Example 1
I = 0.5, ω = 10 → E = 25 J.
Rotational kinetic energy calculator
Type I [kg·m²] and ω [rad/s], or f [Hz] instead of ω. The calculator computes Ek = ½Iω²: 0.5 and 10 rad/s is 25 J, and at f = 2 Hz the same I is about 39.5 J, because ω = 4π.
I from mass and r: I = m·r². Translation, the same joule: Eₖ = ½mv².
Result
Moment of inertia I, Angular speed ω (rad/s) and Frequency f (Hz, optional → ω = 2πf). The result shows up here.
How it works
Rotational kinetic energy is Eₖ = ½Iω². The analogy with ½mv² is plain: mass becomes I, linear speed becomes ω. I = 0.5 kg·m² and ω = 10 rad/s give Eₖ = 25 J. I = 4 and ω = 1 give 2 J. I = 2 and ω = 3 give 9 J.
The I field is kg·m². The ω field is radians per second, not rpm. Instead of ω you can type f in hertz: then the calculator uses ω = 2πf and f wins. I = 0.5 and f = 2 Hz is ω = 4π ≈ 12.57 rad/s and Eₖ ≈ 39.5 J.
Energy grows with the square of ω. I = 1 and ω = 5 give 12.5 J; the same I at ω = 10 gives 50 J, four times more. I = 0.05 and ω = 100 give 250 J. I = 0.01 and f = 50 Hz is ω = 100π and Eₖ ≈ 493 J.
Point-mass I is on the neighbouring page, I = m r². The calculator example: m = 2 kg, r = 0.5 m gives I = 0.5, and here at ω = 8 you get Eₖ = 16 J. Rigid bodies (disk, cylinder) have other factors; here I is already a given.
Type I and either ω or f, then Calculate. The 0.5 and 10 example is 25 J. A comma in 0.5 works. ω and f together: the calculator listens to f. Angular momentum L = Iω is a different quantity, kilogram-meter-squared per second, not a joule.
Translational ½mv² sits on kinetic energy. Same joule, different motion. Torque M = F r sinα twists, but it is not this energy. ω = 2π/T, when you know the period, lives on angular velocity; paste that ω here.
How to use
- Type moment of inertia I in kg·m², for example 0.5. Point I = m r² lives on its own page; here I is a given.
- Type ω in rad/s, for example 10, or leave ω blank and enter f in hertz. If f is present, the calculator uses ω = 2πf and f wins.
- Click Calculate. I = 0.5 and ω = 10 give 25 J. I = 0.5 and f = 2 Hz give about 39.5 J, because ω = 4π.
- I and the ω you use must describe a rotation: I positive, and ω or f nonzero if you want nonzero energy.
- Compare with ½mv² on kinetic energy. Get I from m and r on point I. L = Iω is not this card.
Formula
Ek = ½Iω2
ω = 2πf (when f is given)
Ek, I, ω, and f for rotation
Rotation: Ek = ½Iω². I = 0.5 kg·m² and ω = 10 rad/s give 25 J. At f = 2 Hz the same I is about 39.5 J, because ω = 4π.
- Eₖ
- Rotational energy [J]. ½ × 0.5 × 100 = 25 J. Not ½ m v² from translation.
- I
- Moment of inertia [kg·m²]. 0.5 at ω = 10 is 25 J. Get I from m and r next door.
- ω
- Angular speed [rad/s], not rpm. 10 rad/s. Typed f overrides ω.
- f
- Optional frequency [Hz]. 2 Hz gives ω = 4π and Ek about 39.5 J at I = 0.5.
Real-life examples
Example 2
I = 0.5, f = 2 Hz → ω = 4π.
Example 3
I = 2 kg·m², ω = 3.
Example 4
I = 0.05 kg·m², ω = 100.
Example 5
I = 0.01 kg·m², f = 50.
Example 6
I = 10 kg·m², ω = 2.
Example 7
I = 4 kg·m², ω = 1 → E = 2 J.
Example 8
I = 0.5 from m=2,r=0.5; ω = 8.
Example 9
I = 1, ω = 10 vs 5: E ×4.
Ways to use this calculator
- A flywheel problem: I and ω give the stored joules.
- An f → ω conversion at known I, for example 50 Hz and I = 0.01 ≈ 493 J.
- A comparison of ½Iω² with ½mv² at the same unit J.
Frequently asked questions
How much Eₖ at I = 0.5 kg·m² and ω = 10 rad/s?
Energy is 25 J. That is ½ × 0.5 × 100. At I = 2 and ω = 3 you get 9 J. At I = 4 and ω = 1 you get 2 J.
How much at I = 0.5 and f = 2 Hz?
First ω = 4π, about 12.57 rad/s, so Ek about 39.5 J. Typed f wins over a blank or typed ω.
Can I type ω in rpm?
No. The field wants radians per second. One revolution per second is f = 1 Hz and ω = 2π, about 6.28 rad/s, not 60 in field ω.
Where does I come from if I know m and r?
For a point mass, I = m r² on the moment-of-inertia page. Mass 2 kg and r = 0.5 m give I = 0.5; here at ω = 8 rad/s that is 16 J.
Why does doubling ω give four times Eₖ?
Because ω is squared. At I = 1 kg·m² and ω = 5 rad/s you have 12.5 J; at ω = 10 rad/s you have 50 J.
Is ½Iω² the same as L = Iω?
No. L is angular momentum, in kg·m²/s. Ek is in joules. Same I and ω, two formulas and two cards.
Can I type 0.5 with a comma?
Yes. 0.5 and 0,5 are the same I [kg·m²]. A comma and a period mean the same value.
How much at I = 0.05 and ω = 100?
Energy is 250 J. At I = 10 and ω = 2 you get 20 J. Large ω lifts Ek harder than I does.
How much at I = 0.01 and f = 50 Hz?
Here ω = 100π, about 314 rad/s, so Ek about 493 J. Mains 50 Hz at a small I still gives hundreds of joules.
How is this different from ½mv²?
Here rotation: I and ω. There translation: m and v. Same unit, the joule, different fields.
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.