what is the exact significance of radius of gyration?

Question

mani_jha · Answer

Well, what I can say about that is that the radius of gyration of a substance gives you the radius of a ring which which has the same moment of inertia as the body.
It is simply the square root of the coefficient of M(mass) in the formula of the moment of inertia.
Suppose for a rod, the moment of inertia is ML^2/12. Here, the coefficient of M is L^2/12. So, radius of gyration is:
$$\sqrt{L ^{2}/12}$$
$$=L/(2\sqrt{3)}$$
The larger the radius of gyration, the more the moment of inertia. Usage of the radius of gyration mainly makes equations simpler.
If k is the radius of gyration, its moment of inertia can simply be written as:
I=mk^2

anonymous · Answer

radius of gyration is effective radius of a body for which you can  get moment of inerta of that body by using basic formula I=m(effective radius)^2.
for example moment of inertia of hollow sphere is I=2/3mr^2.compare this to above formula.we get that  2/3mr^2=m(effective radius)^2=>hence effective radius =square root of(2/3)*r.and this effective radius is radius of gyration of hollow sphere when sphere rotates an axis which passes through its centre...radius of gyration depends on axis of rotation also.....

anonymous · Answer

suppose I have an unsymmetrical rigid body. how would i define radius of gyration here?

mani_jha · Answer

Radius of gyration is always defined by:
$$\sqrt{I/M}$$
And it would have the same definiton I gave. The radius of a ring that would have the same moment of inertia as the body

anonymous · Answer

I've read in a couple places in the definition of radius of gyration about mass being concentrated at some point. What does this mean?

anonymous · Answer

what you are saying is right but it is not basic concept mani jha ...this is only formula based concept..

mani_jha · Answer

Hey see if this video clears your confusion: http://www.youtube.com/watch?v=UXAwEneKEFM

anonymous · Answer

@abhijitdash  what you have read is obviously true .let  a point mass which has mass m. let the distance of this mass from the axis of rotation is d.. then moment of inertia I=md^2.if the mass of a body is concentrate at a point which is r'(radius of gyration) distance from the axis of rotation of that body.then I=m(radius of gyration)^2..i hope that you have known about radius of gyration....

mani_jha · Answer

See only the first 2 minutes of the video. Taufique is right. Radius of gyration of a body is also the radius of a point where all the body's mass has been concentrated, and which is revolving around the same axis as the body.