Gravity: the earth and moon

The Earth

According to Wikipedia, the earth’s mass is $(5.9722 \pm 0.0006) \times 10^{24}$ kg. Knowing this it is simple to calculate the force of gravity using Newton’s law of gravity:

$$ F = G \dfrac{M_1 \cdot M_2}{r^2}, $$ where $G$ is the gravitational constant $6.6743 \times 10^{-11}$, $M_1$ is the mass (kg) of object 1, $M_2$ is the mass (kg) of object 2 and $r$ is the distance (m) between the two objects.

Let’s say a person weighs 80 kg and is standing on the surface of the earth. What is the distance of the person to the earth? Newton’s formula assumes the objects are points, i.e. they have no dimensions. To get around this, we will simply model the earth as a point located at its center.

The person is approximately 6371 km from the earth’s center, so we can now calculate the gravitational force:

$$ F = 6.6743 \times 10^{-11} \dfrac{5.9722 \times 10^{24} \cdot 80}{6371000^2}, $$ which is approximately 785.6 Newtons.

The Moon

We can do the same thought-experiment for the moon. We’ll still assume the person is standing on the surface of the earth.

The moon’s mass is $7.3477 \times 10^{22}$ kg. The average distance between the earth and moon is 384.400 km. Now we have enough to calculate the moon’s gravitational force on the person.

$$ F = 6.6743 \times 10^{-11} \dfrac{7.3477 \times 10^{22} \cdot 80}{384400000^2}, $$ which is approximately 0.0027 Newtons.

This is more than 5 orders of magnitude smaller.

Conclusion

The gravitational force of the moon is so small compared to earth’s gravity, measuring the effects of the moon on earth is a challenge.