Question

Consider a uniform triangular plate of mass $M$ whose vertices are $ABC$ has lengths $\ell ,\frac{\ell }{\sqrt{2}}$ and $\frac{\ell }{\sqrt{2}}$ shown in the figure. The Moment of Inertia of this plate about an axis perpendicular to the plane of the plate and passing through point $A$ is:

• A. $\frac{m \ell ^{2}}{6}$
• B. $\frac{m \ell ^{2}}{3}$
• C. $\frac{2 m l^{2}}{3}$
• D. $\frac{m \ell ^{2}}{12}$

Answer 1

Answer: (B)

Moment of inertia of a square plate of mass $M$ and side $a$ about an axis perpendicular to the plane and passing through the centre is, $I_{O \bigotimes}=\frac{M a^{2}}{6}$
Now, from parallel axis theorem, Moment of inertia of the square plate about an axis perpendicular to the plane and passing through the corner is,
$I_{A \bigotimes}=I_{O \bigotimes}+M\left(\frac{a}{\sqrt{2}}\right)^{2}\Rightarrow I_{A \bigotimes}=\frac{M a^{2}}{6}+\frac{M a^{2}}{2}=\frac{2 M a^{2}}{3}$
We can see that the triangular plate is half of square plate, $m=\frac{M}{2}$ and $a=\frac{l}{\sqrt{2}}$ .
For triangular plate, $I_{A \bigotimes}=\frac{2 \left(\frac{M}{2}\right) \left(\frac{\ell }{\sqrt{2}}\right)^{2}}{3}=\frac{m \ell ^{2}}{3}$