The moment of inertia of any object can be determined dynamically with the Parallel Axis Theorem. The parallel axis theorem is the theorem determines the moment of inertia of a rigid body about any given axis, given that moment of inertia about the parallel axis through the center of mass of an object and the perpendicular distance between the axes. The first term is I cm,the second term is mr 2and the final term is zero as the origin lies at the centre of mass. Moment of inertia relative to the new axis with its perpendicular distance r along the x-axis, is represented as: The moment of inertia relative to z-axis that passes through the centre of mass, is represented as R is the perpendicular distance between the two axes.Īssume that the perpendicular distance between the axes lies along the x-axis and the centre of mass lies at the origin. I cm is the moment of inertia of an object about its centre of mass Two point masses, m 1 and m 2, 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 the line joining the two particles.The moment of inertia about Z-axis can be represented as: However due to the rotation of the axes ( yz to y. These principal moments of inertia Iy and Iz can then be used in the bending stress formula. 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 construction process is exactly the same as the standard Mohr’s Circle in Solid Mechanics I, except that: initial coordinates are Y (Iy, Iyz) and Z (Iz, -Iyz). In general, the moment of inertia is a tensor, see below. 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.įollowing are scalar moments of inertia. 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. 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). Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass (which determines an object's resistance to linear acceleration).

Moment of inertia of a circle proof