Calculating the moment of inertia

[goomer]

I know that for for solid disks, the inertia is equal to (1/2)mr^2 and for hoops is just mr^2. This only works for if the mass is evenly distributed around though. So what about when the mass isn't equally distributed? How would you solve for the moment of inertia of a hollow disk if a small metal ball inside of it?

 

[gneill]

I know that for for solid disks, the inertia is equal to (1/2)mr^2 and for hoops is just mr^2. This only works for if the mass is evenly distributed around though. So what about when the mass isn't equally distributed? How would you solve for the moment of inertia of a hollow disk if a small metal ball inside of it?

You could determine the moment of inertia of the objects separately and add them. The parallel axis theorem may be of use if the object's individual moments of inertia are not co-aligned on the axis of rotation.

 

[goomer]

You could determine the moment of inertia of the objects separately and add them. The parallel axis theorem may be of use if the object's individual moments of inertia are not co-aligned on the axis of rotation.

We haven't learned about the parallel axis theorem yet, so I don't think I need to use that.

I forgot to mention that there is a rim inside the disk that keeps the ball at a constant distance away from the radius. Sorry! Would you still calculate the moments of inertia separately? How do you do that?

 

[gneill]

We haven't learned about the parallel axis theorem yet, so I don't think I need to use that.

I forgot to mention that there is a rim inside the disk that keeps the ball at a constant distance away from the radius. Sorry! Would you still calculate the moments of inertia separately? How do you do that?

Yes, calculate the moments of inertia separately. The moment of inertia of a spherical ball of mass M about a point a distance R from its center is just like a point mass M located at radius R. That is, MR². The moment of inertia of your other object depends upon exactly what it is; is it a hollow disk or a hoop?

There are tables of Moments of Inertia for various objects to be found on the web if you google for it. If you want to determine them from first principles then you'll have to do the calculus. It involves performing an integration for each mass element dm over the volume of the object, determining the moment of inertia for each dm and summing them all up. Your text should have an example or two.

 

[asteriatic]

Calculating the moment of inertia for a hollow disk with a small metal ball inside of it requires a different approach than the formulas you have mentioned. When the mass is not evenly distributed, we need to take into account the distribution of mass and its distance from the axis of rotation.

To solve for the moment of inertia in this scenario, we can use the parallel axis theorem, which states that the moment of inertia of a body is equal to the moment of inertia of the body's center of mass plus the product of the mass and the square of the distance between the center of mass and the axis of rotation.

In this case, we would need to calculate the moment of inertia for both the disk and the metal ball separately, using their respective formulas (1/2)mr^2 and mr^2. Then, we can add these two values together, taking into account the distance between the center of mass of the disk and the metal ball and the axis of rotation.

Alternatively, we can also use the general formula for moment of inertia, which takes into account the distribution of mass around the axis of rotation. This formula is I = ∫r^2 dm, where r is the distance from the axis of rotation to the element of mass dm.

In conclusion, when the mass is not evenly distributed, we need to use the parallel axis theorem or the general formula for moment of inertia to accurately calculate the moment of inertia. It is important to consider the distribution of mass and its distance from the axis of rotation in order to get an accurate result.