How to Calculate Moment of Inertia for a Disk Using Calculus?

In summary, the moment of inertia of a disk can be calculated using the formula I=1/2 MR^2 for the axis perpendicular to the plane of the disk and I=1/4 MR^2 for the other axis using the perpendicular axis theorem. However, to obtain the second result using calculus, one must use the formula I=1/12 MR^2 for the moment of inertia of a rod and integrate using the mass of a section of the disk, which can become complicated. The correct method is shown in the provided image.
  • #1
jeremy5561
5
0
I know that the moment of inertia of a disk about the axis perpendicular to the plane of the disk is

I=1/2 MR^2

and by the perpendicular axis theorem the moment of inertia about the other axis is

I=1/4 MR^2

I want to get the 2nd result with calculus. I can't get the right answer. What am I doing wrong?

I know the moment of inertia of a rod is 1/12 MR^2
I take the mass of a section of the disc to me pyr dx where y is the depth of the disc
but when I integrate it gets really messy.

What's the correct way to do this?

http://individual.utoronto.ca/jeremyli/moi.jpg
 
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  • #2
I got it

OH nevermind I got it.
 

1. What is the moment of inertia of a disk?

The moment of inertia of a disk is a measure of its resistance to changes in its rotational motion. It is a property that depends on the mass distribution and the shape of the disk.

2. How is the moment of inertia of a disk calculated?

The moment of inertia of a disk can be calculated using the formula I = 1/2 * m * r^2, where I is the moment of inertia, m is the mass of the disk, and r is the radius of the disk.

3. What factors affect the moment of inertia of a disk?

The moment of inertia of a disk is affected by its mass distribution and the shape of the disk. A disk with more mass concentrated towards the outer edge will have a larger moment of inertia compared to a disk with a more uniform mass distribution.

4. How is the moment of inertia of a disk different from that of a solid cylinder?

The moment of inertia of a disk is different from that of a solid cylinder because a disk has all of its mass concentrated along its outer edge, while a solid cylinder has its mass distributed throughout its entire volume.

5. Why is the moment of inertia of a disk important in physics?

The moment of inertia of a disk is important in physics because it is a crucial factor in determining the rotational motion and stability of objects. It is also used in various equations and laws, such as the law of conservation of angular momentum, to calculate and predict the behavior of rotating objects.

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