10.11 : Mass Moment of Inertia

The mass moment of inertia is a measure of the resistance of a rigid body to angular acceleration. This physical quantity can be evaluated and analyzed in various rotational motions, from everyday objects such as fans and wind turbines to complex machines used in challenging industries.

Consider a differential mass element on an arbitrary object. The differential moment of inertia can be calculated by multiplying the mass of the element by the square of its distance from the object's rotational axis. Integrating this expression over the entire mass distribution yields the total moment of inertia for the object.

The differential mass equals the product of density and differential volume. The equation is simplified when the density term is factored out of the integral for constant density. The moment of inertia reduces to a geometrical factor.

Spherical metal balls are widely used in bearings due to their unique physical and mechanical properties. When it comes to solid spheres, their moment of inertia can be estimated by considering a thin elemental disc located at a fixed distance from its center. For uniform density, the differential mass can be represented as the product of the density and the differential volume of the elemental disc. The differential moment of inertia can be expressed as one-half of the product of the differential mass and the square of its radius.

Integrating the expression over the sphere's diameter gives the moment of inertia of the sphere. The disc radius can be expressed in terms of the sphere's radius, and the limits of integration are adjusted to match the sphere's diameter. The obtained result gives the moment of inertia of the solid sphere about its diameter.

The moment of inertia depends on the mass and radius of the sphere.

In conclusion, understanding moments of inertia is essential for mechanical engineers to ensure maximum stability and efficiency during all kinds of rotational movement occurring inside their systems.