnitude and opposite in direction. Therefore, the answer to b is the same number as the answer to a, only in the opposite direction (or "toward the Moon").

c) Here, we're using the equation above for two different situations, the force of the Earth on the Moon (at some unknown distance) and the force of the Earth on the student near the Earth's surface. So, we can set the two equal to each other.

Gm_Earthm_Moon/r₁² = Gm_Earthm_student/r₂² where r₁ is the distance we're looking for (distance between Earth and Moon after moving it so that the forces are equal) and r₂ is the distance between the Earth and the student, which is the radius of the Earth.

G and m_Earth are on both sides and cancel. So,

m_Moon/r₁² = m_student/r_Earth²

Now, it's just algebra. Rearranging...

r₁² = (r_Earth)²(m_Moon)/(m_student)

and substituting the numbers will give us the answer.