Calculating Moment of Inertia for Non-Uniform Mass Contribution

[myer784]

hi everyone,

ive got problem on calculation of moment of inertia for a body that has non-uniform mass contribution . take for example a disc, i believe we can't apply I = (1/2)mr^2 since if we traced back to the derivation of the inertia formula, it assuming constant rho(density)=mass/volume which is not applicable for a body without uniform mass contribution.

Anyone has any idea how calculate the right moment of inertia :uhh:

 

[radou]

Try somehow to apply the general definition of the moment of inertia $$I = \int_{V} r^2 \rho dV$$ to your problem. If you know the mass distribution function, this shouldn't be hard.

 

[kyle8921]

Hello there,

Calculating the moment of inertia for a body with non-uniform mass contribution can be a bit trickier compared to a body with uniform mass distribution. As you mentioned, the formula for moment of inertia, I = (1/2)mr^2, assumes a constant density, which is not applicable in this case.

To calculate the moment of inertia for a body with non-uniform mass contribution, we need to use the integral form of the formula, I = ∫r^2dm, where r is the distance from the axis of rotation and dm is the infinitesimal mass at that distance. This integral takes into account the varying density at different points of the body, allowing for a more accurate calculation of the moment of inertia.

In the case of a disc, the integral would be I = ∫r^2ρ(r)dA, where ρ(r) is the density at a distance r from the axis of rotation and dA is the infinitesimal area element. This integral can be solved using calculus techniques, such as integration by parts or substitution, to find the moment of inertia for the disc.

I hope this helps with your calculation. Remember to always take into account the varying density of the body when calculating the moment of inertia for non-uniform mass contributions.