Long Thin Rod Study Guide

The "Long Thin Rod" scenario appears a bunch of times both in mechanics and E & M. The reason we use that particular term is that requires that we think in one dimension.

The "long" part comes in handy in E & M because it means we can ignore some pesky electromagnetic issues that would happen should we consider a SHORT thin rod...

Let's take a look at a long thin rod with mass density λ (lambda). If the rod is "uniform" that means that the density of the material is constant and we can consider that on the macro (large) scale:

λ = M/L

meaning that mass is constant for each unit of length.

More importantly, the uniformity exists down to the infinitesimal where the differentials live, so we can (and must) consider that as well:

λ = dm/dℓ

Which allows us to write the differential equation:

λdℓ = dm

Which substitutes nicely when we are finding moments of inertia (for example):

I =∫r2dm

and "" is the generic term for length and we can substitute that for "r" which is the distance to the axis of rotation.

Consider the following worked example from the book to find the moment of inertia about the center of a long thin rod of length :

Notice that we can use that very same method to deal with a two dimensional object (we usually use the lower case sigma(σ), to represent the density of a two dimensional object and we use the Greek letter rho (ρ) to represent the density of a 3 dimensional object as shown in the calculation of the m.o.i. of a solid cylinder below: