Experimentally determine the Moment of Inertia of a complex (mathematically indescribable) object I'm a high schooler that's trying to design a robot for a physics project. The robot is a catapult; it will calculate a trajectory and shoot at a certain speed and angle using a large motor. I'm using an arm that is close to a rod, but it uses a "projectile holder", which makes it slightly different. In order to get the torque, I'd suspect I need the moment of inertia. I have the desired speed and angle I need to shoot at, but how could I put that in terms of torque required?
 A: An important thing to understand about physics (and any science, really) is that the only thing we ever do is construct mathematical models of physical systems. We never even try take every detail into account - instead, we start with a simple model and then gradually add complexity until it becomes as accurate as we want it to be, and then we stop.

At the coarsest level of approximation, you could model your catapult simply as a rod rotating about a pivot point. Knowing the mass and length of the rod and the location of the pivot are sufficient to compute its moment of inertia; if the pivot point is a distance $x$ from one end of the rod, the moment of inertia turns out to be
$$I_{rod}=\frac{1}{3}ML^2 \left(1-3\frac{x}{L} + 3\frac{x^2}{L^2}\right) $$
which can be obtained either directly via integration or, if you are not familiar with how to do calculations like that, from the parallel axis theorem.
Of course, this can't be right - you have a projectile holder on one end. So to improve your model, you could model this projectile holder as a point mass $m$ attached to some point a distance $R$ from the pivot. Doing so would make the total moment of inertia
$$I_{total} = I_{rod} + m R^2$$
To be honest, I would be quite surprised if this level of accuracy was insufficient.  To improve the accuracy, you could take the shape of the projectile holder into account by modeling it as some extended object, but I suspect the inaccuracy which arises from modeling it as a point mass would be dominated by uncertainties in the length, mass, and uniformity of the rod and the projectile holder itself.

The calculation above is meant to give you a ballpark figure for what you can expect the moment of inertia to be. From there, your task would be to actually measure it by applying some torque (e.g. via a motor, or a mass on the other end of the catapult), measuring the angular acceleration of the catapult, and then computing the moment of inertia from there.
At the end of the day, the way to experimentally determine what torque you need to get the result you want is to keep increasing the torque until you get your desired result, and then write that number down. Theoretical calculations can get you in the right ballpark, but they only provide a guide; empirical measurement and testing is king when implementing any real design.
