Three point masses m are placed at the vertices of an...

Three point masses $m$ are placed at the vertices of an equilateral triangle of side $a$. Moment of inertia of the system about an axis $C O D$ passing through a mass $m$ at $O$ and lying in the plane of $A O B$ and perpendicular to $O A$ is (a) $2 m a^{2}$ (b) $\frac{2}{3} m a^{2}$ (c) $\frac{5}{4} m a^{2}$ (d) $\frac{7}{4} m a^{2}$

Three point masses $m$ are placed at the vertices of an equilateral triangle of side $a$. Moment of inertia of the system about an axis $C O D$ passing through a mass $m$ at $O$ and lying in the plane of $A O B$ and perpendicular to $O A$ is
(a) $2 m a^{2}$
(b) $\frac{2}{3} m a^{2}$
(c) $\frac{5}{4} m a^{2}$
(d) $\frac{7}{4} m a^{2}$

Key Concepts

Moment of Inertia

Moment of inertia quantifies the resistance of an object or a collection of masses to rotational acceleration about a given axis and is defined as the sum of each mass times the square of its perpendicular distance from the axis (I = ?mr²). It plays a role analogous to mass in linear motion but for rotational dynamics.

Point Mass Systems

In systems where masses are considered as points, the total moment of inertia is computed by summing the individual contributions of each mass located at a distinct distance from the rotation axis. This concept simplifies the problem by reducing continuous mass distributions to discrete masses.

Determining Perpendicular Distances

A key step in calculating moment of inertia involves finding the perpendicular distance of each point mass from the rotation axis. Understanding these distances is crucial, as the rotational effect of each mass is proportional to the square of this distance.

Equilateral Triangle Geometry

When masses are arranged in an equilateral triangle, the inherent symmetry simplifies the calculation of distances between vertices and distances from geometric features such as centers or axes. Familiarity with the properties of equilateral triangles, including side lengths and altitudes, is essential for setting up the moment of inertia calculation.

Summation of Contributions

The moment of inertia of a system is obtained by summing the contributions (m * r²) from each individual mass. This technique is a fundamental aspect of solving problems in rotational dynamics, particularly when dealing with discrete mass distributions arranged in geometrically symmetric configurations.

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