Formula used
For a constant torque acting through an angular displacement, rotational work is: W = τ × θ
- W is work (joules, J)
- τ is torque (newton-meters, N·m)
- θ is angular displacement (radians, rad)
Note: If torque varies with angle, the more general form is W = ∫ τ(θ) dθ. This tool uses the constant-torque approximation.
How to use this calculator
- Enter the torque value and select its unit.
- Choose whether torque assists or opposes rotation.
- Enter angular displacement and select its unit.
- Press Calculate Work to view results above the form.
- Use Download CSV or Download PDF for reports.
Example data table
| Case | Torque | Angle | Direction | Work (J) |
|---|---|---|---|---|
| 1 | 10 N·m | 2 rad | Assisting | 20 |
| 2 | 5 lbf·ft | 90 deg | Assisting | 5 lbf·ft × π/2 ≈ 10.64 |
| 3 | 120 N·cm | 0.5 rev | Opposing | −(1.2 N·m) × (π) ≈ −3.77 |
Use your own measurements for accurate engineering results.
Rotational work in real systems
1) Why torque produces work
When a moment causes rotation through an angle, energy is transferred. For constant torque, work scales linearly with angular displacement. Doubling torque or doubling angle doubles the work output.
2) Core relationship and units
The calculator uses W = τ × θ, with τ in N·m and θ in radians. The product gives joules, the same unit as linear work. Angle must be in radians for direct SI consistency.
3) Useful unit conversions
Common workshop values appear in lbf·ft, lbf·in, or N·cm. The tool converts these to N·m before computing work. Angles can be entered as degrees or revolutions and are converted to radians.
Helpful reference points: 1 lbf·ft ≈ 1.3558 N·m and 1 lbf·in ≈ 0.1130 N·m. Also, 1 revolution equals 2π radians, and 90° equals π/2 radians. These benchmarks help catch unit-entry mistakes early.
4) Sign convention and interpretation
Work is positive when torque assists the direction of rotation. It becomes negative when torque resists motion, such as braking. Negative work indicates energy removed from the rotating system.
5) Typical magnitudes and quick checks
A 10 N·m torque over 2 rad gives 20 J. A small motor delivering 1 N·m over one revolution produces about 6.283 J. These quick checks help validate inputs before exporting results.
6) Engineering applications
Use rotational work for flywheels, crankshafts, winches, and robotic joints. It supports estimating energy per cycle, evaluating actuator sizing, and comparing different gearing strategies for the same load requirement.
For example, 25 N·m applied through 0.25 rev equals about 39.27 J. In repetitive mechanisms, multiplying work per cycle by cycles per minute gives an energy throughput estimate. If a shaft delivers τ = 2 N·m at ω = 10 rad/s, power is P = τω = 20 W. Over 5 s, that corresponds to 100 J, consistent with work when torque and speed remain steady.
7) Measurement tips and variable loads
Torque can come from a torque wrench, load cell arm, or motor data. Angle may be read from encoders, protractors, or rotation counts. Keep units consistent, and record the direction of torque carefully.
Real torque can vary with angle due to friction, springs, or changing loads. In that case, work is the area under the τ–θ curve. A practical approach is to average torque over the rotation interval.
This page reports joules, kilojoules, and ft·lbf to match common workflows. Exporting helps document lab runs, maintenance checks, and design assumptions.
FAQs
1) Why must the angle be in radians?
Radians make the work equation consistent in SI units. Degrees must be converted first, otherwise the computed joules will be incorrect by a factor of 180/π.
2) Can rotational work be negative?
Yes. If torque opposes the rotation direction, the work is negative. This represents energy extracted from the rotating system, such as during braking or damping.
3) What if torque changes during rotation?
The exact work is the integral of torque over angle. If you only have average torque for the interval, this calculator provides a practical estimate using constant-torque behavior.
4) Is N·m the same as a joule?
They share the same base units, but represent different concepts. N·m is torque, while J is work or energy. Work equals torque multiplied by angular displacement in radians.
5) How do I use revolutions as input?
Select “rev” and enter the number of turns. The tool converts revolutions to radians using 2π per revolution before calculating work.
6) What is the difference between work and power?
Work is energy transferred over an angle. Power is the rate of doing work. If you know time, power equals work divided by time, or τ times angular speed.
7) When should I trust ft·lbf output?
Use ft·lbf when your torque data is in lbf·ft and you want consistent imperial reporting. The calculator converts from joules using a standard factor for clear comparisons.