# What is center of inertia?

What is the center of inertia? Is it the same as the center of mass?

Doug Huffman
Gold Member
What other component of inertia can have space-like - center - dimensions?

I don't understand.

mathman
"A https://www.physicsforums.com/wiki/point [Broken], near or https://www.physicsforums.com/wiki/within [Broken] a https://www.physicsforums.com/wiki/body [Broken], at which the body's https://www.physicsforums.com/wiki/mass [Broken] can be https://www.physicsforums.com/wiki/assumed [Broken] to be https://www.physicsforums.com/wiki/concentrated [Broken] when considering its rotational motion and https://www.physicsforums.com/wiki/moment_of_inertia [Broken]. This may be different from its https://www.physicsforums.com/wiki/centre_of_mass [Broken] which is the equivalent for linear motion."

Above from wikidictionary.

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jtbell
Mentor
Mathman, all of the links in the body of your text point to nonexistent pages. PF doesn't have a wiki.

How to calculate it?

FactChecker
Gold Member
Can anyone give an example where it is easy to see that the center of mass is not the center of inertia?

jbriggs444
Homework Helper
As I understand the concept from the rather terse dictionary definition...

Suppose that you have a turntable. On this turntable you have an object. The "center of inertia" of the object is where you could place its entire mass and wind up with the same moment of inertia as the original object.

Suppose, for instance that the object is a thin hoop of mass m, radius r placed flat on the turntable with its center R units from the center of the turntable. The moment of inertia of this hoop with respect to the center of the turntable is ##mr^2 + mR^2##.

Its "center of inertia", C, is at distance ##\sqrt{r^2+R^2}## from the center of the turntable so that the moment of inertia works out to ##mC^2 = mr^2 + mR^2##

Shouldn't moment of inertia be used instead of the masses?

Have a look at radius of gyration (mechanics), it assumes all the mass is concentrated at a single point and radius.
For instance a cylinder with a mass of 10 kg and a radius of 0.1 metres, rotating about its longitudinal axis has a moment of inertia of
0.04 kg - m², the radius of gyration = 0.0632 metres

Is center of inertia a valid concept? I think inertia is quantitative concept.