Example 1
One turn takes 8 s.
Angular Velocity Calculator
Type period T [s]. The calculator computes ω = 2π/T: 8 s is about 0.785 rad/s, 0.125 Hz and 7.5 rpm, and 6 s is about 1.047 rad/s. Zero seconds does not divide.
From ω to v: v = ω·r. Angular momentum: L = I·ω. Energy: Eₖ = ½Iω².
Result
Period T (s). The result shows up here.
How it works
Angular velocity from the period is ω = 2π / T, and frequency is f = 1 / T. At T = 8 s you get ω = π/4 ≈ 0.785 rad/s, f = 0.125 Hz and 7.5 rpm, because rpm = 60 f. At T = 6 s you get ω ≈ 1.047 rad/s and f ≈ 0.167 Hz. One turn per second is T = 1 s and ω = 2π ≈ 6.283 rad/s.
One field: T in seconds, the time of a full turn. Result ω is in rad/s, f in hertz, and rpm sits with the result. Do not type rpm in the T field. A comma and a period in 8.5 are the same period.
T must be positive. Zero rejects the model: there is no period. The field needs a number before 0.785 rad/s can appear. ω is always in radians per second, not degrees: 360°/s is exactly 2π rad/s.
Rim speed is v = ω r on its own card. L = Iω takes this same ω. Rotational energy ½Iω² is joules, not rad/s.
At T = 1.8 s, ω ≈ 3.49 rad/s and f ≈ 0.556 Hz. At T = 2 s, ω = π ≈ 3.142 rad/s and f = 0.5 Hz, or 30 rpm. At T = 0.5 s, ω ≈ 12.57 rad/s and f = 2 Hz.
Type 8, click Calculate, and match about 0.785 rad/s plus 7.5 rpm. Then try 6 s. With that ω, open v = ω r or L = Iω.
How to use
- Type period T in seconds, the time of one full turn, for example 8.
- Click Calculate. The calculator computes ω = 2π/T and f = 1/T. 8 s gives ω ≈ 0.785 rad/s and f = 0.125 Hz.
- Read rpm too: at T = 8 s it is 7.5 rpm, because rpm = 60 f.
- T must be greater than zero. Zero rejects a period. Do not type rpm in the T field.
- Want rim speed? Open v = ω r. L = Iω takes this same ω.
Formula
ω = 2π / T
f = 1 / T
T in seconds, ω in rad/s, f in Hz. Also rpm = 60·f.
ω, T, and f for rotation
From the period of one turn: ω = 2π/T and f = 1/T. T = 8 s is about 0.785 rad/s, 0.125 Hz and 7.5 rpm.
- ω
- Angular speed [rad/s]. 2π/8 ≈ 0.785. Input to v = ω r on the next card.
- T
- Period [s]. 8 s for a full turn. T must be positive. Not work time t = W/P.
- f
- Frequency 1/T [Hz]. 0.125 Hz. rpm = 60·f = 7.5.
- π
- The constant in 2π/T. A full turn is 2π radians.
Real-life examples
Example 2
Period ≈ 1.8 s per turn.
Example 3
T = 0.05 s per turn.
Example 4
Second hand: T = 60 s.
Example 5
Minute hand: T = 3600 s.
Example 6
While driving: nominal T = 0.2 s per wheel turn.
Example 7
T = 0.02 s.
Example 8
Full circle in 120 s.
Example 9
Earth: T ≈ 86164 s.
Example 10
T = 0.02 s per shaft turn.
Example 11
One turn in 6 s.
Frequently asked questions
How much ω at T = 8 s?
Angular speed is about 0.785 rad/s. That is 2π / 8. The calculator also shows f = 0.125 Hz and 7.5 rpm.
Which units do I type?
Period T [s]. Result ω [rad/s], plus f [Hz] and rpm. Type a minute as 60, not as 1.
What if T = 0?
The calculator refuses zero seconds. ω = 2π/T at zero period does not exist.
Where is v = ω r?
On the linear-from-angular page. There you multiply this ω by r. Here you only get ω from the period.
How much at T = 6 s?
You get about 1.047 rad/s, f ≈ 0.167 Hz. Six seconds per turn, close to one radian per second.
Is ω in degrees per second?
No. The field and the result are in radians per second. A full turn is 2π rad, not 360 in the denominator.
Does a comma in 8.5 work?
Yes. 8.5 and 8,5 are the same T [s]. A comma and a period mean the same value.
How do I get rpm from f?
rpm = 60 · f. At T = 8 s you have f = 0.125 Hz, so 7.5 rpm. The calculator computes that beside ω.
How much at T = 1.8 s?
Angular speed is about 3.49 rad/s, f ≈ 0.556 Hz. A fast turn, more than three radians per second.
Where is L = Iω?
On the angular-momentum page. There you multiply I by this ω. Here only ω = 2π/T.
Knowledge sources
The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.
Page updated in 2026.