Physics · Classical Mechanics · Dynamics & Forces
Torque Calculator
Calculates the torque produced by a force applied at a given distance and angle from a pivot point.
Calculator
Formula
τ (tau) is the torque in newton-meters (N·m); r is the length of the lever arm (distance from the pivot to the point of force application) in meters; F is the magnitude of the applied force in newtons; θ (theta) is the angle between the force vector and the lever arm. Torque is maximized when the force is perpendicular to the lever arm (θ = 90°).
Source: Halliday, Resnick & Krane, Physics (5th ed.), Chapter 11 — Rolling, Torque, and Angular Momentum.
How it works
Torque — also called the moment of force — measures how effectively a force causes an object to rotate about a given axis or pivot point. Unlike a linear force, which accelerates an object along a straight line, torque produces angular acceleration. The greater the torque applied to a rigid body, the faster it will angularly accelerate, according to Newton's second law for rotation: τ = Iα, where I is the moment of inertia and α is the angular acceleration. Understanding torque is essential in virtually every branch of mechanical and structural engineering.
The standard torque formula is τ = r · F · sin(θ), where τ (tau) is the torque in newton-meters (N·m), r is the lever arm — the perpendicular distance from the axis of rotation to the line of action of the force — measured in meters, F is the magnitude of the applied force in newtons, and θ is the angle between the direction of the force vector and the lever arm. When θ = 90°, sin(θ) = 1 and the torque is at its maximum for a given force and lever arm. When the force is applied parallel to the lever arm (θ = 0° or 180°), sin(θ) = 0 and no torque is produced, regardless of how large the force is.
Practical applications of this formula span an enormous range. Automotive engineers calculate engine torque to determine power output and gear ratios. Structural engineers assess the torque on bolts and fasteners to ensure they meet specification. Biomedical engineers model joint torques in the human body to design prosthetics and rehabilitation devices. Robotics engineers program servo motors using torque requirements derived from load and arm length. Any time a rotating mechanism is involved — from a simple door hinge to a wind turbine — torque is the central quantity of interest.
Worked example
Suppose a mechanic applies a force to a wrench to tighten a bolt. The wrench handle is 0.35 m long (lever arm, r), the mechanic pushes with a force of 80 N, and the force is applied at an angle of 75° relative to the wrench handle.
Step 1 — Identify variables:
r = 0.35 m, F = 80 N, θ = 75°
Step 2 — Apply the torque formula:
τ = r · F · sin(θ)
τ = 0.35 × 80 × sin(75°)
Step 3 — Compute sin(75°):
sin(75°) ≈ 0.9659
Step 4 — Calculate torque:
τ = 0.35 × 80 × 0.9659
τ ≈ 27.04 N·m
If the mechanic were to apply the same force perfectly perpendicular (θ = 90°), the torque would be 0.35 × 80 × 1.0 = 28.00 N·m — the theoretical maximum for this configuration. The 75° angle yields about 96.6% of the maximum possible torque, demonstrating why perpendicular force application is generally preferred in fastener work.
Limitations & notes
This calculator assumes a rigid, massless lever arm and a single point of force application, which is an idealization not always met in real-world scenarios. The formula gives the magnitude of the torque vector; in three-dimensional analysis, torque is a vector quantity defined by the cross product τ = r × F, and direction matters for problems involving multiple axes of rotation. The calculator does not account for dynamic effects such as centrifugal forces, bearing friction, or the moment of inertia of the lever arm itself. For multi-force systems, the net torque must be found by summing individual torques with their appropriate signs (clockwise vs. counterclockwise conventions). Additionally, at very small or very large angles near 0° or 180°, floating-point precision may introduce minor rounding errors, though these are negligible for all practical engineering purposes. Always cross-check critical fastener torque values against manufacturer specifications and use calibrated torque wrenches in safety-critical applications.
Frequently asked questions
What is torque and how is it different from force?
Force is a linear push or pull that causes straight-line acceleration, measured in newtons. Torque is the rotational equivalent — it measures how much a force tends to rotate an object about a pivot or axis, measured in newton-meters (N·m). A large force applied close to or parallel to the pivot produces little torque, while the same force applied farther away and perpendicularly produces much greater torque.
Why does the angle matter in the torque formula?
Only the component of force perpendicular to the lever arm contributes to rotation. The sin(θ) factor extracts this perpendicular component. At 90°, the full force contributes to torque. At 0° or 180°, the force is entirely along the lever arm and produces zero torque. This is why you instinctively push perpendicular to a wrench handle for maximum effect.
What units is torque measured in?
In the SI system, torque is measured in newton-meters (N·m). In imperial units, it is commonly expressed in pound-feet (lb·ft) or pound-inches (lb·in). To convert from N·m to lb·ft, divide by 1.35582; to convert to lb·in, divide by 0.11298. This calculator outputs results in N·m.
What is a lever arm in the context of torque?
The lever arm (also called the moment arm) is the perpendicular distance from the axis of rotation to the line of action of the applied force. In this calculator, r represents the straight-line distance from the pivot to the point where the force is applied, and the angle θ accounts for any non-perpendicular application. A longer lever arm always produces more torque for the same applied force.
How do I calculate net torque when multiple forces are applied?
When multiple forces act on a body, calculate the torque produced by each force individually using τ = r · F · sin(θ), then sum them algebraically — assigning positive values to counterclockwise torques and negative values to clockwise torques (or vice versa, depending on your sign convention). The net torque determines whether the system rotates, and in which direction. For static equilibrium, the net torque about any point must equal zero.
Last updated: 2025-01-15 · Formula verified against primary sources.