I = M m M + m x 2 = μ x 2, and the object is a hollow sphere. Two point masses, M and m, with reduced mass μ and separated by a distance, x about an axis passing through the center of mass of the system and perpendicular to line joining the two particles. The parallel axis theorem-which is quite general-states that if is the moment of inertia of a given body about an axis passing through the centre of mass of that body, then the moment of inertia of the same body about a second axis which is parallel to. Point mass m at a distance r from the axis of rotation.Ī point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. The second useful theorem regarding moments of inertia is called the parallel axis theorem. In general, the moment of inertia is a tensor, see below. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.įollowing are scalar moments of inertia. It should not be confused with the second moment of area, which is used in beam calculations. Mass moments of inertia have units of dimension ML 2( × 2). In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.įor simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression.
![second moment of inertia of a circle second moment of inertia of a circle](https://www.engineeringtoolbox.com/docs/documents/1328/geometric_sections_nonsymmetrical_shape-Model.png)
![second moment of inertia of a circle second moment of inertia of a circle](https://mathalino.com/sites/default/files/reviewer-mechanics/000_moment_of_inertia.gif)
Mass moments of inertia have units of dimension ML2( × 2).
![second moment of inertia of a circle second moment of inertia of a circle](https://listimg.pinclipart.com/picdir/s/157-1572985_moment-of-inertia-of-a-circle-second-moment.png)