Rings Vs Disks In Moment of Moment of Inertia Integral

• teroenza
In summary, the conversation discusses the use of different volume elements (dv) in center of mass and moment of inertia calculations. The teacher uses a disk-shaped volume element for the center of mass calculation, but switches to a ring-shaped element for the moment of inertia calculation. This is because the moment of inertia calculation is dependent on the distance from the axis of rotation (r), while the center of mass calculation is dependent on the position along the z-axis. The different choices of volume elements result in different integrals being set up, but both methods can lead to the correct answer.
teroenza

Homework Statement

Not really homework, but putting it here to be safe. While doing a center of mass calculation of a cone centered about the origin (tip touching origin, z axis through it's center) my teacher put on the board that the volume element was

dv= (pi*r^2)*dz using the area of a disk. The variable r was then eliminated by substituting r as a function of z.

However, when doing a moment of inertia problem, he used

dv= (2*pi*r)*dr*dz using rings of width dr

I asked him why the differentials were different and he said, because when doing moment of inertia problems, we use rings because they have their mass concentrated at one radius.

Wouldn't the same argument require us to use rings for the center of mass calculation, which is also dependent upon r ?

He is changing the differentials to avoid making people do triple integrals. Which I prefer here because I can get the correct answer in both cases by setting up the proper triple integral and working it out. I just want to understand it his way in case a test is more easily worked that way.

Thank you

In the center of mass calculation, you're finding ##\langle z \rangle## whereas in the moment of inertia calculation, you're finding ##\langle r^2 \rangle##. If you set up the triple integrals and then integrate so that z or r is the last variable of integration, you'll see you end up with the two different volume elements.

Ok thank you.

1. What is the difference between a ring and a disk in the moment of inertia integral?

A ring is a circular object with all of its mass concentrated at the same distance from the axis of rotation, while a disk is a circular object with its mass distributed evenly throughout its radius. This difference in mass distribution affects how the moment of inertia is calculated for each object.

2. How does the shape of an object affect its moment of inertia?

The shape of an object directly affects its moment of inertia. Objects with a greater mass located farther from the axis of rotation have a higher moment of inertia, while objects with a smaller mass located closer to the axis of rotation have a lower moment of inertia.

3. Which has a higher moment of inertia, a ring or a disk?

In general, a ring has a higher moment of inertia than a disk due to its mass being located farther from the axis of rotation. However, this can vary depending on the specific dimensions and mass distribution of each object.

4. How does the moment of inertia affect an object's rotational motion?

The moment of inertia is a measure of an object's resistance to changes in rotational motion. Objects with a higher moment of inertia require more torque to change their rotational speed, while objects with a lower moment of inertia require less torque.

5. Can the moment of inertia of an object be changed?

Yes, the moment of inertia of an object can be changed by altering its mass distribution or its shape. For example, adding mass farther from the axis of rotation or changing the object's shape can increase its moment of inertia, while removing mass or changing the shape to be more compact can decrease its moment of inertia.

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