Torque Calculator

Definition of torque

The torque about an axis is the cross product of the position vector and the force. The position vector r runs from the axis to the point where the force is applied. The magnitude is M = F·r·sin α, where α is the angle between those two vectors. The unit is N·m. In a workshop the same quantity is called a moment of force. This page computes that quantity for one force.

At a right angle the sine is 1, so the formula reduces to M = F·r. That is the moment in Archimedes’ law of the lever: two such magnitudes balance when F₁·r₁ = F₂·r₂. Archimedes stated that condition for weights in the treatise On the Equilibrium of Planes. The sine in the general formula is the later way of writing the length of the cross product r × F. This calculator does not check the balance of two sides.

Two equivalent forms

The same product can be read in two ways. The perpendicular component is F⊥ = F·sin α, and then M = F⊥·r. The perpendicular arm is r⊥ = r·sin α, the distance from the axis to the line of action, and then M = F·r⊥. In this calculator the field r is the distance to the point of application, not the perpendicular arm already. The calculator computes the perpendicular arm separately.

For 20 N, 0.3 m, and 90° both forms agree: F⊥ = 20 N, r⊥ = 0.3 m, and M = 6 N·m. At 30° the sine is 1/2, so F⊥ = 10 N, r⊥ = 0.15 m, and M = 3 N·m. The torque at a right angle, which is F·r alone, stays 6 N·m. The panel shows these numbers with decimal places: 3.000 N·m, 10.00 N, and 0.150 m.

Numbers from the reference example

Step 1. Convert the angle to radians only in order to compute the sine: degrees times π/180. Step 2. Multiply F·r·sin α. Step 3. For 20 N, 0.3 m, and 90°: sin 90° = 1, M = 6 N·m, and the panel shows 6.000 N·m. Step 4. For 30°: sin 30° = 0.5, M = 3 N·m, and the panel shows 3.000 N·m. Step 5. A force of 10 N on an arm of 0.6 m at a right angle gives the same product, 6 N·m. Step 6. At 60°, with 30 N and 0.2 m, the sine on the panel is 0.8660 and the torque is 5.196 N·m. At 10°, with 40 N and 0.4 m, the sine is 0.1736, the torque is 2.778 N·m, and r⊥ = 0.069 m.

Why the torque is largest at 90°

The sine rises from 0 to 1 between 0° and 90° and falls from 1 to 0 between 90° and 180°. At a fixed F and r the torque is therefore largest at a right angle and equals F·r. At 0° and at 180° the torque is zero. The chart on this page plots that curve for the F and r you entered, at 0°, 30°, 45°, 60°, 90°, 120°, 150°, and 180°.

The calculator does not report whether the body turns clockwise or counterclockwise. In the allowed range the sine is not negative, and F and r are not negative either, so the result does not fall below zero. The direction belongs on the diagram. The calculator rejects 181° and a negative angle.

What this calculator does not compute

One force does not replace a system of forces. Varignon’s theorem says that the torque of the resultant equals the sum of the torques of the components, but this page does not form that sum. The balance of two arms is on the lever calculator and requires a right angle. Angular acceleration from M = I·ε and the energy ½Iω² are on separate pages. Work W = F·s has the dimension of a joule. The same written unit N·m does not turn a torque into energy.

The model leaves out thread friction, flex in the wrench, and the fact that a hand may not grip the end of the arm. The number on this page is the torque of the force you entered about the axis you entered. It is not the tightening torque from a bolt specification.

Units

Enter the force in newtons, the arm in meters, and the angle in degrees. The torque comes out in N·m. A centimeter is 0.01 m. After you switch the header units, force is in lbf, the arm is in feet, and the torque is in ft·lbf. The SI value stays beside it. The factors in the page code are 1 lbf = 4.4482216 N, 1 m = 3.2808399 ft, and 1 ft·lbf = 1.3558179 N·m.