Hong Life CalculatorsUnderstanding the torque and lever calculator — concept and principle
Torque is the turning effect of a force. It tells you how strongly a push or pull on a wrench, door handle, pedal or lever tends to rotate something about a pivot. This calculator gives the torque from a force, its lever arm and its angle, and also treats the same numbers as a simple lever to show the load force and mechanical advantage.
How it is calculated
- Torque: τ = r × F × sin θ, where F is the force in newtons (N), r is the distance from the pivot to the point where the force acts in metres (m), and θ is the angle between the lever arm and the force. The result is in newton-metres (N·m).
- Lever load force: F₂ = F₁ × d₁ ÷ d₂, from the lever balance F₁d₁ = F₂d₂. Here d₁ is the lever (effort) arm r and d₂ is the load arm.
- Mechanical advantage: MA = d₁ ÷ d₂, how many times the lever multiplies the input force.
The lever outputs use the full force, so they assume the force is applied at right angles to the lever.
The principle
Only the part of a force perpendicular to the lever arm makes it turn. That component is F sin θ; the part along the arm just pushes or pulls on the pivot. At θ = 90° sin θ = 1 and all of the force is used, while at 0° or 180° the force points through the pivot and produces no torque. A longer arm gives more torque from the same force, which is why a long wrench loosens a stuck bolt more easily.
A lever in balance (Archimedes' law of the lever) has equal torques on both sides of the fulcrum: F₁d₁ = F₂d₂. The trade-off is distance: the load end moves d₂ ÷ d₁ times as far as the effort end, so the work done is the same on both sides (ignoring friction).
Worked example
- Inputs: F = 50 N, r = 0.3 m, θ = 90°, load arm d₂ = 0.1 m.
- sin 90° = 1, so τ = 0.3 × 50 × 1 = 15 N·m.
- Lever load force: F₂ = 50 × 0.3 ÷ 0.1 = 150 N.
- Mechanical advantage: 0.3 ÷ 0.1 = 3.
| Force (N) | Torque (N·m) |
|---|---|
| 12.5 | 3.75 |
| 25 | 7.5 |
| 37.5 | 11.25 |
| 50 | 15 |
| 62.5 | 18.75 |
| 75 | 22.5 |
| 100 | 30 |
How to read the result
Torque grows linearly with both force and arm length. If the force is applied at 30°, sin 30° = 0.5 and the same 50 N on 0.3 m gives only 7.5 N·m. A mechanical advantage above 1 multiplies force (a crowbar or bottle opener); below 1 the lever trades force for speed and range of motion (a fishing rod or your forearm).
Common mistakes
- Measuring the arm to the end of the handle instead of to where the hand actually pushes.
- Using the angle between the force and the perpendicular rather than between the force and the arm.
- Entering centimetres instead of metres, which makes the torque 100 times too large.
Frequently asked questions
What is the difference between N·m and J?
Both equal kg·m²/s², but torque is a turning effect with a direction, while the joule measures energy. Torque is never written in joules.
How do I convert N·m to kgf·m or lbf·ft?
1 kgf·m = 9.80665 N·m, and 1 lbf·ft ≈ 1.3558 N·m. So 15 N·m is about 1.53 kgf·m or 11.06 lbf·ft.
Sources: Halliday, Resnick and Walker, Fundamentals of Physics, chapter on rotation and torque · Archimedes, On the Equilibrium of Planes, Book I (law of the lever)