Question

Calculate the moment of inertia of a uniform rod of mass M and length l about an axis 1,2,3 and 4.

Answer 1

$\left(I_1\right)=\int \left(dm\right)r^2={\int }_0^l\left(\frac{M}{l}dx\right)x^2=\frac{M{l}^2}{3}$

$\left(I_2\right)=\int \left(dm\right)r^2={\int }_{-l/2}^{l/2}\left(\frac{M}{l}dx\right)x^2=\frac{M{l}^2}{12}$
$\left(I_3\right)$ = 0 (axis 3 passing through the axis of rod)
$\left(I_4\right)=d^2\int \left(dm\right)=M{d}^2$