Let I be the moment of inertia of a uniform square plate about an axis AB that passes through it

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Let I be the moment of inertia of a uniform square plate...

Let I be the moment of inertia of a uniform square plate about an axis $\mathrm{AB}$ that passes through its centre and is parallel to two of its sides $\mathrm{CD}$ is a line in the plane of the plate that passes through the centre of the plate and makes an angle of $\theta$ with $\mathrm{AB}$. The moment of inertia of the plate about the axis $\mathrm{CD}$ is then equal to.... $\{\mathrm{A}\} \mathrm{I}$ $\{B\} I \sin ^{2} \theta$ $\{C\} I \cos ^{2} \theta$ $\{\mathrm{D}\} I \cos ^{2}(\theta / 2)$

Let I be the moment of inertia of a uniform square plate about an axis $\mathrm{AB}$ that passes through its centre and is parallel to two of its sides $\mathrm{CD}$ is a line in the plane of the plate that passes through the centre of the plate and makes an angle of $\theta$ with $\mathrm{AB}$. The moment of inertia of the plate about the axis $\mathrm{CD}$ is then equal to....
$\{\mathrm{A}\} \mathrm{I}$
$\{B\} I \sin ^{2} \theta$
$\{C\} I \cos ^{2} \theta$
$\{\mathrm{D}\} I \cos ^{2}(\theta / 2)$

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Key Concepts

Symmetry in Rigid Bodies

Symmetry plays a key role in simplifying the calculation of the moment of inertia. For shapes like a square, the symmetry ensures that moments of inertia about certain axes (for instance, those parallel to its sides) are equal, which leads to invariance under rotations about the center in the plane. This results in the same moment of inertia for any in-plane axis through the center, provided the mass distribution is uniform.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on both the mass distribution of the body and the axis about which the body rotates. Understanding this concept is crucial in rotational dynamics, as it determines how much torque is needed for a desired angular acceleration.

Rotational Inertia Transformation

When the axis of rotation is changed, especially rotated within the plane of a rigid body, the moment of inertia about this new axis can be determined using transformation rules applied to the inertia tensor. For a body with symmetry, such as a square plate, the moment of inertia about any in-plane axis through the center can be derived by projecting the original inertia components using the cosine and sine of the rotation angle.

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