Angular Displacement Calculator
Calculate angular displacement from radius, angular velocity, or angular acceleration
Calculate Angular Displacement
Arc length along circular path
Distance from center to edge
Angular Displacement Results
Formula used:
θ = s/r = 0.00/0.00 = 0.000 rad
Rotational Motion Analysis
Example Calculation
Jogging on Circular Track
Scenario: Runner jogging around a circular park track
Track radius: 9 meters (from fountain to edge)
Distance covered: 185 meters (measured by smartwatch)
Application: Sports analysis, exercise tracking
Calculation
θ = s/r
θ = 185 m / 9 m
θ = 20.556 radians
θ = 20.556 × (180°/π) = 1177.2°
Number of rotations = 20.556 / (2π) = 3.27 rotations
Result: The runner completed 3.27 full rotations around the track.
Angular Units
Angular Motion Facts
Angular displacement is a vector quantity with magnitude and direction
Radians are the natural unit for angular measurements
θ = s/r relates arc length to radius
Angular velocity is the rate of angular displacement change
Angular acceleration causes changes in angular velocity
Understanding Angular Displacement and Rotational Motion
What is Angular Displacement?
Angular displacement (θ) is the angle through which an object rotates about a fixed axis. It's measured from the initial position to the final position and is typically expressed in radians, degrees, or revolutions. Unlike linear displacement, angular displacement describes rotational motion.
Key Properties
- •Vector quantity with magnitude and direction
- •Positive for counterclockwise rotation
- •Negative for clockwise rotation
- •Independent of object's size or mass
Angular Displacement Formulas
θ = s/r
From arc length and radius
θ = ω × t
From angular velocity
θ = ω₀t + ½αt²
From angular acceleration
θ: Angular displacement (rad)
s: Arc length (m)
r: Radius (m)
ω: Angular velocity (rad/s)
α: Angular acceleration (rad/s²)
t: Time (s)
Rotational Kinematics
Angular Motion Equations (analogous to linear motion)
Linear: v = u + at
Angular: ω = ω₀ + αt
Linear: s = ut + ½at²
Angular: θ = ω₀t + ½αt²
Rotational motion follows similar patterns to linear motion, with angular quantities replacing linear ones. The relationship between linear and angular quantities depends on the radius of rotation.
Applications and Examples
Engineering
Motor design, gear systems, rotating machinery analysis, and mechanical engineering calculations.
Sports & Exercise
Track analysis, spinning motions, gymnastics rotations, and athletic performance measurement.
Astronomy
Planetary rotation, orbital mechanics, satellite motion, and celestial body positioning calculations.