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The moment of inertia (MOI) of a rod that rotates around its center is $\frac{1}{12} m l^2$, while a rod that rotates around its end is $\frac{1}{3} ml^2$, as listed here.

That doesn't sound right - please see the image. A rod that rotates around its center can be viewed as two rods rotating around a common end point. Each rod's MOI is $\frac{1}{3} ml^2$, so two rods have MOI $\frac{2}{3} ml^2$

What is wrong?enter image description here

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Start with the moment of inertia (about one end) of a rod of length $L/2$ and mass $m/2$:

$$ I = \frac{1}{3}\frac{m}{2}\left(\frac{L}{2}\right)^2 = \frac{mL^2}{24} $$

Multiply by two, to get a rod of length $L$ and mass $m$ pivoted about the middle and you get:

$$ I = \frac{mL^2}{12} $$

You forgot to allow for the doubling/halving of the mass.

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The rod is characterised by a constant linear mass density $\lambda$, whence $I = \frac1{12}\lambda L^3$ for the one rotating around its centre. Take two rods of length $L/2$ to get $2\frac13\lambda(L/2)^3 = \frac1{12}\lambda L^3$.

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