# 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.

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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. *

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 direction of OP = displacement vector = Tan⁻¹ PP' / OP'

= tan⁻¹ 2R/πR = tan⁻¹ 2/π

OR,