How to find the Moment of Inertia for a Solid Disk

We have explicitly found the moments of inertia for point masses and also for sticks, and we have even defined the parallel-axis theorem. But what about more complicated objects, such as solid disks or cylinders that spin or roll? We have been given the inertia expression of (1/2)MR^2, and have used it quite a bit in problems, but where in the world does it come from? How do we use the integral definition of inertia to solve this? That is what this video is about, as a number of people have asked about where the inertia expressions for rolling objects come from. I hope this helps and makes sense.

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.

How to Use the Parallel Axis Theorem to Find Moments of Inertia

The parallel axis theorem is a neat shortcut that allows us to find moments of inertia for objects when the axis of rotation is somewhere other than the center of mass of the object. If you know the inertia for objects when going through the center of mass, you can quickly find the new value of I for any axis that is parallel to the center of mass axis and displaced by some distance from the center of mass, d. The theorem says I_new = I_cm + Md^2. We do not have to use the integral to apply the theorem, which is why it is such a nice shortcut.

This video shows a couple quick examples of how to apply the theorem. Hope it helps!

Rotational Motion - New Concepts

For the 3 Chem-Phys sections, we are into rotational motion. This is typically a challenging topic because it is brand new. Keep in mind what makes it new revolves (ha, ha) around 3 new concepts:
- Torque
- Moment of inertia
- Angular momentum

Torques are produced by forces, and specifically those forces that cause a change in rotational motion. In other words, torques produce angular accelerations (analogous to forces causing linear accelerations). Mathematically, individual forces cause a torque = F(r)[sin(theta)]. Torque is a cross product vector, t = r x F.

Moment of inertia is analogous to mass in linear motion. It is a 'resistance to a change in rotational motion.' The higher the inertia, the smaller the angular acceleration from the same torque. Together, torque, t, and moment of inertia, I, are related through the 2nd law for rotations:
t = I(alpha)

The moment of inertia has units of kg m^2, and numerically tells us about the distribution of mass about the axis of rotation of the object or system.

Angular momentum, L, is also a cross product vector, L = r x p. The direction is found with the curly RHR, as we do in class. Remember the conservation of linear momentum? Momentum is conserved for a system if no external forces act on the system. Here is the analogy: angular momentum is conserved for a system if there is no external torques acting on the system. Individual objects can have angular impulse in collisions, but for the system it is conserved. We will get into this in a big way, and I'll soon have some how to videos up for rotations.

Let's have some fun with it!