Angular speed
Positive angular speed magnitude.
Physics · Mechanics · Motion & Energy
Uniform circular motion calculator
Convert frequency, period, angular speed or tangential speed into the complete circular-motion quantities. Choose the mass mode to also calculate centripetal force.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Tangential speed
Speed magnitude on the circular path.
Centripetal acceleration
Inward acceleration magnitude; tangential acceleration is zero.
Frequency
Number of complete cycles per second.
Period
Duration of one complete cycle.
Centripetal force
Required inward resultant for the supplied mass.
Relation and numerical substitution
f = 1 / T—
Uniform circular motion; all radius and speed quantities strictly positive.
01Formulas and symbols
Formulas used
f
f = 1 / TNumber of complete cycles per second.ω
ω = 2π × fPositive angular speed magnitude.ω
ω = v / rPositive angular speed magnitude.f
f = ω / (2π)Number of complete cycles per second.T
T = 1 / fDuration of one complete cycle.v
v = ω × rSpeed magnitude on the circular path.a_c
a_c = ω² × rInward acceleration magnitude; tangential acceleration is zero.F_c
F_c = m × a_cRequired inward resultant for the supplied mass.02Assumptions and limits
Scope of validity
- Uniform circular motion; all radius and speed quantities strictly positive.
03Validation example
Reference numerical case
- r = 0.5 m, f = 2 Hz: T = 0.5 s, ω ≈ 12.5663706 rad/s, v ≈ 6.2831853 m/s and a_c ≈ 78.9568352 m/s². With m = 2 kg, F_c ≈ 157.9136704 N.
04References
Technical references
FAQ
What is the difference between angular speed and linear speed?
Angular speed ω measures how fast the angle changes, while linear speed v measures distance travelled along the circular path per unit time. They are related by v = ωr.
Why is there acceleration when the speed magnitude is constant?
Velocity includes direction as well as magnitude. In uniform circular motion the direction changes continuously, producing centripetal acceleration a_c = v²/r toward the centre even when the speed magnitude is constant.
05How it works
A revolution covers 2π radians and a path length 2πr. Uniform speed still requires acceleration because velocity direction changes. Centripetal acceleration points inward and grows with the square of angular speed; doubling radius at fixed angular speed doubles it.
06Limits of the model
Uniform circular motion with positive radius, frequency and speed magnitudes. No tangential acceleration is included. The force is the required inward resultant, not an additional force to add to tension, friction or gravity. Mass is only required in the force mode.
07Common mistakes
Do not confuse Hz (revolutions per second) with rad/s: ω = 2πf. At fixed tangential speed, increasing radius reduces centripetal acceleration; at fixed ω it increases it.