\section{Continuous mass}

Now let's extend this idea to a \textit{continuous distribution} of masses rather than discrete point masses. This isn't so different; a continuous distribution of mass is really just a lot of point-masses, only that there are so many of them so close together that you can't even count them\footnote{For example, your pencil might seem like a continuous distribution of mass, but it's really just a whole lot of atoms.}. In general, finding the CoM requires integral calculus, but not always...\footnote{Many of the following problems can be solved with integration even though you're meant to solve them without it. But remember, in math, whenever you accomplish the same task two different ways, that really means that they're somehow the same thing.}


\problem{}
You are given a cardboard cutout of Figure \ref{seahorse} and some office supplies. How might you determine the CoM? Does your strategy also work in 3D?

\vfill

\problem{}
Where is the CoM of a right isosceles triangle? What about any isosceles triangle?

\vfill

\problem{}
How can you easily find the CoM of any triangle? Why does this work?

\vfill

\pagebreak


\problem{}
Consider Figure \ref{soda} depicting a simplified soda can.
If you leave just the right amount,
you can get it to balance on the beveled edge, as seen in Figure
\ref{soda filled}.

\begin{figure}[htp]
\centering
\begin{minipage}{0.5\textwidth}
    \centering
    \includegraphics[width=0.6\linewidth]{img/soda.png}
    \caption{}
    \label{soda}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
    \centering
    \includegraphics[width=0.6 \linewidth]{img/soda_filled.png}
    \caption{}
    \label{soda filled}
\end{minipage}
\end{figure}

\problem{}
See Figure \ref{soda filled}. Let's take the can to be massless and initially empty. Let's also assume that we live in two dimensions. We start slowly filling it up with soda to a vertical height $h$. What is $h$ just before the can tips over?

\vfill

\problem{}<3D soda>
Think about how you might approach this problem in 3D. Does $h$ become larger or smaller?

\vfill

\pagebreak

So far we've made the assumption our shapes have mass that is \textit{uniformly distributed}. But that doesn't have to be the case.

\problem{}
A mathemagical wizard will give you his staff if you can balance it horizontally on your finger. The strange magical staff has unit length and it's mass is distributed in a very special way. It's density decreases linearly from $\lambda_0$ at one end to $0$ at the other. Where is the staff's balancing point?

\vfill
\pagebreak
