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(D) The rotational kinetic energy of a body is given by $K = \frac{1}{2} I \omega^2$,where $I$ is the moment of inertia and $\omega$ is the angular sp

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Through one corner normal to the plate

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(D) The rotational kinetic energy of a body is given by $K = \frac{1}{2} I \omega^2$,where $I$ is the moment of inertia and $\omega$ is the angular speed.Since $\omega$ is constant for all cases,the kinetic energy $K$ is directly proportional to the moment of inertia $I$ $(K \propto I)$.Therefore,the kinetic energy will be largest for the axis about which the moment of inertia $I$ is maximum.For a square plate of side $a$ and mass $M$:$1$. Axis through the centre,normal to the plate: $I_1 = \frac{Ma^2}{6}$$2$. Axis along a diagonal: $I_2 = \frac{Ma^2}{12}$$3$. Axis along an edge: $I_3 = \frac{Ma^2}{3}$$4$. Axis through a corner,normal to the plate: Using the perpendicular axis theorem,$I_4 = I_{cm} + M(r^2) = \frac{Ma^2}{6} + M(\frac{a}{\sqrt{2}})^2 = \frac{Ma^2}{6} + \frac{Ma^2}{2} = \frac{4Ma^2}{6} = \frac{2Ma^2}{3}$.Comparing the values,$I_4$ is the largest. Thus,the kinetic energy is largest when the plate is rotated about an axis through one corner normal to the plate.

$A$ solid square plate is spun around different axes with the same angular speed. In which of the following choices of axis of rotation will the kinetic energy of the plate be the largest?

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