Analyzing Circular Motion r(t) = <R cos(ωt), R sin(ωt), h>

Calculus 3 · Axiom Academy

EXAMPLE Analyzing Circular Motion Differentiate a uniform circular-motion path to find velocity and acceleration, prove the speed is constant, and identify the centripetal acceleration. A particle moves along the space curve , where R > 0 is the radius, is the constant angular speed, and h is a fixed height. Find the velocity and acceleration , show that the speed is constant, and describe the direction and magnitude of . The path is a circle of radius R in the plane z = h . The velocity (green) is tangent to the circle; the acceleration (pink) points straight back toward the center. You'll derive both below. Nice work. You differentiated a uniform circular-motion path twice and read the physics straight out of the vectors. Velocity is tangent: is perpendicular to in the plane, so it runs tangent to the circle. Constant speed: the Pythagorean identity collapses the magnitude to , with no t left — the motion is uniform. Centripetal acceleration: points from the particle straight back toward the central axis. Magnitude: , which grows with both the radius and the angular speed. Perpendicular pair: since is tangent and is radial, at every instant — the hallmark of constant-speed motion. These relationships describe circular motion everywhere it appears — from a satellite's orbit to a particle in a magnetic field.

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