Radius of Gyration: Dist to Axis & Mass Concentration

[Zubair Ahmad]

For radius of gyration we say distance from axis to a point where would mass is concentrated. isn't that center of mass?

 

[Gene Naden]

Not exactly. Both center of mass and radius of gyration are measures of where the mass is concentrated, but they are different. To compute radius of gyration, you have to integrate $\int r^2 dm$ where dm is the mass. To get center of mass you have to integrate $\int \vec{r} dm$. Center of mass has three components; it is like a vector. Radius of gyration is a scalar (a simple number, not a vector).

 

[Zubair Ahmad]

But physically we have a single point where we say whole mass is concentrated.

 

[PhanthomJay]

For radius of gyration we say distance from axis to a point where would mass is concentrated. isn't that center of mass?

No, that is not the correct definition of radius of gyration. The radius of gyration, r, is the perpendicular distance to the axis of rotation of a point mass whose moment of inertia (I=mr^2) is the same moment of inertia of the actual object having that same mass. So say an object having 10 kg of mass with an I of 1000 kg-m^2 is equivalent to a point mass of 10 Kg located10 meters away from the axis, so r = 10 m. In other words, r = sq rt (I/m).

 

[PhanthomJay]

But physically we have a single point where we say whole mass is concentrated.

No, we have a single point where IF the whole mass was concentrated there, the moment of inertia would be equivalent to the moment of inertia of the object. A symmetrical object for example has its actual center of mass at the center of it at the rotation axis, but its radius of gyration is not 0, it is sq rt (I/m)

 

[Zubair Ahmad]

So can we say physically there are two such points

 

[PhanthomJay]

So can we say physically there are two such points

No, sir. The center of mass is real; the radius of gyration is imaginary