Universal Gravitation

Universal Gravitation is likely the most basic principle you will learn in Gravitation, which, in hindsight, is rather depressing, to say the least. It is based on the following equation, which has cemented itself as one of the most iconic ever in recent years.

\[\vec F_g = - G \frac{Mm}{r^2} \hat r\]

Let's first get into the definitions of each of the variables in this equations.

The Gravitational Constant, \(G\)

The Gravitational Constant is pretty much just a constant that scientists measured empirically, that has really no relevance to the solution. The only reason it exists is for calculation, and the value we use isn't necessarily accurate.

Empirically determined, \(G\) is defined as follows:

\[G = 6.6743015 \times 10^{-14} \frac{m^3}{kg \cdot s^2}\]

Yes, the units are indeed as cancerous as they seem. We usally use a rounded down value of \(G = 6.67 \times 10^{-14}\) in normal mathematical calculations, since beyond this term, the values approach redundancy due to Significant Figure and Negligibility considerations.

The Masses of the Objects, \(M\) and \(m\)

Gravitational Force is dependent of mass, or in this case, the two masses interacting.

Picture two objects that are next to each other.

Here, you have two masses \(m_1\) and \(m_2\), which are spherical in nature, which are currently located in arbitrary space.

We note that \(m_1 > m_2\), hence we usually do the following assignments:

\[ \begin{align*} M &= max(m_1, m_2) = m_1 \\ m &= min(m_1, m_2) = m_2 \end{align*} \]

This is simply nomenclature but representing the larger mass as \(M\) and the smaller mass as \(m\) makes it clearer as to what exactly your variables refer to.

Often, we use \(M_S\) or \(M_\odot\) to represent the Mass of the Sun, or the Solar Mass. We also use \(M_E\) or \(M_\bigoplus\) to represent the Mass of the Earth.

The following is a list of the variables we use for masses, which are usually represented with \(M\) since they are traditionally huge masses.

Property	Value
Mass of the Sun, \(M_S\) or \(M_\odot\)	\(1.989 \times 10^{30}\)
Mass of Earth, \(M_E\) or \(M_\bigoplus\)	\(5.972 \times 10^{24} \approx 5.97 \times 10^{24}\)
Mass of Moon, \(M_{moon}\)	\(7.348 \times 10^{22}\)

Traditionally, you will use these values in questions designed by normal Astronomy and Physics teachers, although some may intend to be deliberately provocative and ask for the values of rather strange functions. Luckily in the case, you will get a constants sheet to make sure you are on the right page. We shall not show the