Question

what's the difference between the two? If they're not different, what are the relationships between the two? Someone told me that a figure skater, who is spinning and brings their arm inside from an outstretched position, is a good example of a decrease in radius of gyration.

Answer 1

I = mk^2

I = moment of inertia

Moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment of inertia depends on the shape of the body and may be different around different axes of rotation.

k = radius of gyraton

Radius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean squaredistance of the objects' parts from either its center of gravity or a given axis.

Imagine you took a normal, real-world object and condensed all the particles into a hollow sphere. Now imagine this hollow sphere (with no wall thickness) has the same rotational inertia as the original object. The radius of the sphere is the radius of gyration of the original object. (At least that's what wikipedia said. If I remember from dynamics class, though, I'd say it'd be better to imagine a hollow cylinder with no end caps or an extremely thin torus/ring. You avoid the error from the mass at the poles this way.)

Sit in a chair and spin with your arms out. When you pull them in, rotational momentum is conserved, and you spin faster because your rotational inertia decreased. This decrease in rotational inertia with the same mass was caused by a decrease in the radius of gyration.