Find the moment of inertia of a half uniform ring (mass m, radius r) about its center.

Explanation

To find the moment of inertia of a half uniform ring about its center, we can follow these steps:

### Step 1: Understand the Geometry
We have a half uniform ring with mass \( m \) and radius \( r \). The ring is symmetric about the y-axis, and we will calculate the moment of inertia about the center of the ring, which is at the origin.

### Step 2: Consider a Small Element
We can consider a small element of the ring at an angle \( \theta \) with a small angular width \( d\theta \). The length of this small arc element is given by:
\[
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