Torque Calculator

Definition (cross product)

For a force \vecF applied at a position \vecr measured from the pivot, the torque is the cross product

\vec\tau = \vecr × \vecF

Magnitude in 2-D

In a flat (2-D) problem, the position and force vectors lie in the plane. The magnitude of the resulting torque reduces to the perpendicular component of the force times the lever-arm length — equivalently, the lever-arm times the perpendicular component of the force:

\tau = r \, F \, \sinθ = F \, d_\perp = r_\perp \, F

Sign convention (right-hand rule)

Positive torque rotates counter-clockwise (out of the page) when viewed from the +z axis. Negative torque rotates clockwise (into the page). The convention is arbitrary but must be applied consistently. A common choice is "CCW positive, CW negative".

Newton's second law for rotation

Net torque about a pivot equals the moment of inertia of the body about that pivot times the angular acceleration:

\tau_\mathrmnet = I \, α

Units and distinctions

Torque is measured in \mathrmN· m. The same units describe work and energy (joules), but torque and work are conceptually different: work is energy transferred along a path, while torque is a turning effort. The vector nature of torque is what separates it from scalar energy.

Static Equilibrium and Choosing a Pivot

A body at rest requires both that the forces balance and that the torques balance. The second condition is what determines load sharing in beams, brackets, cranes and levers.

The practical trick is that the pivot is yours to choose. Since the body is not rotating about any point, the torque sum vanishes about *every* point — so pick the location where the most unknown forces act, and those unknowns drop out of the equation immediately. For a plank on two supports, taking moments about one support removes that support's reaction and solves for the other in a single line. This freedom is the single most useful technique in statics.

Σ \vecF = 0 \quad \textand \quad Σ \vec\tau = 0

Torque, Power and Gearing

A torque applied through an angular speed delivers power: P = \tau\,\omega, with \omega in radians per second. This is the relationship behind every engine specification sheet — peak torque and peak power occur at different engine speeds precisely because power is their product.

Gears trade one for the other. An ideal gearbox conserves power, so reducing output speed by a factor increases output torque by the same factor. A 4:1 reduction quadruples torque and quarters speed, which is why a low gear climbs a hill and a high gear delivers speed on the flat. The same trade appears in a bicycle's chainrings, a cordless drill's clutch settings, and a wind turbine's gearbox.

P = \tau \omega, \qquad \tau_\textout = \tau_\textin × \frac\omega_\textin\omega_\textout

Practical Magnitudes and Common Errors

Some reference points: a typical car wheel nut is torqued to roughly 110 N·m; a household door handle takes a couple of N·m; a small petrol car engine peaks near 150 N·m, while a heavy truck diesel can exceed 2000 N·m. Torque wrenches are commonly marked in N·m, lbf·ft (1\ \mathrmlbf· ft ≈ 1.356\ \mathrmN· m) or kgf·cm on smaller tools.

The three recurring mistakes are all geometric. First, using the full distance to the force rather than the perpendicular lever arm — only the component of force perpendicular to the radius produces torque, which is the \sinθ in \tau = rF\sinθ. Second, measuring the radius from the wrong point: torque is always quoted about a specific axis, and changing the axis changes the answer. Third, forgetting that a force pointing directly at or away from the pivot produces no torque at all, however large it is.

\tau = r F \sinθ

Worked Example

Inputs

• leverArmR: 0.4
• force: 50
• angleDeg: 90

Result: torque: 20 direction: CCW

Explanation

A 50 N force applied perpendicular (θ = 90°) to a 0.4 m wrench produces 0.4 × 50 × sin(90°) = 20 N·m of torque. The force is at right angles to the handle, so all of it contributes — this is the optimal angle for turning a bolt.

Second Scenario

Inputs

• leverArmR: 1.5
• force: 50
• angleDeg: 90

Result: torque: 20 direction: CCW

Explanation

This scenario uses different inputs (leverArmR = 1.5, force = 50, angleDeg = 90) to show how changing one variable affects the torque result. Run the calculator above with these values to get the exact updated output with step-by-step work.