Rotational Work
Energy transferred by a torque acting over an angular displacement.
W = τ · θ
- τ (Tau): Torque (N·m)
- θ (Theta): Angle (Radians)
Rotational Work Calculator
Calculate rotational work from torque and angle or rotational power from torque and angular velocity.
Rotational Work
Calculate work done by Torque over an Angle or Rotational Power.
Physics Formula
W = τ · θ
Rotational Physics
Just as linear work is Force × Distance, Rotational Work is Torque × Angle. This is crucial for understanding motors, turbines, and other rotating machinery.
Rotational Power
The rate at which rotational work is done.
P = τ · ω
- ω (Omega): Angular Velocity (rad/s)
rotational work formulas and interpretation
Rotational work measures energy transferred when torque acts through an angular displacement.
The calculator also preserves the closely related rotational-power mode for energy transfer rate.
How to use the rotational work calculator
- Choose a model: Select the physical relationship that matches the known values.
- Choose the unknown: Select the quantity you need to calculate.
- Enter values and units: Provide every requested measurement using consistent units.
- Calculate: Check the formula, converted result, sign, and units.
Formula and variables
For constant torque aligned with rotation, work equals torque multiplied by angular displacement in radians.
W = τθ- W — Work
- Rotational energy transfer (J)
- τ — Torque
- Turning moment (N·m)
- θ — Angular displacement
- Angle turned (rad)
Shaft work example
A constant 20 N·m torque turns a shaft through 3 radians.
- Torque
- 20 N·m
- Angle
- 3 rad
- W = 20 × 3
- W = 60 J
Result: Rotational work is 60 J.
The torque transfers 60 joules of mechanical energy.
Understanding your results
Interpreting the result
Angular displacement must be expressed in radians when torque is in newton-metres.
A sign indicates direction only when a consistent rotational sign convention is used.
Assumptions
- Rotation is evaluated about a specified axis.
- Inputs are converted through coherent SI units.
- The selected formula adequately represents the physical system.
Limitations
- The calculator does not simulate time-varying inputs.
- Vector directions and multiple axes must be resolved separately.
- Losses such as friction are not added unless represented in the entered net value.
Common mistakes
- Mixing RPM with radians per second.
- Using diameter instead of radius.
- Entering a zero divisor.
- Ignoring the direction represented by a negative value.
Practical use cases
Physics and education
Check rotational kinematics and dynamics exercises.
Machines and mechanisms
Estimate quantities for wheels, shafts, rotors, and rotating equipment.
Frequently asked questions
Why are radians used in rotational formulas?
Radians make angular and linear relationships dimensionally coherent without an extra conversion factor.
Can the result be negative?
Yes. A negative value means the quantity points opposite the direction chosen as positive.
Sources and review
- SI Brochure, 9th edition — BIPM. Accessed 2026-07-11.
- Special Publication 811 — NIST. Accessed 2026-07-11.
Reviewed 2026-07-11.
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