Prof. Walter Lewin Teach Moment of Inertia at MIT

Added: 2026-05-26

[00:00:00]If I have a rotating disk, I can ask myself the question now, which we have never done before, what kind of kinetic energy, how much kinetic energy is there in a rotating disk? We only dealt with linear motions with 1/2 mv squared, which we never considered rotating objects and the energy that they contain.

[00:00:18]So, let's work on that a little. I have here a disk and the center of the disk is C and this disk is rotating with angular velocity omega that could change in time.

[00:00:29]And the disk has a mass M and the disk has a radius R.

[00:00:35]And I want to know at this moment how much kinetic energy of rotation is stored in that disk.

[00:00:40]I take a little mass element here m of I and this radius equals r of I and the kinetic energy of that element I alone equals 1/2 m of I times v of I squared and v of I is this velocity. This angle is 90 degrees.

[00:01:00]This is v of I.

[00:01:02]Now, v equals omega r. That always holds for these rotating objects. And so, I prefer to write this as 1/2 m of I omega squared r of I squared. The nice thing about writing it this way is that omega, the angular velocity, is the same for all points of the disk, whereas the velocity is not. Because the velocity of a point very close to the center is very low. The velocity here is very high. And so, by going to omega we don't have that problem anymore.

[00:01:33]So, what is now the kinetic energy of rotation of the disk?

[00:01:38]The entire disk. So, we have to make a summation.

[00:01:41]And so, that is omega squared over two times the sum of m of I r I squared over all these elements m I which each have their individual radii r of I.

[00:01:55]And this now is what we call the moment of inertia I.

[00:02:01]Don't confuse that with impulse. It has nothing to do with impulse. And this is moment of inertia.

[00:02:11]So, the moment of inertia is the sum of m i r i squared.

[00:02:18]In So, this can also be written as 1/2 I.

[00:02:24]I put it C there. You will see shortly why because the moment of inertia depends on which axis of rotation I choose times omega squared.

[00:02:32]And when you see that equation, you say, "Hey, that looks quite similar to 1/2 mv squared." And so, I add to this list now, if you go from linear motions to rotational motions, you should change the mass in your linear motion to the moment of inertia in your rotational motion, and then you get back to your 1/2 mv squared. You can see that.