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Angular Motion Concepts

Jul 21, 2025

Overview

This lecture covers example problems using angular motion equations, focusing on calculating angular position, angular velocity, and angular acceleration in various rotational scenarios.

Angular Position Calculation

  • The initial angular position of the wrench is Ï€/2 radians.
  • The wrench rotates counterclockwise (positive direction) by 1/3 of a revolution.
  • Convert 1/3 revolution to radians: (1/3) × 2Ï€ = 2Ï€/3 radians.
  • The final angular position = initial position + angular displacement = Ï€/2 + 2Ï€/3 = 7Ï€/6 radians.

Angular Velocity in RPM

  • Vinyl record rotates clockwise (negative direction) by 1,350° in 5 seconds.
  • Angular velocity ω = Δθ / Δt = -1,350° / 5 s = -270°/s.
  • Convert °/s to revolutions/min: -270°/s × (1 rev/360°) × (60 s/1 min) = -45 RPM.
  • Negative sign indicates clockwise rotation.

Angular Acceleration Calculation

  • Lab centrifuge spins counterclockwise at 80 rad/s and increases speed to 450 rad/s in 6 seconds.
  • Initial angular velocity ω₀ = 80 rad/s; final angular velocity ω = 450 rad/s; time Δt = 6 s.
  • Angular acceleration α = (ω - ω₀) / Δt = (450 - 80) / 6 = 61.67 rad/s².

Finding Initial Angular Velocity

  • Carousel ride spins counterclockwise (positive direction) with unknown initial angular velocity ω₀.
  • Acceleration α = 2 rad/s², time t = 6 s, angular displacement Δθ = 40 rad.
  • Use Δθ = ω₀t + ½αt² → ω₀ = (Δθ - ½αt²)/t = (40 - 0.5×2×36)/6 = 0.67 rad/s.

Angular Acceleration from Rest

  • CD starts from rest (ω₀ = 0), reaches ω = -30 rad/s, and rotates Δθ = -90 rad.
  • Use ω² = ω₀² + 2αΔθ → α = (ω² - ω₀²)/(2Δθ) = ((-30)² - 0)/(2×-90) = -5 rad/s².

Key Terms & Definitions

  • Angular Position (θ) — The angle describing an object's rotation, measured in radians.
  • Angular Displacement (Δθ) — The change in angular position.
  • Angular Velocity (ω) — The rate of change of angular position (radians/sec or RPM).
  • Angular Acceleration (α) — The rate of change of angular velocity (radians/sec²).
  • Counterclockwise Rotation — Positive direction for angles and angular quantities.
  • Clockwise Rotation — Negative direction for angles and angular quantities.

Action Items / Next Steps

  • Practice similar angular motion problems to reinforce concepts.
  • Review and memorize key angular motion equations and unit conversions.
  • Prepare any questions for the next class or office hours.

Full transcript