For a thin square plate A B C D as shown, consider three...

For a thin square plate $A B C D$ as shown, consider three axes (marked 1,2 , and 3 ) in plane of the plate. Moment of inertia of the plate are $I_{1}, I_{2}$ and $I_{3}$ about axis 1,2 , and 3 respectively. (a) $I_{1}=I_{2}=I_{3}$ (b) $I_{1}>I_{2}>I_{3}$ (c) $I_{3}>I_{2}>I_{1}$ (d) $l_{1}=I_{3}>I_{2}$

For a thin square plate $A B C D$ as shown, consider three axes (marked 1,2 , and 3 ) in plane of the plate. Moment of inertia of the plate are $I_{1}, I_{2}$ and $I_{3}$ about axis 1,2 , and 3 respectively.
(a) $I_{1}=I_{2}=I_{3}$
(b) $I_{1}>I_{2}>I_{3}$
(c) $I_{3}>I_{2}>I_{1}$
(d) $l_{1}=I_{3}>I_{2}$

Key Concepts

Perpendicular Axis Theorem

This theorem states that for a flat, planar object, the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane. Although it directly applies to out?of-plane axes, understanding this theorem provides insight into how planar mass distributions contribute differently to rotation about various axes.

Moment of Inertia

This is a measure of an object’s resistance to rotational acceleration about an axis, determined by the distribution of its mass relative to that axis. In the context of rigid body dynamics, knowing the moment of inertia helps predict the response of an object when a torque is applied, making it a fundamental concept in rotational motion analysis.

Axis of Rotation

An axis of rotation is an imaginary line about which a body rotates. In problems involving different moments of inertia, the choice of axis is crucial because the distribution of mass relative to each axis will affect the resulting moment of inertia, determining how easily the object can rotate about that axis.

Thin Plate Approximation

A thin plate is an idealized two?dimensional object where one dimension (thickness) is negligible compared to the other dimensions. This approximation simplifies calculations by allowing the use of planar mass distributions and enables the application of the perpendicular axis theorem in analyzing moments of inertia.

Symmetry

Symmetry in a body, such as a square plate, plays an essential role in determining moments of inertia. When a body has high symmetry, its mass distribution about certain axes can be identical or simply related, which simplifies the analysis of its rotational properties and can lead to equal or proportional moments of inertia about those axes.

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