Torque Calculator: τ = r·F·sin(θ), lever, wrench, motor power
Torque calculator τ = r·F·sin(θ). Lever arm, force, angle. Torque-to-power conversion (HP/kW). Wrench, car wheel bolts, engine, gears examples. Units N·m, ft·lb, kgf·m.
Report a calculation error
Help us improve. Describe what is wrong with the calculation.
Result
| Voce | Valore |
|---|
Show full details
Share result
Disclaimer: this calculation is for informational purposes only. For important decisions, consult a qualified professional.
Torque: τ = r × F
Torque (or moment, couple) is the rotational analog of linear force. When you apply force F to an object at distance r from a rotation axis, it tends to rotate. Effective rotation force: τ = r × F = r·F·sin(θ), where θ is angle between lever arm r and force F. SI unit: N·m (newton-meter). This calculator solves 5 modes: τ, F, r, θ, and torque-to-power conversion (for engines).
Fundamental formulas
- τ = r · F · sin(θ) — torque magnitude
- τ = r × F — vector (cross product)
- τ = I · α — rotational Newton's law
- W = τ · θ — work done by torque
- P = τ · ω — power (ω = angular velocity rad/s)
- L = ∫τ dt — angular impulse = angular momentum change
The angle θ: geometry matters
Torque is maximum when force is perpendicular to lever arm (θ = 90°): τ = r·F. Zero when force is parallel (θ = 0° or 180°).
| Angle θ | sin(θ) | Effectiveness |
|---|---|---|
| 0° | 0 | NO torque |
| 30° | 0.5 | 50% |
| 45° | 0.707 | 71% |
| 60° | 0.866 | 87% |
| 90° | 1.0 | 100% — OPTIMAL |
| 180° | 0 | NO torque |
Right-hand rule
Torque is a pseudovector with direction from right-hand rule. Thumb points in τ direction; fingers curl in rotation direction.
Units and conversions
| Unit | Conversion | Use |
|---|---|---|
| N·m | SI | Scientific, European cars |
| kN·m | ×1000 | Construction, big engines |
| kgf·m | ×9.81 | Technical (old) |
| ft·lb | ×1.356 | US cars, engines |
| in·lb | ×0.113 | Fine mechanics USA |
Practical examples
Wrench
30 cm wrench, 50 N perpendicular: τ = 15 N·m ≈ 11 ft·lb.
Car wheel
100 N·m spec, 25 cm wrench: F = 400 N. With 40 cm: F = 250 N.
Car engine
300 N·m at 3,000 RPM: P = 94.2 kW = 126 HP.
Door hinge
80 cm door, 50 N end perpendicular: τ = 40 N·m. At 10 cm from hinge: only 5 N·m.
Bike pedal
70 kg cyclist, 17.5 cm crank: τ_max = 120 N·m.
Typical bolt torques
| Bolt | N·m |
|---|---|
| M4 | 2-3 |
| M6 | 10-12 |
| M8 | 25-30 |
| M10 | 50-60 |
| M12 (car wheel) | 80-100 |
| M14 | 130-150 |
| M16 | 200-250 |
| M20 | 400-500 |
τ·ω = Power
P = τ·ω for rotating motor. Fundamental in automotive engineering.
Typical car engines
| Engine | τ (N·m) | RPM | P (kW) |
|---|---|---|---|
| City gas | 100 | 4000 | 55 |
| Diesel sedan | 300 | 2000 | 100 |
| V8 sport | 550 | 5000 | 330 |
| F1 V8 2.4L | 250 | 18,000 | 560 |
Lever arm
Lever arm is perpendicular distance from axis to force line. If F is oblique, only perpendicular component creates torque: τ = r·F·sin(θ).
Lever classes
Class 1
Fulcrum between force and load. Seesaw, scissors, pliers.
Class 2
Load between fulcrum and force. Wheelbarrow, nutcracker, bottle opener.
Class 3
Force between fulcrum and load. Tweezers, baseball bat, fishing rod.
Rotational dynamics
Newton's second law for rotation:
τ_net = I · α
I = moment of inertia (kg·m²). Depends on mass distribution:
- Solid sphere (center): I = (2/5)·m·r²
- Solid cylinder: I = (1/2)·m·r²
- Thin rod (center): I = (1/12)·m·L²
- Thin rod (end): I = (1/3)·m·L²
- Thin ring: I = m·r²
Angular momentum conservation
L = I·ω. Without external torques:
I₁·ω₁ = I₂·ω₂
- Ice skater pulling arms in (I ↓, ω ↑)
- Stellar collapse: slow star → fast pulsar
- Gyroscope stability
- Diver somersault
Applications
Automotive
- Engines (torque-RPM curve)
- Transmission
- Torque wrenches for bolts
Mechanical engineering
- Drive shaft design
- Gears (τ_out = τ_in × ratio)
- Bearings
Tools and DIY
- Wrenches, screwdrivers
- Impact wrenches
- Torque screwdrivers
Biomechanics
- Joint torques
- Orthoses, prostheses
Sports
- Tennis, baseball (racket/bat torque)
- Golf (swing torque)
- Cycling (crank)
Construction
- Structural stability
- Crane lifting
Astronomy
- Planetary rotation
- Tidal effects
- Precession
Torque wrench
Essential mechanic tool for precise torque:
- Beam: visual indicator
- Click: clicks at target
- Digital: LCD, ±1-4% accuracy
- Ultrasonic: measures bolt tension
History
Archimedes (3rd c. BC): lever law.
Galileo (17th c.): beam moments.
Newton (1687): τ = I·α.
Euler (18th c.): rigid body dynamics.
Lagrange, Hamilton (19th c.): modern formalism.
FAQ
Torque formula?
τ = r·F·sin(θ). Max at θ = 90°. Unit N·m.
Difference between torque and force?
Force causes linear acceleration (F=ma), torque causes angular acceleration (τ=Iα). Different units (N vs N·m).
Why longer wrench unscrews better?
τ = r·F: larger r means less F needed. 40 cm wrench halves required force vs 20 cm.
Engine power in HP or kW?
P = τ·ω. 1 HP = 745.7 W. 100 kW ≈ 134 HP.
Car wheel bolt torque?
Typically 90-140 N·m for common cars. Too tight: breaks. Too loose: wheel loosens. Use torque wrench.
Torque and work same units?
Dimensionally yes (N·m = J). Conceptually different: torque is rotational vector, work is scalar energy. W = τ·θ.
What is moment of inertia?
Rotational mass analog: I quantifies resistance to angular acceleration change. Depends on mass distribution.
Static moment
Body in static equilibrium: Σ τ_i = 0. Combined with Σ F = 0, determines constraint reactions. Foundation of structural engineering.
Common errors
- Confusing torque with force
- Using distance from force not axis
- Ignoring angle (sin factor)
- Confusing with moment of inertia
- Using HP as torque (HP is power)
- Ignoring direction (right-hand rule)
Updated July 2026
Comments (0)
Login to leave a comment.
No comments yet. Be the first!