What is the purpose of radius of gyration

[TheBlueDot]

Hello,

I'm given a problem of a mass rolling down an incline with mass 'm', radius 'r', and radius of gyration Rg, and I need to write the Lagrangian for the motion. I'm confused on why both r and Rg are given. Don't I just need one of the two for the moment of inertia?
Thanks!

 

[jambaugh]

The radius of gyration gives the moment of inertia via the equivalence of the given object and an object with all its mass concentrated at a distance of the radius of gyration from the center of mass. So $I = m R_g^2$.

You can treat the object as a massless cylinder of the given physical radius with a hollow cylinder of mass m and radius $R_g$ embedded within it.

 

[TheBlueDot]

Hi jambaugh,

Thank you for your response. The part I'm confused on is if I use Rg the moment of inertia (assume a cylinder) would be I= mRg^2, but if I use 'r', then I=1/2 *mr^2. I guess I don't understand where the question is leading...

 

[jambaugh]

Why would $I = \frac{m}{2} r^2$? Is $R_g$ given to be equal to $r/\sqrt{2}$? Did the problem specifically say the object was a solid disk or cylinder? Maybe the object is not what you are assuming with the $\frac{m}{2}r^2$ formula.