Question

Find the displacement vector of the lower most point of a wheel (initially at origin ) when the wheel (initially at origin ) rolls along +X axis by half a revolution in the XY plane.

Answer 1

see diagram.

The point P is initially at the origin O.  The circle, initially on the left, rotates right side.    The point P follows the curved locus as shown.  Reaches the top of the circle at P on the right.  The diameter PP' gets inverted  on the right side.

The distance rolled by the wheel is  OP' on the x axis:  π R ,  R = radius.
The diameter PP' = 2 R.

The triangle OPP' is a right angle triangle.
     OP² = P'O² + P'P²
           = (πR)² + (2R)²
             = (π²+4) R²

   The magnitude of displacement vector  OP = √(4+π²) * R

The direction of  OP = displacement vector = Tan⁻¹  PP' / OP'
        = tan⁻¹ 2R/πR = tan⁻¹ 2/π  = 32.48 deg.

OR,  the slope of displacement vector = 2/π.