EGR 280 Mechanics 9 – Particle Kinematics II

Curvilinear motion of particles Let the vector from the origin of a fixed coordinate system to the particle be the position vector r The time derivative of position is velocity: P x y z r s

The magnitude of velocity is speed, and is the time rate of change of arc length. Speed is a scalar quantity. The time derivative of velocity is acceleration:

Motion of several particles r B = r A + r AB = r A + r B/A v B = v A + v B/A a B = a A + a B/A A B x y z rArA rBrB r AB =r B/A

Intrinsic coordinate system Define a coordinate system that moves with the particle: e t = unit tangent vector. Always tangent to the path of the particle e n = unit normal vector. Perpendicular to e t,, always points into the curve As the particle moves along the curve, the unit tangent vector moves in the direction of the unit normal vector: de t /dθ = e n etet enen dθdθ

Intrinsic coordinate system The velocity, by definition, is always tangent to the curve: v = v e t The acceleration is the time rate of change of velocity: The intrinsic coordinate system is used often to describe circular motion.