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The radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there.
Although they have different units, the gyration tensor is related to the moment of inertia tensor. The key difference is that the particle positions are weighted by mass in the inertia tensor, whereas the gyration tensor depends only on the particle positions; mass plays no role in defining the gyration tensor.
In general, given an object of mass m, an effective radius k can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is =, where k is known as the radius of gyration around the axis.
In many cases the moment of inertia, and hence the angular momentum, can be simplified by, [15] =, where is the radius of gyration, the distance from the axis at which the entire mass may be considered as concentrated.
The radius of this circle, , can be determined by equating the magnitude of the Lorentz force to the centripetal force as = | |. Rearranging, the gyroradius can be expressed as = | |. Thus, the gyroradius is directly proportional to the particle mass and perpendicular velocity, while it is inversely proportional to the particle electric charge ...
The moment of inertia of an object, symbolized by , is a measure of the object's resistance to changes to its rotation. The moment of inertia is measured in kilogram metre² (kg m 2). It depends on the object's mass: increasing the mass of an object increases the moment of inertia.
The moments of inertia of a mass have units of dimension ML 2 ([mass] × [length] 2). It should not be confused with the second moment of area, which has units of dimension L 4 ([length] 4) and is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.
= / is the radius of gyration, is the second moment of area (area moment of inertia), is the area cross section. For slender columns, the critical buckling stress is usually lower than the yield stress.