question_answer

Consider a uniform square plate of side \[a\] and mass \[M\]. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its comers is

A) \[\frac{5}{6}M{{a}^{2}}\]     

B) \[\frac{1}{12}M{{a}^{2}}\]

C) \[\frac{7}{12}M{{a}^{2}}\]                

D) \[\frac{2}{3}M{{a}^{2}}\]

Correct Answer: D

Solution :

[d] For a rectangular sheet moment of inertia passing through O, perpendicular to the plate is \[{{I}_{O}}=M\left( \frac{{{a}^{2}}+{{b}^{2}}}{12} \right)\] For square plate it is\[\frac{M{{a}^{2}}}{6}\] (as a = b) \[r=\sqrt{\frac{{{a}^{2}}}{4}+\frac{{{a}^{2}}}{4}}=\frac{a}{\sqrt{2}}\] \[\therefore {{r}^{2}}=\frac{{{a}^{2}}}{2}\] \[\therefore \]Moment of inertia about B parallel to the axis through O is \[{{I}_{B}}={{I}_{O}}=M{{r}^{2}}=\frac{M{{a}^{2}}}{6}+\frac{M{{a}^{2}}}{2}=\frac{2M{{a}^{2}}}{6}\] \[I=\frac{2}{3}M{{a}^{2}}\]