# What are moment of inertia, mass moment of inertia, and radius of gyration?

#### ShawnD

I'm learning how to find these things in school but I have no idea what they are. Moment of inertia's units are distance^4 such as in^4 or mm^4. Mass moment of inertia has units of mass*distance^2 such as kgmm^2. Radius of gyration is a distance such as mm or in.

So what are they exactly and how are they used.

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#### jamesrc

Gold Member
The area moment of inertia [L4] (aka second moment of area) is a measure of the resistance to bending. You'll see it when considering things like the bending stress (due to a bending moment, M) in a cross section ($$\sigma = \frac{Mc}{I}$$) or vibrations in continuous systems. To compute the area moment of inertia about, say, the x-axis, you would compute the following:

$$I = \int y^2 dA$$ (where dA = dxdy)

There is also an analagous polar moment of area that indicates a cross section's resistance to torsion.

Mass moment of inertia [ML2] is the rotational analog of mass. (Some people use J rather than I here to distinguish the to types of moments of inertia, but J is also used for the polar area moment of inertia. As long as you're aware of the context, you shouldn't end up confusing them.) You'll see this moment of inertia in the calculation of kinetic energy or in Newton's laws, for example. The calculation for mass moment of inertia about an axis is $$I = \int \rho^2 dm$$ where &rho; is the distance from the axis.

There is a radius of gyration [L] for mass and area (they are similar concepts). For the mass radius of gyration, it is the number k such that I = m*k2, where I is the mass moment of inertia and m is the mass of the body. It is the distance from a given axis at which the mass of the body would have to be concentrated so that its moment of inertia would remain unchanged. Similarly, the area radius of gyration is the number k such that I = A*k2.