Force & Arm
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Torque is the rotational equivalent of linear force. Whether you're designing a gearbox, sizing an electric motor, tightening a cylinder head, or selecting a servo for a robotic arm — accurate torque calculation is foundational to safe, efficient mechanical design.
*Approximate net torque acting on Earth from tidal braking forces.
Torque (symbol: τ, from Latin torquere— "to twist") is the rotational counterpart of linear force. While a linear force pushes or pulls an object along a straight line, torque causes an object to rotate about an axis. It is formally defined as the cross product of the position vector (lever arm) and the applied force: τ = r × F.
When the force is applied perpendicularly to the lever arm — which maximises rotational effect — the scalar form simplifies to τ = F × r, where F is force in Newtons and r is the perpendicular distance from the axis of rotation to the line of action of the force (the moment arm), measured in metres. The result is expressed in Newton-metres (N·m), the SI unit of torque.
Torque and work (energy) share the same dimensional analysis — both equal N·m — but they are fundamentally different quantities. Torque is a vector (direction matters); energy is a scalar. Torque is sometimes called the moment of force or simply moment in structural engineering contexts.
τ = F × r × sin(θ)The fundamental definition. F = force (N), r = lever arm length (m), θ = angle between force vector and lever arm. When θ = 90° (perpendicular), sin(90°) = 1 and τ = F × r.
τ = P / ω = (P × 60) / (2π × RPM)P = power (Watts), ω = angular velocity (rad/s). Practical shortcut for motors: τ (N·m) = [P (kW) × 9,549] / RPM. Imperial: τ (lb·ft) = [HP × 5,252] / RPM.
τ = I × α (Newton's 2nd Law for rotation)I = moment of inertia (kg·m²), α = angular acceleration (rad/s²). Used for dynamic analysis: sizing a motor to accelerate a flywheel or robotic joint from rest to operating speed in a specified time.
| Object | Axis | Formula |
|---|---|---|
| Solid cylinder / disk | Central (longitudinal) | I = ½mr² |
| Thin cylindrical ring | Central (longitudinal) | I = mr² |
| Solid sphere | Any diameter | I = ⅖mr² |
| Thin spherical shell | Any diameter | I = ⅔mr² |
| Thin rod | Perpendicular through centre | I = (1/12)mL² |
| Thin rod | Perpendicular through end | I = (1/3)mL² |
| Rectangular plate | Through centre (parallel to width) | I = (1/12)mL² |
The Newton-metre (N·m) is the SI unit of torque. In the United States and the automotive industry, pound-foot (lb·ft) is standard. Note: "pound-foot" (force × distance) is the correct term for torque; "foot-pound" technically denotes energy, though the terms are used interchangeably in everyday mechanical work.
| From | To | Multiply by | Example |
|---|---|---|---|
| N·m | lb·ft | 0.73756 | 100 N·m = 73.8 lb·ft |
| lb·ft | N·m | 1.35582 | 300 lb·ft = 406.7 N·m |
| N·m | kgf·m | 0.10197 | 100 N·m = 10.2 kgf·m |
| kgf·m | N·m | 9.80665 | 10 kgf·m = 98.1 N·m |
| lb·in | N·m | 0.11298 | 100 lb·in = 11.3 N·m |
| N·m | lb·in | 8.8508 | 100 N·m = 885 lb·in |
| oz·in | N·m | 0.00706 | 100 oz·in = 0.706 N·m |
Engine torque (N·m or lb·ft) is rated at a specific RPM on a dynamometer. A diesel engine producing 400 N·m at 1,800 RPM delivers strong low-speed pulling force for towing. A sports engine producing 350 N·m at 7,000 RPM prioritises high-RPM power. Drivetrain gear ratios multiply torque: a 4:1 gear ratio transforms 400 N·m of engine torque into up to 1,600 N·m at the driven wheel (minus friction losses). Wheel-hub motors in EVs eliminate the drivetrain entirely, applying torque directly to each wheel.
Electric motors are characterised by their stall torque (maximum torque at zero speed), continuous torque (sustainable without overheating), and peak torque (brief maximum). Industrial servo motors and stepper motors (used in CNC machines and robotics) are rated in N·m or oz·in. The torque constant Kt (N·m/A) lets engineers calculate torque from motor current: τ = Kt × I. This is critical for servo sizing: motor torque must overcome friction, load inertia, and gravity torque of the mechanical system.
Correct torque is critical for fastener integrity. The torque-tension equation is T = K × D × F, where T is tightening torque, K is the nut factor (0.15–0.20 for dry steel; 0.10–0.15 lubricated), D is nominal diameter, and F is clamping force. Automotive wheel lug nuts typically require 100–150 N·m. Cylinder head bolts are often torque-angle tightened: first torqued to a specified value, then rotated an additional angle (e.g., 90°) to achieve a controlled bolt stretch beyond the yield point for consistent clamping.
In structural engineering, torque (bending moment) analysis is essential for beam design. Torsional stress in circular shafts is calculated using τ = T·r/J, where T is applied torque, r is shaft radius, and J is the polar moment of inertia (J = πr⁴/2 for solid circular sections). Bridge designs must account for torsional loading from eccentric traffic loads. Wind turbine blades transfer aerodynamic torque to the main shaft — modern 15 MW offshore turbines generate shaft torques exceeding 20,000 kN·m.
Robot joint torque requirements are calculated from the link geometry, payload mass, and desired acceleration. For a planar robot arm: τ_joint = m × g × L × cos(θ) + I_link × α, where m is the combined mass of all links and payload distal to the joint, L is the distance to the centre of mass, θ is the joint angle relative to horizontal, and α is angular acceleration. Safety factors of 2–3× are applied because motors must handle dynamic loads, friction, and manufacturing tolerances.
Torque wrenches are essential for home mechanics. A torque wrench measures applied torque and provides an audible click or visual indicator when the set value is reached. Common applications: spark plugs (15–30 N·m), oil drain plug (20–35 N·m), wheel lug nuts (100–150 N·m), and bicycle components (5–15 N·m for carbon fibre parts — over-tightening crushes carbon fibre layup). Digital torque adapters allow any standard ratchet to measure torque, making precise fastening accessible for all skill levels.
Torque and power are related but measure fundamentally different things. Torque is a static or instantaneous rotational force — how hard the engine twists. Power is the rate at which work is done — how fast torque is delivered. They are linked by the formula:
P (W) = τ (N·m) × ω (rad/s)
where ω = 2π × RPM / 60
For acceleration from a standstill, torque matters most — which is why electric vehicles (which produce maximum torque from zero RPM) accelerate explosively. For sustained high speed, power at high RPM determines performance. A 4-cylinder economy engine may match a V8's torque at 5,000 RPM but cannot sustain it because of lower displacement and thermal limits. In industrial machinery, motors are selected for their torque speed curve — the relationship between available torque and operating RPM across the full operating range.
| Vehicle / Machine | Peak Torque | Peak Power | Torque RPM |
|---|---|---|---|
| Tesla Model S Plaid | 1,420 N·m (1,047 lb·ft) | 760 kW (1,020 hp) | 0 RPM (instant) |
| Formula 1 engine (2024) | ~450 N·m (332 lb·ft) | 1,000 kW+ (1,340 hp) | ~10,500 RPM |
| Diesel semi-truck engine | 2,800 N·m (2,064 lb·ft) | 450 kW (600 hp) | 1,100 RPM |
| Economy sedan (2.0L) | 200 N·m (148 lb·ft) | 110 kW (148 hp) | 4,000 RPM |
| 15 MW offshore wind turbine | 20,000+ kN·m | 15,000 kW | 8–12 RPM |
Proper fastener torque ensures correct clamping force and prevents joint failure from both under- and over-tightening. Always verify torque specifications with the manufacturer's service manual for safety-critical applications. Values below are typical for dry (unlubricated) conditions; reduce by 15–25% when lubricant or thread locker is applied.
| Application | Torque (N·m) | Torque (lb·ft) | Notes |
|---|---|---|---|
| Wheel lug nut (passenger car) | 100–150 | 74–111 | Always torque in a star pattern |
| Wheel lug nut (truck/SUV) | 150–200 | 111–148 | Re-check after 50 km |
| Spark plug (aluminum head) | 20–30 | 15–22 | Apply anti-seize to threads |
| Oil drain plug | 20–35 | 15–26 | New crush washer each time |
| Cylinder head bolt (typical) | 80–120 | 59–89 | Use torque-angle method for accuracy |
| Bicycle stem clamp | 5–7 | 4–5 | Critical for carbon fibre components |
| M8 Grade 8.8 bolt | 25 | 18 | Standard structural fastener |
| M10 Grade 8.8 bolt | 49 | 36 | Standard structural fastener |
| M12 Grade 8.8 bolt | 86 | 63 | Standard structural fastener |
Archimedes formulates the principle of the lever: "Give me a long enough lever and a fulcrum, and I will move the world." His treatise On the Equilibrium of Planes describes the moment of a force — the precursor to torque.
Isaac Newton publishes Principia Mathematica, establishing F = ma. The rotational equivalent — τ = I × α — follows directly, though Newton worked primarily with linear motion.
Augustin-Louis Cauchy and Siméon Denis Poisson develop the mathematical framework for stress tensors, linking internal material stresses to applied torque in shafts — foundational to mechanical engineering design.
James Watt's unit of "horsepower" (550 ft·lbf/s) becomes the de facto standard for rating steam engine power. The relationship between horsepower, torque, and RPM (HP = τ × RPM / 5,252) originates from this era.
The International System of Units (SI) establishes the Newton-metre (N·m) as the official unit of torque, replacing the multitude of regional units (kgf·m, lbf·ft, ozf·in) with a coherent global standard.
Tesla Motors introduces the Model S with 440 lb·ft (597 N·m) of instant electric torque, redefining consumer expectations for vehicle acceleration. Modern EVs with dual motors produce over 1,000 N·m, surpassing most supercars.
In physics, "torque" and "moment of force" (or "moment") are synonymous — both describe a rotational force. In structural and civil engineering, "moment" is the preferred term, particularly for bending moments (forces causing beam bending). In mechanical and automotive engineering, "torque" is standard. The mathematics are identical: τ = F × r × sin(θ).
Only the component of force perpendicular to the lever arm creates rotation. When force is applied at angle θ, the effective perpendicular component is F × sin(θ). At 90° (perpendicular), sin(90°) = 1 — maximum torque. At 0° or 180° (parallel to lever arm), sin(0°) = 0 — zero torque regardless of force magnitude. This is why door handles are designed for perpendicular pushing, and why wrenches are most efficient at 90° to the bolt axis.
Yes. Use: HP = (τ_lb·ft × RPM) / 5,252, or equivalently: kW = (τ_N·m × RPM) / 9,549. Example: 300 lb·ft at 5,252 RPM = exactly 300 HP. Note that the 5,252 constant is derived from converting RPM to rad/s and watts to horsepower: 1 HP = 550 ft·lbf/s = 745.7 W. This relationship means torque and horsepower curves always cross at 5,252 RPM on a dyno chart.
Stall torque is the maximum torque a motor produces at zero RPM (when the rotor is held stationary). It equals the motor's torque constant (Kt, in N·m/A) multiplied by the locked-rotor current. Operating at stall torque for extended periods causes thermal damage due to I²R losses. Most motor ratings specify continuous torque (safely maintainable indefinitely) and peak torque (brief bursts, typically 2–3× continuous).
Calculate torque from force and lever arm, power and RPM, or moment of inertia and angular acceleration. Supports multiple calculation methods with unit conversions.
90° = perpendicular (maximum torque)
Enter values above to see results.
This calculator is part of a comprehensive guide
Rotational Mechanics
Calculate torque from force and lever arm, power and RPM, or moment of inertia and angular acceleration for mechanical design.
Force-Arm
τ = F × r
Power-Speed
τ = P / ω
Inertia-Accel
τ = I × α
Unit
N·m (SI)
Reviewed by: CalculatorApp Mechanical Engineering Team
Torque is the rotational equivalent of linear force — it causes or resists angular acceleration. Every rotating machine involves torque: engines and motors produce it, gearboxes transmit and transform it, fasteners require it for proper clamping, and brakes oppose it. Understanding torque is essential for sizing motors, designing drive trains, specifying fasteners, and analyzing mechanical failures.
Force & Arm
τ = F × rPower & Speed
τ = P / ω (ω = 2πn/60)Inertia
τ = I × αPower Link
P = τ × ω| Application | Typical Torque Range | Design Focus |
|---|---|---|
| Threaded fasteners | 1 N·m – 2000 N·m | Clamp load, yield limit |
| Electric motors | 0.1 N·m – 10 kN·m | Starting torque, efficiency |
| Automotive engines | 100 – 2000 N·m | Peak torque vs RPM curve |
| Wind turbines | 10 kN·m – 10 MN·m | Gearbox ratio, shaft design |
Antiquity: Archimedes formalizes the lever principle — the geometric foundation of torque.
1687: Newton's laws of motion establish the rotational analogs: τ = Iα follows from F = ma.
1800s: Steam engines make torque a central practical engineering quantity for power transmission.
1882: De Laval's steam turbine demonstrates high-speed rotational torque in continuous operation.
1900s: Automotive and aerospace industries drive precise torque standards for fasteners and driveshafts.
Modern era: Torque sensors, dynamometers, and smart wrenches enable real-time torque control in manufacturing.
Standard mechanical design and power transmission codes.
Torque rating standards for electric motors.
Bolt tightening torque specifications from ISO.
Rotating equipment guarding and safety standards.
Myth: Higher torque always means a more powerful machine.
Fact: Power = torque × speed. A high-torque, low-speed motor can have the same power as a low-torque, high-speed motor.
Myth: Tightening a bolt harder always improves joint strength.
Fact: Over-torquing yields the bolt shank, reducing or eliminating clamp load and causing premature failure.
Myth: Torque and moment are different quantities.
Fact: Torque and moment are both force × distance; torque specifically applies to a rotational axis.
Myth: Gear reduction reduces power.
Fact: Ideal gears preserve power; torque increases as speed decreases by the gear ratio (minus friction losses).
Torque (τ) is a rotational force: τ = F × r. It is measured in Newton-meters (N·m) in SI units.
P = τ × ω, where ω is angular velocity in rad/s. For rpm: P = τ × 2πn/60.
τ = I × α, where I is moment of inertia (kg·m²) and α is angular acceleration (rad/s²).
1 N·m = 0.7376 ft·lb. Multiply Newton-meters by 0.7376 to get foot-pounds.
Torque wrenches apply a specified tightening torque to fasteners, preventing over- or under-tightening that can cause joint failure.
A gear reduction multiplies torque by the gear ratio while reducing speed proportionally, maintaining constant power (minus losses).
At constant power, torque decreases as RPM increases: τ = P / ω. High-torque motors run at low RPM.
Bolt tightening torque depends on bolt size, grade, and lubrication. Typical M10 8.8 bolt: ~45-50 N·m dry.
Both describe rotational force × distance; torque specifically refers to a twisting force about a rotational axis.
L = I × ω. Torque equals the rate of change of angular momentum: τ = dL/dt, analogous to force = dp/dt.
τ = P/ω from the shaft power required; then select a motor with torque rating exceeding this with a service factor.
Excess torque stresses the bolt beyond its yield strength, causing permanent stretch or fracture of the threaded section.
Pair torque analysis with motor sizing, gear ratio, and kinetic energy calculators for complete drive system design.
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