Calculator
Pick a mode: forward energy, inverse solvers, or the rolling comparison.
Rolling without slipping: ω = v/r, Etotal = ½mv² + ½Iω².
Worked example. Solid cylinder, m = 2 kg, r = 0.5 m → I = ½·2·0.5² = 0.25 kg·m².
At 100 RPM = 10.471975512 rad/s, E = ½·0.25·10.471975512² = 13.7077838904 J
(0.0137077839 kJ, 0.0038077177 Wh). Try the defaults above: they reproduce this exactly.
Moment of inertia library
Every formula is checked for dimensions (kg·m²) and for the hoop limit: a hollow cylinder with ri → ro gives I → m·ro², exactly the hoop.
| Shape (axis) | Formula |
|---|---|
| Solid cylinder / disc, central axis | I = 1/2 m r² |
| Hollow cylinder / pipe, central axis | I = 1/2 m (r_o² + r_i²) |
| Hoop / thin ring | I = m r² |
| Solid sphere | I = 2/5 m r² |
| Hollow sphere / thin shell | I = 2/3 m r² |
| Rod, axis through center | I = 1/12 m L² |
| Rod, axis through one end | I = 1/3 m L² |
| Point mass at distance r | I = m r² |
Frequently asked questions
Every worked number below was recomputed with 50-digit arithmetic.
What is rotational kinetic energy?
Rotational kinetic energy is the energy stored in a spinning rigid body: E_rot = 1/2 x I x omega^2, where I is the moment of inertia in kg m^2 and omega is the angular velocity in rad/s. Worked example: a solid cylinder with m = 2 kg and r = 0.5 m has I = 0.25 kg m^2; spun at 100 RPM (10.471975512 rad/s) it stores E = 13.7077838904 J, which is 0.0137077839 kJ or 0.0038077177 Wh.
What are the moment of inertia formulas for common shapes?
The calculator's built-in library: solid cylinder/disc I = 1/2 m r^2; hollow cylinder/pipe I = 1/2 m (r_outer^2 + r_inner^2); hoop/thin ring I = m r^2; solid sphere I = 2/5 m r^2; hollow sphere/thin shell I = 2/3 m r^2; rod about its center I = 1/12 m L^2; rod about one end I = 1/3 m L^2; point mass I = m r^2. Example: a hollow cylinder with m = 3 kg, r_outer = 0.4 m, r_inner = 0.3 m gives I = 0.375 kg m^2.
How do I convert RPM or degrees per second to rad/s?
Use omega(rad/s) = RPM x 2 x pi / 60, and omega(rad/s) = (deg/s) x pi / 180. Reference values: 60 RPM = 6.2831853072 rad/s; 1800 RPM = 188.4955592154 rad/s; 3000 RPM = 314.1592653590 rad/s; 360 deg/s = 6.2831853072 rad/s (exactly one revolution per second); 90 deg/s = 1.5707963268 rad/s. The calculator accepts rad/s, RPM, and deg/s directly.
How much of a rolling object's energy is rotation vs translation?
For rolling without slipping, omega = v / r, so E_total = 1/2 m v^2 + 1/2 I omega^2. Worked example in the Rolling tab: a solid sphere with m = 1 kg, r = 0.1 m rolling at v = 5 m/s has E_trans = 12.5 J and E_rot = 5 J, for a total of 17.5 J - rotation is 28.571% of the total (exactly 2/7). The tab shows this split for solid cylinders, hoops, and solid or hollow spheres.
Why does a hoop roll slower than a solid cylinder down a ramp?
A hoop concentrates its mass at the rim, so a larger share of its energy goes into rotation: 1/(1+1) = 50% for a hoop, versus 1/(1+0.5) = 33.333% for a solid cylinder and 40% for a hollow sphere. Energy locked in rotation cannot drive forward motion, so the hoop is slower. The Rolling tab reports each shape's rotational share.
Can I solve for omega or I instead of energy?
Yes. The inverse solvers use omega = sqrt(2E / I) and I = 2E / omega^2, and every direction round-trips exactly. Example: E = 13.7077838904 J with I = 0.25 kg m^2 gives omega = 10.471975512 rad/s, which is 100 RPM or 600 deg/s.
I need flywheel energy storage engineering, not textbook physics. Is this the right tool?
For storage engineering, use our flywheel energy storage calculator instead: it covers disc and rim rotors, specific energy in Wh/kg, and the honest material hoop-stress limit (sigma = rho x v^2 vs tensile strength). This tool is the general-physics rotational kinetic energy calculator: 8 rigid-body inertia shapes, rad/s / RPM / deg/s conversion, inverse solvers, and the rolling-vs-sliding comparison.
Does this handle relativistic or recoil kinetic energy?
No - this tool is classical (Newtonian) rotational kinetic energy only. For particles moving near the speed of light, use our relativistic kinetic energy calculator; for firearm/gun recoil energy, use our recoil energy calculator.
What are the most common mistakes when computing rotational KE?
Four traps: (1) entering deg/s as rad/s gives a 57.2958x error - always convert; (2) moment of inertia is kg m^2, not kg m; (3) for a hollow cylinder the inner radius cannot exceed the outer radius; (4) a hoop is the limit of a hollow cylinder as r_inner approaches r_outer - 1/2 m (r_o^2 + r_i^2) becomes m r_o^2 - and the calculator enforces this consistency.
Related tools
Same physics family, different jobs: pick the one that matches your question.
Flywheel Energy Storage CalculatorDisc/rim rotors, Wh/kg, and the honest material hoop-stress limit for storage engineering.
Relativistic KE CalculatorKinetic energy of particles near light speed, where 1/2 mv^2 breaks down.
Recoil Energy CalculatorFirearm/gun recoil energy from projectile and powder momentum.