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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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Answer» gram.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) * RThe 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. |
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