Rotational Inertia Of A Hoop

Moment of Inertia - 1

Definition of Moment of Inertia

Y'all might want to take a look at this table of moments of inertia for various objects.

Moment of Inertia of a Rod - Axes through Middle

We volition summate the moment of inertia of a sparse rod of length 50 about an centrality which passes through the center of the rod and is perpendicular to the rod. The rod has mass M.

Await at the contribution of a small-scale piece of the rod (length dx) located a altitude x away from the centrality. First discover the mass of this piece:

Now find the moment of inertia:

Notice that the limits of intergration are from 0 to L/2 - that's just half the rod so nosotros multiply by ii.

Finishing up we find:

Moment of Inertia of a Rod - Axes through One End

Now let's expect at this arrangement:

The integral at present is:

The moment of inertia is larger in this example - makes sense considering in that location'south more mass farther away from the centrality than in the first case.

Moment of Inertia of a Hoop - Axes through Middle

Let's find the moment of inertia of a hoop about an axis through the center of the hoop and perpendicular to the plane of the hoop. What is a hoop? Think of a hula-hoop:

All the mass is located the same altitude from the origin so

Since r = R = constant I can pull it out of the integral.

Moment of Inertia of a Disk - Axes through Center

Let's now switch to a disk. Here's an example of a disk:

To detect the moment of inertia of a disk, interruption it up into many hoops - we'll phone call these annuli of thickness dr as shown below.

So the moment of inertia of a disk is smaller than that of a hoop of the same mass and radius - makes sense considering for the hoop all the mass is every bit far from the axis as it can be.