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

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

Key Concepts

Transformation of Moments of Inertia

When an axis of rotation is rotated within the plane of a symmetric body, the moment of inertia about the new axis can be found using a coordinate transformation of the inertia tensor. For a uniform square plate, the symmetry about the center means that the moments of inertia about orthogonal in-plane axes are identical, resulting in the moment of inertia being independent of the specific in-plane orientation of the axis. This underlies the result that the moment of inertia about any such axis remains constant.

Moment of Inertia

The moment of inertia is a measure of how the mass of an object is distributed with respect to a given axis of rotation. It quantifies the resistance of the object to changes in its rotational motion. In problems like this, it is computed by integrating the mass distribution multiplied by the square of the perpendicular distance to the axis, and it plays a similar role in rotational dynamics as mass does in linear dynamics.

Inertia Tensor

The inertia tensor is a mathematical representation that encapsulates all moments of inertia about various axes through a body’s center of mass, including products of inertia. For objects with symmetrical mass distributions, like a uniform square plate, the tensor simplifies because many off?diagonal terms vanish and diagonal terms may be equal. This is crucial when determining the moment of inertia about any arbitrary axis, as the tensor’s symmetry can lead to rotation invariance in certain planes.

Transcript

Please give Ace some feedback