Question

I have a stick spinning around a perpendicular axis in the center. What's its rotational inertia?\[\int r^2\,dm\]?

Answer 1

That is the definition of the moment of inertia. The trick now is the evaluate that integral.

If the stick is of length a and uniform mass density with a total mass m, then
\[ dm = \frac{m}{a} dr \]
and the moment of inertia is
\[ I = \int_{-a/2}^{a/2} \frac{m}{a} r^2 dr \]
Now evaluate.

Answer 2

Can you run me through how we arrived at \[dm=\frac{m}{a}dr\]?