Torque Calculator
Calculate torque using force and lever arm (τ = r × F × sinθ) or rotational dynamics (τ = Iα).
Input Values
Torque
50.0000 N·m
Formula Used:
τ = r × F × sin(θ)
τ = 0.5 × 100 × sin(90°) = 50.0000 N·m
What is Torque?
Torque, also known as moment of force, is a measure of the rotational force applied to an object. It determines how effectively a force can cause an object to rotate around an axis. Torque depends on the magnitude of the force, the distance from the axis of rotation, and the angle at which the force is applied. The SI unit for torque is Newton-meters (N·m).
Torque Formulas
From Force
τ = r × F × sin(θ)
Where: τ = torque, r = distance, F = force, θ = angle
Rotational Form
τ = I × α
Where: τ = torque, I = moment of inertia, α = angular acceleration
What Is Torque?
Torque (also called moment of force) is the rotational equivalent of linear force—it measures the tendency of a force to cause rotation about an axis. Just as force causes linear acceleration, torque causes angular acceleration. Every time you turn a doorknob, use a wrench, or pedal a bicycle, you're applying torque.
Torque depends on three factors: the magnitude of force, how far from the rotation axis the force is applied (lever arm), and the angle at which force is applied. Maximum torque occurs when force is perpendicular to the lever arm.
| Quantity | Linear Motion | Rotational Motion |
|---|---|---|
| Cause of motion | Force (F) | Torque (τ) |
| Inertia | Mass (m) | Moment of inertia (I) |
| Displacement | Distance (d) | Angle (θ) |
| Velocity | v (m/s) | ω (rad/s) |
| Acceleration | a (m/s²) | α (rad/s²) |
| Newton's 2nd Law | F = ma | τ = Iα |
Torque Formula
Where:
- τ= Torque (Newton-meters, N·m)
- r= Lever arm length (meters)
- F= Applied force (Newtons)
- θ= Angle between r and F
The Lever Arm Principle
The lever arm (or moment arm) is the perpendicular distance from the rotation axis to the line of action of the force. Longer lever arms multiply your force's turning effect—this is why long wrenches make loosening bolts easier, and why door handles are placed far from hinges.
Archimedes famously said, "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." This captures the mechanical advantage of torque.
| Tool/Situation | Typical Lever Arm | Force Applied | Resulting Torque |
|---|---|---|---|
| Door handle | 0.8 m from hinge | 10 N | 8 N·m |
| Short wrench | 0.15 m | 100 N | 15 N·m |
| Long wrench | 0.5 m | 100 N | 50 N·m |
| Breaker bar | 1.0 m | 100 N | 100 N·m |
| Car steering wheel | 0.18 m | 50 N | 9 N·m |
| Bicycle pedal | 0.17 m (crank length) | 200 N | 34 N·m |
Pro tip: If a bolt won't budge, use a longer wrench or attach a pipe to extend the lever arm rather than applying more force (which risks injury or breaking the tool).
Torque Units and Conversions
Torque is measured in force × distance units. Different regions and industries use different conventions, which can cause confusion, especially in automotive contexts where engine torque specifications vary between markets.
| Unit | Symbol | Used In | Conversion to N·m |
|---|---|---|---|
| Newton-meter | N·m | SI standard, engineering | 1 (reference) |
| Pound-foot | lb-ft (or ft-lb) | US automotive | 1.356 N·m |
| Kilogram-meter | kg·m | Some European specs | 9.807 N·m |
| Kilogram-centimeter | kg·cm | Servo motors | 0.0981 N·m |
| Ounce-inch | oz·in | Small motors, RC | 0.00706 N·m |
| Dyne-centimeter | dyn·cm | Scientific | 10⁻⁷ N·m |
Important: Don't confuse lb-ft (torque) with ft-lb (work/energy). While dimensionally equivalent, the convention is lb-ft for torque and ft-lb for work.
Rotational Equilibrium and Balance
An object is in rotational equilibrium when the sum of all torques equals zero—it has no tendency to rotate. This principle is essential for designing balanced structures, analyzing seesaws, and understanding how cranes stay upright.
For equilibrium: Σ τ_clockwise = Σ τ_counterclockwise
| System | Balance Condition | Application |
|---|---|---|
| Seesaw | m₁r₁ = m₂r₂ | Children's playground |
| Crane | Load × arm = Counterweight × arm | Construction safety |
| Lever | F_in × r_in = F_out × r_out | Mechanical advantage |
| Mobile (art) | Σ τ = 0 at each pivot | Balanced hanging sculpture |
| Human body | Muscles counteract gravity | Posture, lifting |
The center of mass concept is closely related—an object balanced at its center of mass has zero net torque from gravity.
Engine Torque and Power
In automotive contexts, engine torque measures the rotational force an engine produces at the crankshaft. Torque determines acceleration feel and towing capacity, while power (torque × RPM) determines top speed capability. The relationship is: Power = Torque × Angular Velocity.
| Vehicle Type | Engine | Peak Torque | Peak Power |
|---|---|---|---|
| Economy car | 1.5L 4-cyl | 145 N·m @ 4,500 RPM | 80 kW (107 hp) |
| Family sedan | 2.0L turbo | 350 N·m @ 2,000 RPM | 185 kW (248 hp) |
| Sports car | 3.0L twin-turbo | 500 N·m @ 3,000 RPM | 300 kW (402 hp) |
| Pickup truck | 5.0L V8 | 570 N·m @ 4,000 RPM | 290 kW (389 hp) |
| Diesel truck | 6.7L turbo diesel | 1,050 N·m @ 1,800 RPM | 330 kW (443 hp) |
| Electric car | Dual motor | 660 N·m @ 0 RPM | 350 kW (469 hp) |
Electric vehicle advantage: Electric motors produce maximum torque at 0 RPM (instantaneous), giving EVs excellent acceleration from a standstill.
Power-Torque Relationship
Where:
- P= Power (Watts)
- τ= Torque (N·m)
- ω= Angular velocity (rad/s)
- RPM= Revolutions per minute
Torque Specifications and Fastening
Proper torque specifications are critical for mechanical assemblies. Under-tightening can cause loosening and failure; over-tightening can strip threads or crack components. Torque wrenches ensure fasteners are tightened correctly.
| Application | Typical Torque | Why It Matters |
|---|---|---|
| Wheel lug nuts (car) | 80-140 N·m | Safety: wheel must not come loose |
| Engine head bolts | 40-80 N·m (multiple stages) | Sealing: head gasket compression |
| Spark plugs | 12-28 N·m | Seal without cracking ceramic |
| Bicycle pedals | 35-40 N·m | Secure but removable |
| Bicycle stem bolts | 4-6 N·m | Clamp carbon without crushing |
| Electronics screws | 0.3-0.5 N·m | Secure delicate components |
Torque sequence matters: For multi-bolt assemblies like cylinder heads, tighten in a specific pattern (usually center-out) in multiple stages to ensure even clamping force.
Rotational Dynamics: τ = Iα
Newton's second law for rotation states that torque equals moment of inertia times angular acceleration: τ = Iα. This is analogous to F = ma for linear motion. Moment of inertia (I) plays the role of mass—it measures how hard it is to change an object's rotation.
| Object Shape | Moment of Inertia | Axis of Rotation |
|---|---|---|
| Solid cylinder/disk | I = ½MR² | Through center, perpendicular |
| Hollow cylinder | I = MR² | Through center, perpendicular |
| Solid sphere | I = ⅖MR² | Through center |
| Hollow sphere | I = ⅔MR² | Through center |
| Thin rod (center) | I = 1/12 ML² | Perpendicular, through center |
| Thin rod (end) | I = ⅓ML² | Perpendicular, through end |
Key insight: Mass farther from the rotation axis contributes more to moment of inertia. This is why figure skaters spin faster when they pull their arms in—they reduce I, and angular momentum (L = Iω) is conserved, so ω increases.
Newton's Second Law for Rotation
Where:
- τ= Net torque (N·m)
- I= Moment of inertia (kg·m²)
- α= Angular acceleration (rad/s²)
Worked Examples
Wrench Torque Calculation
Problem:
You apply 80 N of force at the end of a 0.25 m wrench, perpendicular to the handle. What torque do you apply to the bolt?
Solution Steps:
- 1Identify values: F = 80 N, r = 0.25 m, θ = 90° (perpendicular)
- 2Apply torque formula: τ = r × F × sin(θ)
- 3Since sin(90°) = 1: τ = r × F
- 4Calculate: τ = 0.25 × 80 = 20 N·m
- 5Convert to lb-ft: 20 ÷ 1.356 = 14.7 lb-ft
Result:
You apply 20 N·m (14.7 lb-ft) of torque. If the bolt requires 50 N·m, you need either more force (200 N) or a longer wrench (0.625 m).
Seesaw Balance Problem
Problem:
A 30 kg child sits 2 m from the pivot of a seesaw. Where must a 45 kg child sit to balance?
Solution Steps:
- 1For balance: τ_clockwise = τ_counterclockwise
- 2Torque from child 1: τ₁ = m₁gr₁ = 30 × g × 2
- 3Torque from child 2: τ₂ = m₂gr₂ = 45 × g × r₂
- 4Set equal: 30 × g × 2 = 45 × g × r₂
- 5Simplify (g cancels): 60 = 45 × r₂
- 6Solve: r₂ = 60/45 = 1.33 m
Result:
The 45 kg child must sit 1.33 m from the pivot to balance the seesaw. Heavier children sit closer to the pivot.
Engine Power from Torque
Problem:
A car engine produces 350 N·m of torque at 4,500 RPM. What is the power output?
Solution Steps:
- 1Use power formula: P = τ × ω
- 2Convert RPM to rad/s: ω = 4500 × 2π/60 = 471.2 rad/s
- 3Calculate power: P = 350 × 471.2 = 164,920 W
- 4Convert to kW: P = 164.9 kW
- 5Convert to hp: P = 164,920 ÷ 746 = 221 hp
Result:
The engine produces 164.9 kW (221 hp) at peak torque. This is why automakers advertise both torque and power—they describe different aspects of performance.
Tips & Best Practices
- ✓Maximum torque occurs when force is perpendicular to the lever arm (sin 90° = 1).
- ✓To increase torque without more force, increase the lever arm length—use a longer wrench or pipe extension.
- ✓Always use a torque wrench for critical fasteners like wheel lug nuts and engine components.
- ✓Torque and energy have the same units (N·m) but are different concepts—torque is a force moment, not energy.
- ✓For equilibrium problems, choose a convenient pivot point to simplify calculations (one that eliminates unknown forces).
- ✓Convert RPM to rad/s by multiplying by 2π/60 before using P = τω.
- ✓Remember the parallel axis theorem: I = I_cm + Md² when calculating moment of inertia about an offset axis.
Frequently Asked Questions
Sources & References
Last updated: 2026-01-22