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. $\frac{2m{l}^2}{3}$
• 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, $I_{O⨂}=\frac{M{a}^2}{6}$
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,
$I_{A⨂}=I_{O⨂}+M{\left(\frac{a}{\sqrt{2}}\right)}^2\Rightarrow I_{A⨂}=\frac{M{a}^2}{6}+\frac{M{a}^2}{2}=\frac{2M{a}^2}{3}$
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, $I_{A⨂}=\frac{2\left(\frac{M}{2}\right){\left(\frac{\ell }{\sqrt{2}}\right)}^2}{3}=\frac{m{\ell }^2}{3}$