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Calculus Based Section: Rotational Inertia of Solid Bodies

TopicsCalculus Based Section: Rotational Inertia of Solid Bodies

In our study of rotational dynamics we skipped over exactly how to calculate the
rotational inertia of a solid body. The process for calculating this quantity is quite
complicated, and requires quite a bit of calculus. Thus we devote a section to calculating
this quantity.

Consider a small section of a rod, a radius r from the axis of rotation, and with a mass
δm, as shown below:

Because the volume of the section of the rod is sufficiently small, we can calculate the
moment of inertia of this single piece: I = δmr^{2}. To find the moment of inertia of
the entire rod, we sum over all pieces of a similar size that compose the rod:

Ir_{k}^{2}δm_{k}

To get an exact answer for the moment of inertia, we take the limit as the δm gets
smaller; as the rod is broken up into more and more pieces. Thus:

I

=

r_{k}^{2}δm_{k}

=

r^{2}dm

This integral equation is the basic equation for the moment of inertia of a solid body.

Even with this equation, it is quite difficult to calculate the moment of inertia of a solid
body. We will go through an example to show how it is done. Let us simply return to the
example of the solid rod of length L, and mass M, rotated about its center, as shown
below.

Let us denote the cross sectional area of the rod by A. Thus the volume of the small
element of mass, dV = Adx, where dx is the length of the small element of mass. Thus,
if we denote the density of the rod by ρ, then we can describe dm in terms of
dx:

dm = ρdV = ρAdx

However, we can also express ρ in terms of measured quantities:
ρ = M/V = M/AL. Thus we can plug all this into our integral equation:

I

=

r^{2}dm

=

x^{2}(ρAdx)

=

x^{2}(Adx)

=

x^{2}dx

Thus we now have an integral we can evaluate. We simply need to determine the limits.
If we denote the axis of rotation to be at x = 0, then we simply integrate from -L/2 to
L/2:

I

=

x^{2}dx

=

[]_{-L/2}^{L/2}

=

ML^{2}

This is the equation for the moment of inertia of a thin rod, and it agrees with measured
values.

In general, the moment of inertia of a solid body varies with MR^{2}, where R is the
measure of the radius, or length of a given object. To find the exact value of the moment
of inertia, however, the complicated calculus is required.