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Two Ways to Find the Moment of Inertia for a Sphere
What is the moment of inertia and why would you wan to find it for a sphere? Will this even be fun? The answer to the last question — YES. Trust me, I’m going to make it fun.
What is the moment of inertia?
This is just a quick review. I mean, if you have read this far then I suspect you already have an idea about the moment of inertia. First, let me be clear — there are two different moments of inertia (and they are both represented by I). If you are rotating a rigid object about a FIXED axis then I is a scalar value (that’s the one we will calculate here). However, if the object is free to rotate in any direction then I is a tensor. If you are looking for stuff about the tensor version, then this other post is for you.
Now back to the scalar version of I. If you have a bunch of masses connected together and rotating about a fixed axis with an angular speed ω, then it’s possible to write the rotational kinetic energy as:
Where for a finite number of point mass, I would be:
Yes, I can also be used to define the angular momentum — but I like the kinetic energy definition because it’s easier to show that the kinetic energy of a bunch of masses is the same as the rotational kinetic energy above. In short, I like to think of I as the “rotational” mass. It’s the property of a rigid object that makes it more difficult to change the rotational motion just like normal mass makes it more difficult to change an object’s linear motion.
Oh, but what if you don’t have discrete point masses? What if you have a continuous mass distribution like with a solid sphere? In that case we have an infinite sum of infinitely small point masses. Yes, the sum becomes an integral.
Here I put a triple integral just to emphasize that we are integrating over the whole volume (in 3D) of the object. OK, let’s get to some calculations.
