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 diameters, we can use the perpendicular axis theorem. Here’s a step-by-step solution:

### Step 1: Understand the Geometry of the Ring
A uniform circular ring has a mass \( M \) and a radius \( R \). The ring lies in a plane, and we want to find the moment of inertia about an axis that passes through its diameter.

### Step 2: Use the Perpendicular Axis Theorem
The perpendicular axis theorem states that for a planar body (like our ring), the moment of inertia about an axis perpendicular to the plane (z-axis) is equal to the sum of the moments of inertia about two orthogonal axes (x-axis and y-axis) lying in the plane:
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