What is the moment of inertia of a uniform circular ring about its diameters ?

Explanation

To find the moment of inertia of a uniform circular ring about its diameter, we can use the perpendicular axis theorem. Here’s a step-by-step solution:

### Step 1: Understand the Configuration
We have a uniform circular ring with radius \( r \) and mass \( M \). We need to calculate the moment of inertia about an axis that passes through its center and is perpendicular to the plane of the ring.

### Step 2: Apply the Perpendicular Axis Theorem
The perpendicular axis theorem states that for a planar body, the moment of inertia about an axis perpendicular to the plane (let's call it \( I_z \)) is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane (let's call them \( I_x \) and \( I_y \)):
\[
...