What Is Work Done by Torque?
When you push a box across the floor, you do work: force times distance. When you turn a wrench, spin a wheel, or drive a motor, you also do work — but now the "force" is torque and the "distance" is an angle. That rotational version of work is what this calculator measures.
Work done by torque is the energy transferred to an object as it rotates. It shows up everywhere a shaft turns: tightening a bolt, pedaling a bike, spinning a hard drive, or driving a car's wheels. Engineers use it to size motors, estimate energy use, and understand how much usable motion a rotating system produces.
The Formula
The relationship is a single multiplication:
[
W = \tau \times \Delta\theta
]
Where:
- W is the work done (Joules)
- τ is the applied torque (Newton-meters)
- Δθ is the change in angular position (radians)
The units check out neatly: 1 N·m multiplied by a dimensionless angle in radians gives 1 Joule — because a Joule is exactly one Newton-meter of energy.
Important: the angle must be in radians, not degrees. If your angle is in degrees, convert first by multiplying by π/180 — or just select degrees in the calculator above and let it handle the conversion.
Worked Example
Suppose a mechanic applies a steady 25 N·m of torque while a bolt turns through a change in angular position of 50 radians:
[
W = 25 \text{ N·m} \times 50 \text{ rad} = 1250 \text{ J}
]
So the torque does 1250 Joules of work. To put that in perspective: 50 radians is almost eight full rotations, so even a moderate torque adds up quickly once the object keeps turning.
Try a degree-based case too: the same 25 N·m turning through exactly one quarter-turn (90° = π/2 ≈ 1.5708 rad) does 25 × 1.5708 ≈ 39.27 J. Same torque, much smaller sweep of the wrench — much less work.
What the Result Tells You
| Result | Interpretation |
|---|---|
| Positive W | The torque drives the rotation — energy flows into the object, spinning it up. |
| Zero W | Either no net torque acts, or nothing actually rotates. Holding a bolt still does no work, however hard you push. |
| Negative W | The torque opposes the rotation — like brakes or friction — and removes energy from the object. |
Two practical takeaways follow directly from the formula:
- Doubling either factor doubles the work. More torque or more rotation means proportionally more energy transferred.
- Holding position does no work. Since Δθ = 0, a static torque transfers no energy at all — a stalled motor draws current yet does zero mechanical work on the shaft.
Quick Recap
- Work done by torque: (W = \tau\Delta\theta), with torque in N·m and the angle in radians.
- Convert degrees to radians with ×π/180; one full turn is 2π radians.
- Positive results add rotational energy; negative results take it away.
If you want to see where that rotational energy ends up, the rotational kinetic energy calculator is a natural next step.