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Q & A: Radius of Gyration

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Most recent answer: 03/08/2010
Q:
Please could you define for me the Radius of Gyration in the context of a Flywheel? Thankyou.
- Daniel Stewart (age 19)
Electrolux Outdoor Products, Darlington UK
A:
Daniel -

The radius of gyration is defined as (I/M)^1/2, or the square root of the moment of inertia divided by mass. The moment of inertia is an important thing to know when solving problems that have to do with how things rotate. The equation for moment of inertia is different depending on the shape of the object, but for a flywheel (basically a solid disk), it’s I=(1/2)MR^2, where M is the mass and R is the radius of the flywheel. Putting this into the equation from before, we find that the radius of gyration for a flywheel is R*(1/2)^1/2, or 0.707*R.

The units of moment of inertia are mass*distance^2, so if you divide by mass and take the square root, you get something with units of just distance (a.k.a. the radius of gyration). Conceptually, the radius of gyration is the distance that, if the entire mass of the object were all packed together at only that radius, would give you the same moment of inertia. That is, if you were to take the entire mass of the disk-shaped flywheel with some radius and pack it into a narrow donut whose radius is the flywheel’s radius of gyration, they’d both have the same moment of inertia. With the same moment of inertia, they will behave very similarly when you spin them.

-Tamara

(published on 10/22/2007)

Follow-Up #1: Radius of gyration of a spoked flywheel?

Q:
i have spoke type of flywheel .so how could i know the radius of gyration???
- kull (age 23)
hyderabad,a.p,india
A:
I am assuming you are referring to the radius of gyration about a central axis perpendicular to the plane of the flywheel.  The reason is, that many references to the radius of gyration involve a planar configuration where the axis of rotation is about a line lying in the plane of the object.

In both cases the definition is Rg2 = I/M where I is the moment of inertia about the axis and M is the total mass of the object.  A flywheel of radius R with spokes can be broken into the sum of two parts: the rim of the wheel and the spokes.   The rim is easy:  Irim = Mrim R2.  Consider an individual spoke: its contribution is  Ispoke = Mspoke R2/12.   The total moment of inertia is
Itot = Irim + N*Ispoke . The total mass is Mtot = Mrim + N*Mspoke  where N is the number of spokes.  There you have it:  Rg2 = Itot/Mtot.


(published on 03/08/2010)

Follow-up on this answer.