Circular Motion: Uniform and Non-uniform

Introduction

Circular motion is fundamental in mechanics, describing everything from electron orbits to planetary motion to rotating machinery. The key feature is that velocity is always tangent to the circle while acceleration points toward the center (for uniform motion) or has both radial and tangential components (for non-uniform motion).

Angular Variables

Angular position θ (radians) describes orientation. Angular velocity ω = dθ/dt (rad/s) describes rotation rate. Angular acceleration α = dω/dt = d²θ/dt² (rad/s²). Relations to linear variables: arc length s = rθ, linear speed v = rω, tangential acceleration a_t = rα. These connect rotational and translational descriptions of circular motion.

Uniform Circular Motion

In uniform circular motion (constant speed), the particle travels equal arcs in equal times. While speed is constant, velocity direction changes continuously. The centripetal (center-seeking) acceleration has magnitude a_c = v²/r = ω²r and is always directed toward the center. Period T = 2π/ω = 2πr/v. Frequency f = 1/T = ω/(2π). No tangential acceleration since speed is constant.

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Introduction

Circular motion is fundamental in mechanics, describing everything from electron orbits to planetary motion to rotating machinery. The key feature is that velocity is always tangent to the circle while acceleration points toward the center (for uniform motion) or has both radial and tangential components (for non-uniform motion).

Angular Variables

Angular position θ (radians) describes orientation. Angular velocity ω = dθ/dt (rad/s) describes rotation rate. Angular acceleration α = dω/dt = d²θ/dt² (rad/s²). Relations to linear variables: arc length s = rθ, linear speed v = rω, tangential acceleration a_t = rα. These connect rotational and translational descriptions of circular motion.

Uniform Circular Motion

In uniform circular motion (constant speed), the particle travels equal arcs in equal times. While speed is constant, velocity direction changes continuously. The centripetal (center-seeking) acceleration has magnitude a_c = v²/r = ω²r and is always directed toward the center. Period T = 2π/ω = 2πr/v. Frequency f = 1/T = ω/(2π). No tangential acceleration since speed is constant.

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Non-uniform Circular Motion

When speed varies, acceleration has two components: (1) Centripetal a_c = v²/r toward center (changes direction), (2) Tangential a_t = dv/dt = rα tangent to circle (changes speed). Total acceleration magnitude: a = √(a_c² + a_t²). The direction is at angle β = tan^-1(a_t/a_c) from the radial direction toward the tangential direction.

Equations for Constant Angular Acceleration

Analogous to linear motion with constant a: (1) ω = ω₀ + αt, (2) θ = θ₀ + ω₀t + 1/2αt², (3) ω² = ω₀² + 2α(θ - θ₀). These apply to rotating wheels, decelerating flywheels, and any rigid body rotation with constant angular acceleration. All angular quantities use radians for these equations.

Examples and Applications

Examples: turning car (friction provides centripetal force), banked curves (normal force component provides centripetal force), electron in magnetic field (Lorentz force causes circular motion), artificial gravity in rotating space stations, centrifuges, Ferris wheels, roller coasters with loop-the-loops. The loop-the-loop requires minimum speed at top: v > √(gr) to maintain contact.

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Solved Example: Banking Angle

A curve of radius 100 m is to be banked so a car can take it at 20 m/s without friction. Find the banking angle θ. Solution: Forces on car: weight mg down, normal force N perpendicular to road. Vertical: Ncosθ = mg. Horizontal (centripetal): Nsinθ = mv²/r. Dividing: tanθ = v²/(gr) = 400/(9.8×100) = 0.408. Therefore θ = 22.2°. This is the ideal banking angle where no friction is needed.

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