Why Do Different Objects Share Similar Moments of Inertia?

[Cha0t1c]

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!

 

[BvU]

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