Show that the moments of inertia of a regular hexagonal...

Show that the moments of inertia of a regular hexagonal molecule made up of six identical atoms of mass $m$ are given by
\[
I_{\|}=6 m r^{2} \quad \text { and } \quad I_{\perp}=3 m r^{2}
\]
where $r$ is the bond distance.

Key Concepts

Moment of Inertia

This concept involves quantifying the resistance a body has to angular acceleration about a given axis and is calculated by summing (or integrating) the product of each mass element with the square of its distance from that axis. For discrete systems such as molecules, the total moment of inertia is obtained by summing over the contributions of individual atoms, which makes it possible to analyze the rotational dynamics of the system.

Discrete Mass Distribution

In analyzing molecules composed of individual atoms, each atom is treated as a point mass. This simplification allows one to compute the moment of inertia by summing the products of each atom’s mass and the square of its distance from the chosen rotation axis, providing insights into the rotational behavior of the system.

Regular Polygon Geometry

The arrangement of atoms in a regular polygon, such as a hexagon, means that each vertex is equidistant from the center. This regularity simplifies calculations of the moments of inertia, as it provides uniform distances and angles that can be exploited to derive expressions for rotational properties by symmetry.

Perpendicular Axis Theorem

For a planar object, the theorem states that the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane. This principle is particularly useful for systems with high symmetry, such as molecules arranged in a regular polygon, and allows the determination of in-plane moments from the easier calculation of the out-of-plane moment.

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