About the Moment of Inertia

In summary, the conversation discusses the similar moments of inertia of different objects, specifically a thin disk, cylinder, and hoop. It is found that the moment of inertia for each object is the same when rotating about its own axis. The explanation for this is that the hoop can be thought of as a pile of disks, while also considering the thickness of the disks to be constant. Alternatively, the perpendicular axis theorem can also be used to explain the phenomenon.
  • #1
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Hello, I am a computer science major and Ex-Biology grad student, my knowledge in physics is humble, but I got a little curious when my professor derived the expressions of moment of inertia for different objects.

The moment of Inertia of a thin disk is 1/2MR2, but it is the same as the moment of Inertia for a cylinder and, surprisingly, the same for a thin hoop rotating about its diameter. So, in short:

I disk/perpendicular to axis of rotation = I Cylinder/perpendicular to axis of rotation= I thin hoop/through diameter

QUESTION:
How do we explain the similar moments of Inertia of the different objects?

My hypothesis concerning the moment of inertia of the thin hoop:
The hoop is rotating about its own diameter, if we take any point on the hoop and project its rotation on a plane perpendicular to the axis of rotation, we'll end up with a circle. Now, if we do the to every point on the hoop, we end up with a disk. Hence, the similar moment of inertia between the disk and the hoop!
 
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  • #2
Hi,

$$I\equiv \int r^2\, dm$$
Disk and cylinder are the same because it's the same integral.
You can also think of a cylinder as a pile of disks,

Hoop is a little different. For your scenario you still have to show that the thickness of the 'disks' is a constant ...

Or, you could consider that the hoop is a planar object and use the perpendicular axis theorem
 

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