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

The moment of inertia of the triangle about the base.

$I=\frac{M{h}^2}{6}$

Hence the moment of inertia of the given triangle about base BC.

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

Similarly the moment of inertia of the triangle about AB.

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

Applying the perpendicular axes theorem.

$I_B=I_{BC}+I_{AB}=\frac{M{l}^2}{12}+\frac{M{l}^2}{12}=\frac{M{l}^2}{6}$