# Calculating Moment of Inertia for Non-Uniform Mass Contribution

• myer784
In summary, a disc with non-uniform mass distribution cannot use the formula I = (1/2)mr^2 for calculating moment of inertia. Instead, the general definition I = \int_{V} r^2 \rho dV can be applied by using the mass distribution function.
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

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.

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.

## 1. What is the formula for calculating moment of inertia for non-uniform mass distribution?

The formula for calculating moment of inertia for non-uniform mass distribution is:
I = ∫r^2 dm

## 2. How do I find the mass distribution for a non-uniform object?

To find the mass distribution for a non-uniform object, you will need to divide the object into small infinitesimal elements and calculate the mass of each element. Then, you can integrate the mass distribution over the entire object to find the total mass.

## 3. Can I use the same formula for calculating moment of inertia for uniform and non-uniform objects?

No, the formula for calculating moment of inertia for non-uniform objects is different than the formula for uniform objects. For non-uniform objects, you need to take into account the varying mass distribution, while for uniform objects, the mass is evenly distributed and the formula is simplified.

## 4. What are the units of moment of inertia for non-uniform mass distribution?

The units of moment of inertia for non-uniform mass distribution are kg*m^2. This is because the moment of inertia is a measure of an object's resistance to rotation, and it is dependent on both the mass and the distance from the axis of rotation.

## 5. Is there an easier way to calculate moment of inertia for non-uniform mass distribution?

There are some simplified methods for calculating moment of inertia for certain common shapes, such as cylinders or spheres. However, for more complex shapes, the integral method is typically the most accurate and reliable way to calculate moment of inertia for non-uniform mass distribution.

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