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.