Question

Let 'I' be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle $\theta$ with AB. The moment of inertia of the plate about the axis CD is then equal to

• A. $I$
• B. $I{\sin }^2\theta$
• C. $I{\cos }^2\theta$
• D. $I{\cos }^2\left(\frac{\theta }{2}\right)$

Answer 1

The correct option is A $I$

$I_{AB}=I_{A^{\prime }{B}^{\prime }}=Iand{I}_{CD}=I_{C^{\prime }{D}^{\prime }}$
If $I_0$ be the moment of inertia of the square plate about an axis passing through O and perpendicular to the plate, then by the perpendicular axis theorem
$I_0=I_{AB}+I_{A^{\prime }{B}^{\prime }}=2I_{AB}....\left(1\right)$
or $I_0=I_{CD}=I_{C^{\prime }{D}^{\prime }}=2I_{CD}....\left(2\right)$
From ((1) and (2))
$I_{CD}=I_{AB}=I.$