Power Transmission Formula
Reference for P = T × ω and P = F × v power transmission formulas for shafts, belts, and gears.
Covers torque, angular velocity, and motor sizing efficiency.
The Formula
P = T × ω
P = (2π × T × n) / 60
P (hp) = (T × n) / 5252 (T in ft·lb, n in RPM)
P = (2π × T × n) / 60
P (hp) = (T × n) / 5252 (T in ft·lb, n in RPM)
Power transmission formulas relate the torque on a rotating shaft to the power it delivers. Higher torque or higher speed means more power transmitted.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| P | Power transmitted | Watts (W) or horsepower (hp) |
| T | Torque | N·m or ft·lb |
| ω | Angular velocity | rad/s |
| n | Rotational speed | RPM (revolutions per minute) |
Example 1
A motor shaft delivers 50 N·m at 1500 RPM. Find the power in watts.
P = (2π × 50 × 1500) / 60
= (2 × 3.14159 × 50 × 1500) / 60
= 7,854 W ≈ 7.85 kW
Example 2
An engine produces 200 ft·lb of torque at 3000 RPM. Find the horsepower.
P = (T × n) / 5252 = (200 × 3000) / 5252
= 114.2 hp
When to Use It
Use the power transmission formula when:
- Selecting motors and engines for mechanical systems
- Designing shaft sizes for required power loads
- Calculating belt and chain drive specifications
- Converting between torque, RPM, and power ratings
Key Notes
- These formulas assume 100% mechanical efficiency; real shafts, gears, and bearings have losses — multiply by efficiency (η) to get actual output: P_out = η × T × ω, where η is typically 0.90–0.98 for precision gearboxes
- The constant 5252 in the hp formula comes from (33,000 ft·lb/min ÷ 2π) — horsepower and torque curves always cross at exactly 5252 RPM for any engine, making it a useful checkpoint value
- Unit consistency is critical: ω must be in rad/s (not RPM) for the P = Tω formula in SI units; convert RPM to rad/s with ω = 2π × n / 60
- For belt and chain drives, power is transmitted as P = F × v, where F is the effective tension (tight-side minus slack-side) and v is belt velocity in m/s — this linear form is equivalent to T × ω at the pulley