Question

The figure shows a square plate of uniform mass distribution. $A{A}^{\prime }$ and $B{B}^{\prime }$ are the two axes lying in the plane of the plate and passing through its centre of mass. It $I_0$ is the moment of inertia of the plate about $A{A}^{\prime }$ then its moment of inertia. about the $B{B}^{\prime }$ axis is

• A. $I_{\theta }$
• B. $I_{\theta }\cos \theta$
• C. $I_{\theta }{\cos }^2\theta$
• D. None of these

Answer 1

The correct option is A $I_{\theta }$

Line CC' is perpendicular to AA' and line DD' is perpendicular to BB'.

Also $I_{A{A}^{\prime }}=I_{C{C}^{\prime }}$ and $I_{B{B}^{\prime }}=I_{D{D}^{\prime }}$ (due to symmetry)

Using perpendicular axis theorem about P:

$I_P=I_{A{A}^{\prime }}+I_{C{C}^{\prime }}$ and $I_P=I_{B{B}^{\prime }}+I_{D{D}^{\prime }}$

$⟹I_P=2I_{A{A}^{\prime }}$ and $I_P=2I_{B{B}^{\prime }}$

Thus $I_P=2I_{A{A}^{\prime }}=2I_{B{B}^{\prime }}$

$⟹I_{B{B}^{\prime }}=I_{A{A}^{\prime }}=I_o$