Three-point masses, each of mass, m are placed at the corner...

Three-point masses, each of mass, $m$ are placed at the corner of an equilateral triangle of side, $l$. Then the moment of inertia of this system about an axis along one side of the triangle is:
(a) $3 m l^{2}$
(b) $m l^{2}$
(c) $\frac{3}{4} m l^{2}$
(d) $\frac{3}{2} m l^{2}$

Key Concepts

Equilateral Triangle Geometry

In an equilateral triangle, all sides and angles are equal, with each angle measuring 60 degrees. The geometric properties of an equilateral triangle, such as its height (which is related to the side length by a factor of ?3/2), are essential in determining the distances from the vertices to any given side, which in turn are needed to calculate the moment of inertia.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to angular acceleration about an axis. In systems composed of discrete masses, it is computed as the sum of the products of each mass and the square of its perpendicular distance from the axis of rotation.

Point Masses

Point masses are idealized masses that are assumed to have no spatial extent; all of the mass is concentrated at a single point. This simplification allows for straightforward calculations in mechanics, particularly when summing contributions to the moment of inertia.

Axis of Rotation

The axis of rotation is an imaginary line about which an object or system rotates. The moment of inertia depends on the arrangement of the masses relative to this axis, as only the components of the masses' positions that are perpendicular to the axis contribute.

Perpendicular Distance

The perpendicular distance is the shortest distance from a point mass to the axis of rotation. Since the moment of inertia of a point mass is proportional to the square of this distance, accurately determining it is crucial in rotational dynamics.

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