Rotational Inertia

By: Russell and Malachi Brown and Zachary Beene

Rotational Inertia, I is:Measure of an object’s resistance to changes in rotation.

The moment of inertia, I, of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared.

I = mr2

The moment of inertia of any extended object is built up from that basic definition. An object can be thought of as a sum of particles, each having a mass of dm. Integration is used to sum the moment of inertia of each dm to get the inertia of body.

𝐼=∑ 𝑟2𝑑𝑚=∫𝑟 2𝑑𝑚System of Particles

Continuous Object

Here is an example of a rod’s moment of inertia being integrated.(about an axis in the center)

The center of the rod is Length of 0, so the two ends are –L/2 and L/2. The dm in the original equation is replaced with M/L*dr, because M/L is the proportional mass per length.

For simplicity, here are the equations for inertia of common objects (already integrated).

Parallel axis theoremThe moment of inertia about any axis parallel to the axis through the center of mass is given by:

Here’s an example using a rod.The moment of inertia of a rod about its center is given by the equation

I = (1/12)mL2

To find the moment of inertia of a rod about its end is given by:

I = Icm + md2

I = (1/12)mL2 + m(L/2)2

I = (1/12)mL2 + mL2/4I = (1/3)mL2