Question

A uniform triangular plate of mass $M$ whose vertices are $ABC$ has lengths $l,\frac{l}{\sqrt{2}}$ and $\frac{l}{\sqrt{2}}$ as shown in figure. Find the moment of inertia of this plate about an axis passing through point $B$ and perpendicular to the plane of the plate.

Answer 1

moment Inertia of triangle at base $I=\frac{1}{6}mass\times heigh{t}^2$

$I_{BC}=\frac{1}{6}M{\left(\frac{l}{\sqrt{2}}\right)}^2=\frac{M{L}^2}{12}$

$I_{AB}=\frac{1}{6}M{\left(\frac{l}{\sqrt{2}}\right)}^2=\frac{M{L}^2}{12}$

Apply perpendicular axis theorem

$I=I_{BC}+I_{AB}=\frac{M{L}^2}{12}+\frac{M{L}^2}{12}=\frac{M{L}^2}{6}kg{m}^2$

the moment of inertia of this plate about an axis passing through point BB and perpendicular to the plane of the plate is $\frac{M{L}^2}{6}kg{m}^2$