To determine the displacement of point $P$ when the wheel turns half a revolution forward, we first need to understand a few key concepts.

Step 1: Understand the Problem

• Radius $R$: The radius of the wheel is given as $4$ meters.
• Half Revolution: When a wheel turns half a revolution, the point $P$ originally on the circumference of the wheel will move from one point on the circle to a point directly opposite on the circumference.

Step 2: Calculate the Displacement

Displacement is the straight-line distance between the initial and final positions of point $P$.

• The initial position of $P$ is at some point on the circumference.
• After a half revolution, $P$ moves to the point directly opposite its starting position.

Geometry of the Problem

• The distance between two opposite points on a circle is the diameter of the circle.
• The diameter $D$ is twice the radius $R$.

So, the displacement $d$ of point