You must always choose a set of coordinates when evaluating an integral. It usually makes sense to choose coordinates that match the symmetry of your object or system. For a cube, you might choose regular Cartesian coordinates \(dV = dx dy dz\text{,}\) while for a cylinder you might choose polar coordinates \(dV = s ds d\phi dz\text{.}\) You might even decide that your object or system can be treated as two-dimensional (or even one-dimensional), in which case you do not need all three dimensions. In any case, you will need to integrate over the entire object with respect to every relevant coordinate.
Section 5.15 Chop-Multiply-Add: Calculating Moment of Inertia
For a continuous object, like a solid sphere or cylinder, the moment of inertia can be calculated using
\begin{equation*}
I = \int r^2 dm
\end{equation*}
As with previous integral expressions in physics such as those you saw in Section 2.13, Section 4.5, and Section 5.13, you can use the Chop-Multiply-Add strategy to construct and evaluate any necessary integrals.
Chop: You can start by chopping up the object into small pieces. Each piece of the object is represented by the expression \(dm = \rho dV\text{,}\) where the \(dV\) represents the infinitesimal volume of the piece and \(\rho\) is the volume mass density of the object. In general, \(\rho\) can be either uniform (constant over the entire object) or non-uniform (in which case it will change from place to place).
Note 5.15.1. Choosing Coordinates for Integrals.
Multiply: Once you know the mass of each small piece \(dm\text{,}\) you multiply to find the moment of inertia of that small piece is: \(dI = r^2dm = r^2 \rho dV\text{.}\) Here \(r\) is the straight-line (perpendicular) distance from the axis of rotation to the location of \(dm\text{.}\) It can be highly useful to draw a diagram of the shape, including an example \(dm\text{,}\) and label \(r\) so that it can be easily written in terms of your chosen coordinates.
Add: Last, once you have the infinitesimal \(dI\text{,}\) you add together every \(dI\) over the entire object:
\begin{equation*}
\int dI = \int r^2 dm
\end{equation*}
Note 5.15.2. Density.
Physicists commonly use different letters for density depending on whether it is a one-dimensional (linear), two-dimensional (surface), or three-dimensional (volume) density. The letters typically chosen are, respectively, \(\lambda\) (for linear), \(\sigma\) (for surface), and \(\rho\text{.}\) One reason for this difference is that the different densities have different units, so this makes it easier to check your units at the end!
Exercises Activities
1. Calculate - Baton.
In the previous section, you looked up the moment of inertia for a baton spinning about one of its ends. Use the Chop-Multiply-Add strategy to calculate this moment of inertia for yourself, given the mass \(M\) and length \(L\) of the baton. For simplicity, assume the baton is one dimensional and the mass is uniformly distributed \(\lambda = M/L\text{.}\)
