Four point masses, each of value m, are placed at the...

Four point masses, each of value $m$, are placed at the corners of a square $A B C D$ of side $l$. The moment of inertia of the system about an axis passing through $A$ and parallel to $B D$ is
(a) $\sqrt{3} m l^{2}$
(b) $3 \mathrm{ml}^{2}$
(c) $m l^{2}$
(d) $2 \mathrm{ml}^{2}$

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Key Concepts

Moment of Inertia

This is a measure of an object's resistance to angular acceleration about an axis. For a system of point masses, it is calculated by summing the product of each mass and the square of its perpendicular distance from the axis of rotation.

Point Mass Distribution

In problems involving discrete masses, the distribution refers to the specific placement of masses at various points in a system. This affects the overall moment of inertia since each point mass contributes based on its distance from the axis.

Perpendicular Distance Calculation

Determining the perpendicular distance from each mass to the axis is critical in computing the moment of inertia. Geometry and the use of coordinate systems often aid in calculating these distances accurately in systems with non-standard axes.

Coordinate Geometry in Inertia Calculations

Using a coordinate system to model the positions of masses allows one to apply analytic geometry methods to compute distances and angles. This approach is especially useful when the axis is not aligned with the main geometric features of the mass distribution.

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