If a point moves along a circle with constant speed, prove that its angular speed about any point on the circle is half of that about the centre.

Explanation

To prove that the angular speed of a point moving along a circle about any point on the circle is half of that about the center, we can follow these steps:

### Step 1: Define the Circle and Points
Consider a circle with center \( C \) and radius \( R \). Let \( M \) be a point on the circumference of the circle where a particle is moving with constant speed. Let \( O \) be another point on the circumference, which is directly opposite to \( M \) across the center \( C \).

### Step 2: Define Angular Displacement
Let the particle moving along the circle subtend an angle \( \theta \) at the center \( C \). The angular displacement \( \theta \) corresponds to the arc length \( s \) traveled by the particle, where \( s = R \theta \).

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