How to do NON-Guass's Law Problem - A Stick WITH Ends

Gauss's law provides a relatively easy way to find E-fields for charged spheres, long sticks or cylinders, and large plates. The last two, of course, are approximations in the end, but ends, edges, and corners really make for a difficult math problem. When stuck with such a problem, we have no choice but to stick (heh, heh) with the fundamentals. That is, point charges. We know how to handle point charges, and physically that is all a charged object is - a bunch of extra charged particles. So we need to set up an integral and add up a bunch of small fields or potentials! Check it out.

How to Calculate Moments of Inertia with Integral - Sticks

When it comes to finding moments of inertia, the one thing we can find exactly is the inertia of a point mass, I = mr^2. Here, m is the mass, and r is the distance from the mass to the axis of rotation. But we run into some amount of difficulty when we have real objects that rotate and are made of countless point masses, i.e. atoms. How can we get the total moment of inertia?

We have an integral definition for I. What is is really telling us to do is break up the object into a bunch of small pieces of mass, dm. This is like saying break it up into a bunch of point masses, find each individual inertia, and then add (integrate) them all up to get the total. I will show how to do this with a stick in this video, which is the main case we would ever need to use the integral for class. Keep in mind that the other inertias we use for disks, balls, and so on, are found with this integral, too. I hope this helps. Keep in mind that I also have another video that shows how to use the parallel-axis theorem, which is a way around having to do the integration if you happen to know the moment of inertia of an object with the axis through the center of mass of the object.