What is the Moment of Inertia? A 10 Minute Introduction

[Greg Bernhardt]

Definition/Summary
The moment of Inertia is a property of rigid bodies.
It relates rotational force (torque) to rotational acceleration in the same way that mass relates ordinary (linear) force to ordinary acceleration.
Moment of Inertia has dimensions of distance squared times mass ($ML^2)$.
The moment of Inertia is always relative to a given axis. The same rigid body will usually have different Moments of Inertia for different axes.
Moment of Inertia is additive: the Moment of Inertia of a composite body is the sum of the Moments of Inertia of its parts (relative to the same axis).
Equations
Moment of inertia of rigid body about an axis, where r is the distance from that axis and $\rho$ is the density:
$$I\ \ =\ \int dm\ r^2\ =\ \int \rho dxdydz\ r^2$$
(for comprehensive lists of moments of inertia of specific bodies, see See Also)
Moment of momentum (angular momentum) about centre of mass, C:
[itex]\ \ \ \ \boldsymbol{L}_C =...

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[eljose79]

The moment of inertia is a fundamental property of rigid bodies that describes their resistance to rotational motion. It is defined as the rotational equivalent of mass, in that it relates rotational force (torque) to rotational acceleration in the same way that mass relates ordinary (linear) force to ordinary acceleration. This relationship is described by the equation I = ∫r^2 dm, where I is the moment of inertia, r is the distance from the axis of rotation, and dm is the mass element. The moment of inertia has dimensions of distance squared times mass (ML^2) and is always relative to a given axis. This means that the same rigid body can have different moments of inertia for different axes.

The moment of inertia is additive, meaning that the moment of inertia of a composite body is the sum of the moments of inertia of its individual parts, all relative to the same axis. This makes it a useful property for analyzing the rotational motion of complex systems.

Another important concept related to the moment of inertia is the moment of momentum, also known as angular momentum. This is defined as the product of the moment of inertia and the angular velocity, and it describes the rotational motion of a body around its center of mass. The equation for moment of momentum is L = Iω, where L is the moment of momentum, I is the moment of inertia, and ω is the angular velocity.

In summary, the moment of inertia is a key property that describes the rotational motion of rigid bodies. Its relationship to torque and rotational acceleration is analogous to that of mass and ordinary acceleration, making it an essential concept in the study of rotational dynamics.