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This is a homework question from a precalculus class that I'm a TA for.

The force $F$ between two bodies of mass $m_1$ kilograms and $m_2$ kilograms whose centers of mass are a distance $r$ meters apart is given by the formula \begin{equation*} F = \frac{G m_1 m_2}{r^2} \,, \end{equation*} where $G$ is the gravitational constant of the universe. What are the units of the gravitational constant $G$?

I wanted to write up a thorough solution to this exercise for my class, and figured I'd post it online to help anyone else who may wander across it.

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Since we want to figure out what the units of $G$ are, first we should rewrite the given formula with $G$ isolated: \begin{equation*} F = \frac{G m_1 m_2}{r^2} \quad\implies\quad G = \frac{F r^2}{m_1 m_2} \end{equation*} Since this is an equality, the units on the left-hand side (so the units of $G$) will be the same as the units on the right-hand side. We recall that since $F=ma$ the units of force are $\mathrm{kg}(\mathrm{m}/\mathrm{s}^2)$. So the units of the right-hand side must be \begin{equation*} \frac{\mathrm{kg}\frac{\mathrm{m}}{\mathrm{s}^2}\,\mathrm{m}^2} {\mathrm{kg}\,\mathrm{kg}} \,, \end{equation*} which we can write more cleanly as \begin{equation*} \frac{\mathrm{m}^3}{\mathrm{kg}\,\mathrm{s}^2} \,. \end{equation*} So these are the units of $G$.

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