Use Eq.(9.20) to calculate the moment of inertia of a slender, uniform rod with mass \(M\) and length \(L\) about an axis at one end, perpendicular to the rod.

The formula for the moment of inertia is the "sum of the product of the mass" of each particle by "the square of its distance from the axis of rotation". The moment of inertia formula is expressed as

\(I = {\int r ^2}dm\) ….. (1)

Here,\(r\)and\(dm\)is the distance and mass respectively.

Use the equation to determine the moment of inertia as shown below.

\(I = \int {{r^2}} dm\)

The linear mass density is

\({\lambda _m} = \frac{M}{L}\)

Here, \(M\) is the mass and \(L\) is the length.

Take an infinitesimal piece of the rod that has width \(dx\) over a distance \(x\) from the axis of rotation.

The moment of inertia of an infinitesimal piece is as follows:

\(dI = {x^2}dm\)

Put \({\lambda _m}dx\) for \(dm\) in the above equation as follows:

\(dI = {x^2}{\lambda _m}dx\)

Compute \(I\) for the entire rod as follows:

\(\begin{array}{c}I = \int_0^L {{x^2}{\lambda _m}dx} \\ = {\lambda _m}\int_0^L {{x^2}dx} \\ = {\lambda _m}\left( {\frac{{{x^3}}}{3}} \right)_0^L\\ = {\lambda _m}\frac{{{L^3}}}{3}\end{array}\)

Put \(\frac{M}{L}\) for \({\lambda _m}\) in the obtained equation as follows:

\(\begin{array}{c}I = \frac{M}{L}\frac{{{L^3}}}{3}\\ = \frac{1}{3}M{L^2}\end{array}\)

Hence, the moment of inertia of a slender uniform rod is \(I = \frac{1}{3}M{L^2}\).

Draw slender uniform rod with length \(L\) about an axis at one end perpendicular to the rod as follows: