📂Classical Mechanics

Inertia Moment of a Thin Rod

Formulas

The moment of inertia for a rod with length $a$ and mass $m$ is:

• If the axis of rotation is at the end of the rod, it is $I=\dfrac{1}{3}ma^{2}$.

• If the axis of rotation is at the center of the rod, it is $I=\dfrac{1}{12}ma^{2}$.

Derivation

When the Axis of Rotation is at the End of the Rod

If $\rho$ is defined as the mass per unit length, the mass of the rod is $m=\rho x$. And since $dm=\rho dx$, it follows that:

$$I_{z} = \int_{0}^{a} x^{2}\rho dx = \frac{1}{3}a^{3}\rho$$

But given that the length of the rod is $a$, $\rho=\dfrac{m}{a}$ and we obtain the following result:

$$I_{z}=\frac{1}{3}ma^{2}$$

When the Axis of Rotation is in the Middle of the Rod

$$\begin{align*} I_{z} &= \int_{-\frac{a}{2}}^{\frac{a}{2}}x^{2}\rho dx = \frac{1}{3} \left( \frac{a^{3}}{8}+\frac{a^{3}}{8} \right)\rho \\ &= \frac{1}{12}a^{3}\rho \\ &= \frac{1}{12}ma^{2} \end{align*}$$

Comparison

Comparing the two results shows that the moment of inertia is smaller when the axis of rotation is in the middle of the rod. This means that with the same force, a rod with its axis of rotation in the middle will rotate more.