Angular velocity
Angular velocity around the fixed axis.
Physics · Mechanics · Motion & Energy
Rotational kinematics calculator
Find final angular velocity and angular displacement under constant angular acceleration, or solve the time-free inverse relations.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Angle
Signed angular displacement about the fixed axis.
Angular acceleration
Signed change in angular velocity per unit time.
Relation and numerical substitution
ω = ω₀ + α × t—
Fixed axis; constant α; elapsed time ≥ 0; signed angular quantities.
01Formulas and symbols
Formulas used
ω
ω = ω₀ + α × tAngular velocity around the fixed axis.θ
θ = ω₀ × t + ½ × α × t²Signed angular displacement about the fixed axis.ω
ω = √(ω₀² + 2 × α × θ)Angular velocity around the fixed axis.α
α = (ω² − ω₀²) / (2 × θ)Signed change in angular velocity per unit time.02Assumptions and limits
Scope of validity
- Fixed axis; constant α; elapsed time ≥ 0; signed angular quantities.
03Validation example
Reference numerical case
- ω₀ = 2 rad/s, α = 3 rad/s², t = 4 s: ω = 2 + 3 × 4 = 14 rad/s; θ = 2 × 4 + ½ × 3 × 4² = 32 rad.
04References
Technical references
FAQ
Are rotational kinematic equations the same as linear kinematics?
They have the same mathematical structure for constant acceleration, with displacement x replaced by angle θ, velocity v by angular velocity ω, and acceleration a by angular acceleration α.
When can constant-angular-acceleration equations be used?
Use them when a body rotates about a fixed axis and angular acceleration can be treated as constant over the interval. If α varies significantly with time or angle, a different model is required.
05How it works
These relations are the rotational counterparts of linear constant-acceleration motion. Choose counterclockwise as positive and use that convention for ω₀, ω, α and θ. The time mode handles reversing rotation; the time-free speed mode chooses the positive square root only.
06Limits of the model
Rotation about one fixed axis with constant angular acceleration. θ is signed angular displacement, not an absolute angle or a number of revolutions unless converted. No torque or moment-of-inertia calculation is included.
07Common mistakes
Convert rpm to rad/s and degrees to radians through the selectors. A negative α does not always slow the rotation: compare its sign with ω. The positive time-free root cannot represent a negative final ω.