Question

The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth? Why?

Answer 1

Hint
We use Newton’s law of gravitation to solve this problem. We find the gravitational force with which the earth attracts the moon, then find the gravitational force with which the moon attracts the earth. Then we compare both the gravitational forces to see the relation between the two.
$\Rightarrow F = \dfrac{{GMm}}{{{r^2}}}$, This is the formula of universal law of gravitation.
Where $F, G, M, m, r$ represent gravitational force, universal gravitation constant, and mass of the body (1), the mass of the body (2), and distance between the two objects respectively.

Complete step by step answer
Newton’s law of gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
From the definition, we know that earth attracts the moon with a force of ${F_{earth - moon}} = \dfrac{{GMm}}{{{r^2}_{earth - moon}}}$, here $M$ represents the mass of the earth, and $m$ represents the mass of the moon.
Now calculating the force with which the moon attracts the earth, we get the gravitational force as, ${F_{moon - earth}} = \dfrac{{GMm}}{{{r^2}_{moon - earth}}}$
We take the distance between moons and earth the same in both cases because by definition it is the distance between the centers of both the objects. In this case, it is the distance from the center of the earth to the center of the moon which will be the same either way round. ( ${r_{earth - moon}} = {r_{moon - earth}}$) . The mass of the earth and the mass of the moon will remain the same in both the equations.
$\Rightarrow {F_{earth - moon}} = \dfrac{{GMm}}{{{r^2}_{earth - moon}}}$
$\Rightarrow {F_{moon - earth}} = \dfrac{{GMm}}{{{r^2}_{moon - earth}}}$
$\therefore {F_{moon - earth}} = {F_{earth - moon}}$
From the two equations of the force of gravitation, we can conclude that the moon and earth attract each other with an equal force in the opposite direction.

Note
We can find the magnitude of the force of attraction between the moon and earth by simply substituting the values in the formula. We can also see this in terms of Newton’s third law of motion, every action has an equal and opposite reaction. The force of attraction with which the earth pulls the moon is equal and in the opposite direction to the force of attraction with which the moon pulls the earth.