Why Does the Polar Moment of Inertia Use r^2 in Its Formula?

[Andrea Vironda]

TL;DR Summary

Derivation of the formula for the calculation, among other things, of bending stresses

Hi,
A well-known part of the formula for calculating the deflection stress is ##I_z=\int \int r^2 dA##
Usually a moment of inertia is something related to how difficult is to move an object. In this case is understandable but i don't understand the meaning of the double integral.
Using ##r^4## wouldn't be the same?

 

[Mark44]

Summary:: Derivation of the formula for the calculation, among other things, of bending stresses

Hi,
A well-known part of the formula for calculating the deflection stress is ##I_z=\int \int r^2 dA##

Don't you mean moment of inertia rather than deflection stress?

I read the formula you wrote as the moment of inertia of some two-dimensional region in the x-y plane that is being rotated around the z-axis. dA represents the area of some infinitesimal region, with an implied mass of 1 unit of some kind. If the integral is replaced by an iterated Cartesian or rectangular integral, dA will become dxdy or dydx, depending on the order of integration. If the integral is replaced by an iterated polar integra, dA will be replaced by ##rdrd\theta##, so the iterated integral could look like ##\int_{\theta}\int_r r^2 r dr~d\theta##, or ##\int_{\theta}\int_r r^3 dr~d\theta##, assuming the integration is performed first on r. In both integrals the mass of the region dA would be 1 unit, by implication.

Usually a moment of inertia is something related to how difficult is to move an object. In this case is understandable but i don't understand the meaning of the double integral.
Using ##r^4## wouldn't be the same?

 

[Andrea Vironda]

Don't you mean moment of inertia rather than deflection stress?

Yeah, only part of deflection stress formula.

Ok i think i understood. but why ##r^2## into the integral and not simply ##r##?
If i integrate ##r## on ##dA## i will get something related to the area

 

[willem2]

Ok i think i understood. but why r² into the integral and not simply r?

It is about the momentum of the tension force on dA. You get r² in the integral because the force on the surface dA depends on how much the material has been stressed or compressed, which is proportional to the distance from dA to the center of the beam.
To get the momentum of this force, you have to multiply with this distance again.