Gravity

If you ever jump, you'll notice that you always come back down. This seems really trivial and unimportant since you've been used to this your whole life. However, the reason you always fall back down after jumping up is because of the Earth's gravitational force(gravity). This is the same reason why astronomical bodies, like Jupiter or Pluto, orbit the Sun. The Sun's extremely high mass causes those bodies to move towards it constantly.

The gravitational force is given by the following equation:

The universal gravitational constant, given above by G, is a constant of proportionality for this force. Notice that the force is proportional to the product of both the masses(M1 and M2) involved. This means that gravity isn't just experienced by one body. The gravitational force occurs between two masses. R is the radius between the masses, which means that the farther the two masses are, the weaker the gravitational force is.

The range of the gravitational force is infinite, technically. This means a body like Proxima Centauri is technically pulling on you right now. However, Proxima Centauri is 4.2 light years, or 3.974 * 10¹⁶ m away, so the gravitational force on you from Proxima Centauri is so small that you might as well just neglect it.

If you get rid of M1 and assume M2 is the Earth's mass, then you get a value of 9.81 m/s², which you'll notice is in units of acceleration. This should make sense because we took the gravitational force and divided a mass out of it, giving acceleration. This acceleration of 9.81 is the acceleration we all feel due to the Earth. In fact, when you get rid of the mass of the body experiencing the force, the remaining quantity is always the gravitational acceleration one object feels due to another object.

The gravitational force between two masses is always attractive, meaning objects experiencing it will always experience a force in the direction of the other body. This is a very subtle detail but notice that the gravitational force always acts along a line connecting both bodies.

Since gravity acts between two bodies, this means the gravitational force one body experiences due to another is equal in magnitude and opposite in direction to the gravitational force the other body experiences due to the first body. This is just Newton's Third Law being applied to gravity.

If you're confused, think about this "trippy" scenario. You feel your weight as the Earth pulls on you. However, you are also pulling on the Earth because the gravitational force between you and the Earth is mutual. The difference is that the Earth has a lot more mass than you so its inertia is high, meaning it will resist acceleration. This means the Earth doesn't really move that much from the force you exert on it.

The equation above isn't Newton's Law of Gravitation specifically. It's a simplification based on it, however. You take the mass of the object experiencing the gravitational force and multiply it by g, which is the acceleration due to gravity. g is equivalent to the part of Newton's Law of Gravitation excluding M1. g is just gravitational acceleration, which on Earth's surface is 9.8 m/(s^2). In many real-world problems, while objects do fall, which makes their accelerations due to gravity technically change, the change in gravitational acceleration is minimal.

The nature of the gravitational force is such that it mainly applies to large astronomical bodies. Notice that the radius term involved in the denominator of Newton's Law of Gravitation. But, what radius are we talking about? If you took the gravitational force between you and the Earth and said the radius between you and the Earth is the distance between you and the ground, then the gravitational force would be immensely huge.

So, what radius do we use? We use the radius between the center of masses of the two masses involved in the gravitational attraction. The equation for center of mass of an object is:

This is the position in space where the mass of a body is assumed to be concentrated at.

Remember how gravity acts between two bodies at the same time through Newton's Third Law? Well, this is no different for the Earth-Moon System. The Earth exerts a gravitational force on the Moon, causing it to orbit. However, the Moon also exerts a gravitational force on the Earth, but it's much less noticeable in most ways since the Earth is much more massive. However, an observable effect of the Moon's gravity on Earth is our tides.

However, the Moon isn't one of the only cause of Earth's tides. Another cause of our tides is the Sun, but it has around half the effect on our tides that the Moon does. Below are the possible tidal arrangements between the Moon and the Sun: