Engineering Calculators
● P = T·ω Guide

Torque, Power & Speed: The P = T·ω Relationship

A plain-language guide to the torque–power–speed relationship P = T·ω: why it holds for any rotating shaft, the unit traps that catch people out, and how motor torque relates to current.

Open the Speed / Torque / Power Calculator →

One equation ties them together

Torque, power, and speed on a rotating shaft aren't three independent quantities — fix any two and the third is determined by one exact relationship:

P = T · ω

power equals torque times angular velocity. It holds for any rotating shaft regardless of what's driving it — motor, engine, turbine — because it's just the rotational form of "power = force × velocity." This is the equation behind every "what motor do I need?" conversation.

What it actually tells you

The practical consequence is the trade-off it forces: for a given power, torque and speed are inversely related. A motor delivering a fixed power makes high torque only at low speed and low torque at high speed. That's why gearboxes exist — to trade the speed a motor is good at for the torque a load needs, keeping the power (minus losses) the same on both sides. Want more torque at the wheel? Gear it down, and it spins slower in exact proportion.

The unit trap that catches everyone

The equation is only clean in consistent SI units: power in watts, torque in newton-metres, and angular velocity in radians per second. The mistake people make is plugging in RPM directly. Speed almost always comes as RPM, and you must convert: ω (rad/s) = RPM × 2π / 60. Skip that factor of 2π/60 and your answer is off by ~9.55×. Mixing horsepower, lb-ft, and RPM brings its own conversion constants — the underlying physics is identical, only the bookkeeping changes.

Relating torque to motor current

For a permanent-magnet motor there's a second useful relationship: torque is roughly proportional to current, T = Kt · I, where Kt is the torque constant. It's a handy cross-check — estimate torque from the current you're drawing — but it assumes an ideal linear machine. Real motors deviate near magnetic saturation or at very high current, so treat it as an estimate, not a precise figure.

Worked conversions, efficiency & checklist

A worked RPM→torque calculation, how efficiency turns mechanical output into electrical input, and a checklist so the units never bite you.

A worked example

A motor delivers 10 kW at 3000 RPM — what's the shaft torque? First convert speed: ω = 3000 × 2π/60 = 314.2 rad/s. Then T = P/ω = 10000 / 314.2 = 31.8 N·m. Double the speed to 6000 RPM at the same power and the torque halves to 15.9 N·m — the inverse relationship in action. This is the whole calculation; the only place to go wrong is the RPM→rad/s conversion.

From shaft power to electrical input

P = T·ω gives mechanical shaft power. The electrical power you draw from the battery is higher, because the motor and inverter aren't perfect: P_elec = P_mech / η, where η is the efficiency (motoring). At 95% efficiency, 10 kW of shaft power needs ~10.5 kW electrical, and the missing 0.5 kW becomes heat. For regeneration the relationship inverts — mechanical input becomes electrical output, times efficiency.

Checklist

  1. Convert RPM to rad/s (× 2π/60) before using P = T·ω — this is the #1 error.
  2. Keep SI consistent: watts, N·m, rad/s (or use a tool that tracks the conversions).
  3. Remember torque and speed trade inversely at fixed power — size the gearbox accordingly.
  4. Use T = Kt·I only as an estimate for PM motors; it breaks down near saturation.
  5. Divide by efficiency to get electrical input; the shortfall is heat to be cooled.

← All guides · Open the Speed / Torque / Power Calculator