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How does rigid-body dynamics account for an object’s shape and mass distribution?

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A diagram of a cylindrical vessel containing gas, with a movable piston held down by a weight. Countless small dots beneath the piston represent gas particles in rapid, random motion, illustrating Bernoulli's model of gas pressure. — physics As conceived by Daniel Bernoulli in *Hydrodynamica* (1738), gases consist of numerous particles in rapid random motion. He assumed that the pressure of a gas is produced by the direct impact of the particles on the walls of the container.

Rigid-body dynamics accounts for an object’s shape and mass distribution by treating the body as a collection of mass elements, each contributing to the object's overall inertial properties based on its position relative to an axis of rotation.1

Mass Distribution and Inertia

The primary way dynamics incorporates mass distribution is through the moment of inertia ( $I$ ). While mass ( $m$ ) measures resistance to linear motion, the moment of inertia expresses resistance to rotational motion. Because this value depends on the square of the distance ( $R_{i}^{2}$ ) of each mass bit from the axis, mass located farther from the axis contributes significantly more to the object's rotational inertia than mass located near the axis.1

The Role of the Centre of Mass

The centre of mass serves as a critical reference point for simplifying complex systems. For many calculations, the entire mass of an object can be treated as if it were concentrated at this single point. When an object rotates about an axis that does not pass through its centre of mass, the moment of inertia can be calculated by adding the moment of inertia about a parallel axis through the centre of mass to a term representing the total mass rotating at the distance between the two axes.21