Question

Three dense point size bodies of same mass are attached at three vertices of a light equilateral triangular frame. Identify the increasing order of their moments of inertia about following axes:
I) about an axis perpendicular to plane and passing through a corner
II) about an axis perpendicular to plane and passing through centre
III) about an axis passing through any side in the plane of the triangle
IV) about perpendicular bisector of any side

• A. III, IV, II, I
• B. III, II, IV, I
• C. II, IV, III, I
• D. II, III, IV, I

Answer 1

The correct option is B III, II, IV, I

Let the mass of each be $m$ and side of equilateral triangle be $a$.
Moment of inertia = $M{r}^2$
(I) When axis perpendicular to the plane of the triangle and passes through a corner distance of two masses is 'a' and one is '0'
Moment of Inertia about an axis perpendicular to plane and passing through a corner
= $m{a}^2+m{a}^2+m(0{)}^2=2m{a}^2$
(II) When axis perpendicular to plane and passing through center distance of all three masses is $\frac{a}{\sqrt{3}}$
Moment of Inertia about an axis perpendicular to plane and passing through centre
= $m{\frac{a}{\sqrt{3}}}^2+m{\left(\frac{a}{\sqrt{3}}\right)}^2+m{\left(\frac{a}{\sqrt{3}}\right)}^2=m{a}^2$
(III) When axis about an axis passing through any side distance of two masses is '0' and $\frac{\sqrt{3}}{2}a$
Moment of Inertia about an axis axis passing through any side
$=m{\left(\frac{\sqrt{3}}{2}a\right)}^2+m{0}^2+m{0}^2=\left(\frac{3}{4}\right)m{a}^2$
(IV) When axis about an about perpendicular bisector of any side distance of two masses is 'a/2' and $\frac{\sqrt{3}a}{2}$
Moment of Inertia about an about perpendicular bisector of any side
= $m{\left(\frac{\sqrt{3}a}{2}\right)}^2+m{\left(\frac{a}{2}\right)}^2+m{\left(\frac{a}{2}\right)}^2=\left(\frac{5}{4}\right)m{a}^2$
Therefore (III)<(II)<(IV)<(I)