About

Angular displacement θ quantifies the angle through which a rigid body rotates about a fixed axis. Errors in computing θ propagate directly into torque estimates, gear-ratio sizing, and servo-motor positioning. A miscalculated θ of even 0.5rad in a CNC spindle cycle can scrap an entire production batch. This calculator implements the four standard rotational kinematic equations assuming constant angular acceleration α. You select which variables are known and the tool solves for θ algebraically. Results convert between radians, degrees, and full revolutions in real time.

The underlying model assumes a rigid body with no slip and constant α over the interval. For non-constant acceleration profiles (e.g., cam lobes, servo ramps), you must integrate α(t) numerically. This tool does not handle that case. Pro tip: if you measure ω with a tachometer, convert RPM to rad/s before entering values (multiply by 2π÷60).

Formulas

The four rotational kinematic equations for constant angular acceleration α:

Equation 1 (known: ω₀, α, t):

θ = ω₀t + 12αt²

Equation 2 (known: ω₀, ω, t):

θ = 12(ω₀ + ω)t

Equation 3 (known: ω, α, t):

θ = ωt − 12αt²

Equation 4 (known: ω₀, ω, α):

θ = ω² − ω₀²2α

Where: θ = angular displacement (rad). ω₀ = initial angular velocity (rad/s). ω = final angular velocity (rad/s). α = angular acceleration (rad/s²). t = elapsed time (s).

Unit conversion factors: 1 rev = 2π rad = 360°. To convert degrees to radians: multiply by π÷180.

Reference Data

Quantity	Symbol	SI Unit	Common Alt. Units	Typical Range (Machinery)
Angular Displacement	θ	rad	°, rev	0 - 10⁶ rad
Initial Angular Velocity	ω₀	rad/s	RPM, °/s	0 - 10,000 rad/s
Final Angular Velocity	ω	rad/s	RPM, °/s	0 - 10,000 rad/s
Angular Acceleration	α	rad/s²	°/s², rev/s²	−500 - 500 rad/s²
Time	t	s	ms, min	0 - 3600 s
1 Revolution	-	2π rad	360°	-
1 Degree	-	0.01745 rad	π÷180 rad	-
Earth Rotation Rate	ω_E	7.2921×10⁻⁵ rad/s	1 rev/day	-
Ceiling Fan (Low)	-	≈ 10 rad/s	≈ 100 RPM	-
Car Engine Idle	-	≈ 73 rad/s	≈ 700 RPM	-
Hard Drive (7200 RPM)	-	≈ 754 rad/s	7200 RPM	-
Turbocharger	-	≈ 15,708 rad/s	≈ 150,000 RPM	-
Dental Drill	-	≈ 41,888 rad/s	≈ 400,000 RPM	-
Flywheel Energy Storage	-	≈ 6,283 rad/s	≈ 60,000 RPM	-
Neutron Star (Fastest)	-	≈ 4,500 rad/s	≈ 716 rev/s	-

Frequently Asked Questions

Division by zero makes Equation 4 undefined. When α = 0, angular velocity is constant (ω = ω₀), and displacement reduces to θ = ω₀t. The calculator detects this case and switches to Equation 1 automatically, which degenerates correctly to uniform rotation.

Yes. A negative θ indicates rotation in the clockwise direction (assuming the standard convention where counter-clockwise is positive). This occurs when a body decelerates past zero velocity and reverses, or when initial velocity and acceleration oppose each other. The calculator preserves the sign to convey directional information.

All angular velocity inputs entered in °/s are internally converted to rad/s by multiplying by π÷180. Acceleration in °/s² follows the same conversion. The result is computed in radians and then displayed in all three unit systems simultaneously. The conversion is exact and introduces no floating-point drift beyond IEEE 754 limits (~15 significant digits).

They use different known variables. Equation 1 uses ω₀ (initial velocity), while Equation 3 uses ω (final velocity). If you supply consistent values where ω = ω₀ + αt, both equations yield identical θ. Discrepancies indicate inconsistent inputs. The calculator flags this with a warning.

No. All four equations assume α is constant over the time interval. For variable acceleration (e.g., a servo motor following an S-curve profile), you must integrate α(t) over time. Piecewise-constant approximations work: split the motion into intervals of constant α, compute θ for each, and sum. Each interval's final ω becomes the next interval's ω₀.

Arc length s = rθ, where r is the radial distance from the rotation axis and θ is in radians. A point at r = 0.5 m undergoing θ = 10 rad travels 5 m along its circular path. This is critical for belt-drive calculations and grinding wheel surface speed limits.