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

0.000
rad
Angular Displacement
0.00
rotations
Complete Turns
Small rotation
Less than quarter turn
0.0
degrees
In Degrees

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

Quarter turnπ/2 rad = 90°
Half turnπ rad = 180°
Full turn2π rad = 360°
1 radian≈ 57.3°
1 degree≈ 0.0175 rad
1 revolution= 2π rad

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.