What is the torque formula?

Torque τ = F × r × sin(θ), where F is force in newtons, r is the distance from the pivot in meters, and θ is the angle between the force direction and the lever arm. Maximum torque occurs when θ = 90° (force perpendicular to arm). The cross-product form: τ = r × F.

What units is torque measured in?

SI: newton-meters (N·m). Imperial: pound-feet (lb·ft) or pound-inches (lb·in). 1 N·m ≈ 0.7376 lb·ft. 1 lb·ft = 12 lb·in. Other units: kgf·m, oz·in. Always confirm units in vehicle specs — engine torque often quoted in lb·ft in US, N·m elsewhere.

How does lever arm length affect torque?

Torque is proportional to lever arm length. Doubling arm length doubles torque for the same force. Why long-handled wrenches give mechanical advantage for stuck fasteners. Why door knobs are placed far from hinges (small force makes large torque).

Why does the angle matter for torque?

Only the perpendicular component of force creates torque. τ = F × r × sin(θ). At 90° (perpendicular), sin = 1, full torque. At 0° (parallel to arm), sin = 0, no torque at all. At 45°, sin = 0.707 — 70% of maximum. This is why pushing perpendicular to a wrench works best.

How are torque and power related?

Power = Torque × Angular velocity. P(W) = τ(N·m) × ω(rad/s). Or in US units: P(hp) = τ(lb·ft) × RPM / 5252. A high-torque low-RPM motor can have same power as low-torque high-RPM motor. Trucks: high torque, low RPM. Sports cars: moderate torque, high RPM.

What torque do I need for car lug nuts?

Typically 80-150 N·m (60-110 lb·ft) depending on vehicle. Compact cars: 80-100 N·m. SUVs/trucks: 120-150 N·m. Check owner's manual or dealer for exact spec. Use calibrated torque wrench — overtightening can stretch studs; undertightening can let wheels loosen during driving.

What's the difference between torque and moment?

Often used interchangeably. "Torque" emphasizes a twisting/rotating effect. "Moment" emphasizes a beam-bending or static balance effect. Both have units N·m. In structural engineering, "bending moment" describes static beam stresses; in mechanical engineering, "torque" describes shaft rotation.

How does Newton's second law work for rotation?

Linear: F = ma. Rotational: τ = Iα. Where I = moment of inertia and α = angular acceleration. Larger I = more resistance to angular acceleration for same torque. This determines how quickly motors spin up loads and how rapidly objects can be rotated.

Torque Calculator

Calculate the torque (rotational force) produced by a force applied at a distance from the pivot point. Uses the formula τ = F × r × sin(θ), where F is force, r is the lever arm, and θ is the angle between the force and lever arm.

Torque is the rotational equivalent of force — it measures how effectively a force causes rotation around an axis. Whenever you tighten a bolt with a wrench, open a door, or pedal a bicycle, you're applying torque. The basic formula τ = F × r × sin(θ) captures three things: the magnitude of the applied force, the distance from the pivot point (lever arm), and the angle between force and lever arm.

The product is maximized when force is perpendicular to the lever arm (θ = 90°, sin θ = 1). Pushing on a door at 90° to its surface works best; pushing parallel to the door (θ = 0°) does nothing. This is why wrench handles are long: increasing r multiplies torque for the same applied force, giving mechanical advantage.

Units of torque are newton-meters (N·m) in SI, or pound-feet (lb·ft) in imperial. A typical bolt requires 5-50 N·m for tightening; car engine peak torque ranges from 100-1,000 N·m; large diesel truck engines reach 2,000+ N·m. Industrial machinery can produce millions of N·m. The Saturn V rocket engines produced ~30 million N·m of torque combined.

For rotating systems, torque produces angular acceleration: τ = Iα (rotational analog of F = ma), where I is moment of inertia. This determines how quickly motors spin up loads, how fast brakes stop wheels, and how rapidly satellites can reorient.

Common applications: mechanical engineering (bolt tightening, motor sizing), automotive (engine specs, transmission design), structural engineering (beam moments), robotics (joint actuator sizing), and any analysis involving rotation or twisting.

Inputs

Force (N)

Lever Arm / Radius (m)

Angle θ (degrees)

Results

Torque

50 N·m

Torque (lb·ft)

36.878 lb·ft

Torque (kgf·m)

5.099 kgf·m

Torque Results

Parameter	Value
Force	100 N
Lever Arm	0.5 m (1.640 ft)
Angle	90° (sin θ = 1.0000)
Torque (N·m)	50 N·m
Torque (lb·ft)	36.8780 lb·ft
Torque (kgf·m)	5.0986 kgf·m
Formula	τ = F × r × sin(θ)

Last updated: May 29, 2026

Formula

**Torque:** τ = F × r × sin(θ) Where: - τ = torque (N·m) - F = applied force (N) - r = lever arm from pivot to point of force application (m) - θ = angle between force direction and lever arm (degrees) **Maximum torque (force perpendicular to lever arm):** τ_max = F × r (when θ = 90°) **Vector form:** τ⃗ = r⃗ × F⃗ The cross product gives both magnitude and direction (right-hand rule for axis). **Worked example: torque wrench on bolt** Applying 50 N straight-down force on a 30 cm wrench, perpendicular (θ = 90°). τ = 50 × 0.30 × sin(90°) = 50 × 0.30 × 1 = 15 N·m If you pull at 45° to the wrench instead: τ = 50 × 0.30 × sin(45°) = 50 × 0.30 × 0.707 = 10.6 N·m 70% of maximum — losing effectiveness. **Unit conversions:** | Unit | In N·m | |---|---| | 1 N·m | 1 | | 1 N·cm | 0.01 | | 1 kgf·m | 9.807 | | 1 lb·ft | 1.3558 | | 1 lb·in | 0.1130 | | 1 oz·in | 0.0071 | | 1 dyne·cm | 10⁻⁷ | **Common torque ranges:** | Application | Torque | |---|---| | Small screws (M2-M4) | 0.1-3 N·m | | Standard bolts (M6-M10) | 5-50 N·m | | Large bolts (M14-M20) | 100-500 N·m | | Door knob | 1-5 N·m | | Manual screwdriver | 1-3 N·m | | Electric screwdriver | 5-20 N·m | | Cordless drill (low/high) | 20-65 N·m | | Car wheel lug nuts | 80-150 N·m | | Car engine torque (peak) | 150-500 N·m | | Sport car engine | 400-800 N·m | | Diesel truck | 1,500-3,500 N·m | | Saturn V engines | ~30,000,000 N·m | **Lever arm trick:** If you can't lift heavy load directly: - Long lever arm reduces required force. - Crowbar (r ~ 1 m) on lifting 100 kg object: force at handle = mg × r_load / r_handle. Archimedes: "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." **Torque and rotational motion:** τ = I × α Where: - I = moment of inertia (kg·m²) - α = angular acceleration (rad/s²) Like F = ma but for rotation. **Moment of inertia for common shapes:** | Shape | I | |---|---| | Point mass at radius r | mr² | | Solid disk/cylinder (axis through center) | ½mr² | | Hollow cylinder | mr² | | Solid sphere | (2/5)mr² | | Hollow sphere | (2/3)mr² | | Rod about center | (1/12)ML² | | Rod about end | (1/3)ML² | **Power and torque:** P = τ × ω Where ω = angular velocity (rad/s). For RPM: ω = 2π × RPM / 60 rad/s. P (kW) = τ (N·m) × ω (rad/s) / 1000. Or: P (hp) = τ (lb·ft) × RPM / 5252. **Worked example: car engine** V8 engine producing 400 N·m at 4,000 RPM. ω = 2π × 4000 / 60 = 418.9 rad/s. P = 400 × 418.9 = 167,552 W ≈ 167 kW ≈ 224 hp. Engine power and torque interact: high torque + RPM = high power. **Bolt tightening (proper torque):** Bolt torque specs are critical for proper clamping force. Over-torque can strip or fatigue; under-torque allows loosening. Manufacturers provide specs: - Engine head bolts: precise multi-step torque + angle. - Lug nuts: 80-150 N·m typical for cars. - Architectural bolts: ASTM/ISO standards. Use torque wrench for safety-critical fasteners. **Newton's second law (rotational):** ΣF = m × a (linear) Στ = I × α (rotational) Net torque produces angular acceleration. Zero net torque → constant angular velocity (or rest). **Static equilibrium:** For a rigid body in equilibrium: ΣF = 0 (no translation) Στ = 0 (no rotation) Used in structural analysis (bridges, buildings). **Torque on a current loop (electromagnetism):** τ = μ × B × sin(θ) Where μ = magnetic moment, B = magnetic field. Basis of all electric motors. **Pendulum torque:** For a pendulum at angle θ from vertical: τ = m × g × L × sin(θ) Restoring torque, drives oscillation. **Common mistakes:** - **Forgetting angle factor**: peak torque at 90° only. - **Wrong arm length**: measure from pivot to force application point. - **Mixing units**: N·m and lb·ft easily confused. - **Counterintuitive direction**: right-hand rule for vector direction.

How to use this calculator

Enter applied force in newtons (weight × 9.81 to convert kg-force).
Enter distance from pivot to point of application (lever arm) in meters.
Enter angle between force and lever arm (90° = perpendicular, maximum torque).
Calculator returns torque in N·m.
Convert: 1 lb·ft ≈ 1.356 N·m; 1 kgf·m ≈ 9.81 N·m.
For power: P = τ × ω; multiply torque by angular velocity in rad/s.

Worked examples

Loosening a stuck lug nut

**Scenario:** Car lug nut requires 150 N·m to loosen. Using a 40 cm tire iron. Force needed? **Calculation:** F = τ / r = 150 / 0.40 = 375 N. In weight terms: ~38 kg-force. **Result:** Need to apply 38 kg of force on the end of the wrench — feasible by leaning your weight. Longer breaker bar (60 cm): only 250 N (~25 kg). Even longer (1 m): 150 N (~15 kg). Why long-handled tools work for stuck fasteners.

Car engine torque

**Scenario:** Engine produces 400 N·m peak at 4,000 RPM. Peak power? **Calculation:** ω = 2π × 4000/60 = 418.9 rad/s. P = 400 × 418.9 = 167,552 W = 167 kW ≈ 225 hp. **Result:** ~225 hp peak power. Typical of mid-size sports car. Compare: Tesla Plaid motor: ~1,400 N·m at lower RPM but 1,020 hp total via electric architecture. Diesel trucks: 2,500 N·m at lower RPM, lower hp but huge pulling power.

Opening a door

**Scenario:** Push 5 N on a door, 80 cm from hinges, at various angles. **Calculation:** - 90° (perpendicular): τ = 5 × 0.80 × 1.0 = 4 N·m. - 45°: τ = 5 × 0.80 × 0.707 = 2.83 N·m. - 30°: τ = 5 × 0.80 × 0.5 = 2 N·m. - 0° (parallel): τ = 0 (no rotation). **Result:** Perpendicular force maximizes door opening. Pushing at 30° is half as effective. Pushing parallel doesn't open at all. Same reason: cleat your push perpendicular to the door for maximum effect.

When to use this calculator

**Use torque calculations for:**

- **Mechanical engineering**: bolt tightening, motor sizing. - **Automotive**: engine specifications, transmission ratios. - **Structural engineering**: beam moments, structural connections. - **Robotics**: joint actuator sizing, gripper design. - **Industrial machinery**: gearboxes, conveyors, mixers. - **Power tools**: drill/impact driver capacity. - **Sports**: bicycle pedaling, golf swing analysis. - **Aerospace**: control surface actuation.

**Torque vs power:**

- **Torque**: instantaneous rotational force (N·m). - **Power**: rate of work = torque × angular speed.

A small motor at high RPM can have same power as large motor at low RPM: - 100 N·m at 1,000 RPM = ~10.5 kW. - 1,000 N·m at 100 RPM = ~10.5 kW.

Same power, very different applications. Truck engines: high torque, low RPM. Sports car engines: moderate torque, high RPM.

**Bolt tightening best practices:**

1. Use calibrated torque wrench. 2. Clean threads (oily threads need lower torque to achieve same clamp force). 3. Tighten in stages for large bolts (e.g., 50%, 75%, 100%). 4. Critical bolts: torque + additional rotation angle (more precise). 5. Use lubricant per spec (changes friction coefficient).

**Motor selection:**

For accelerating a load to target speed: - Required torque: τ = I × α + τ_load_resistance. - Required power: P = τ_max × ω_max.

Match motor capability to peak transient demand, not just steady state.

**Common applications:**

- **Automotive**: peak torque rating determines low-end pulling power. - **Cycling**: rider torque on pedals × gear ratio = rear wheel torque. - **Wind turbines**: blade torque drives generator. - **Robotics**: each joint motor sized for max load + safety. - **Cranes**: lifting torque depends on weight × radius. - **Aerospace**: gyroscopes use torque to reorient spacecraft.

**Right-hand rule:**

For determining torque direction in vector form: - Curl fingers in direction of rotation. - Thumb points along torque vector axis.

Counterclockwise rotation (viewed from + axis side) = positive torque.

**Static equilibrium analysis:**

For structures (beams, bridges): - Sum of forces = 0. - Sum of torques about any point = 0.

Used to find reaction forces at supports, beam moments, structural design.

**Torque amplification by gears:**

Output torque = input torque × gear ratio × efficiency.

A small high-RPM motor with 10:1 reduction gearbox produces 10× torque output at 1/10 the speed.

**Software:**

- **CAD with simulation**: SolidWorks, Fusion 360 motion analysis. - **FEA**: ANSYS, Abaqus for stress and deflection from torque. - **MATLAB / Python**: dynamic system modeling. - **Vehicle dynamics**: CarSim, ADAMS.

**Pitfalls:**

- **Forgetting sine of angle**: torque max only at 90°. - **Wrong arm length**: measure perpendicular distance from pivot to line of force. - **Mixing units**: N·m vs lb·ft (factor 1.356). - **Confusing torque with power**: instantaneous vs rate. - **Direction signs**: right-hand rule matters in 3D. - **Friction in threads**: bolt clamp force depends on torque AND friction. - **Dynamic vs static**: rotating systems have inertia (τ = Iα), not just static balance.

Common mistakes to avoid

Forgetting the sine of the angle (peak torque only at 90°).
Measuring lever arm incorrectly (must be perpendicular distance).
Mixing units (N·m vs lb·ft vs kgf·m).
Confusing torque with power (torque is rotational force; power = τ × ω).
Ignoring direction (right-hand rule for 3D vector torque).
Using torque without considering friction in bolts (clamp force varies).
Treating dynamic problems as static (forgetting τ = Iα).
Adding torque from rotating loads as if static.

Frequently Asked Questions

Sources & further reading

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