Torque
τ = r × F = rF sin(θ). Rotational equivalent of force. τ = Iα for angular acceleration. τ = P/ω from power and angular velocity. N·m or lb·ft.
Why This Physics Calculation Matters
Why: Engines, motors, wrenches, and rotating machinery all depend on torque. Bolt tightening specs, motor ratings, and mechanical design.
How: τ = r × F when force perpendicular. τ = P/ω = 9549×P(kW)/RPM for N·m. Maximum torque at 90° between force and lever arm.
- ●τ = rF sin θ; max when θ = 90°.
- ●Engine power: τ (N·m) ≈ 9549 × P(kW) / RPM.
- ●1 N·m = 0.738 lb·ft.
- ●Right-hand rule for torque direction.
Sample Examples
🔧 Wrench on Bolt
100 N force on 0.3m wrench at 90°
🚪 Opening a Door
20 N push at 0.8m from hinge
🚗 Car Engine
200 kW at 5500 RPM
🚴 Bicycle Pedaling
500 N on 17cm crank at 45°
⚡ Electric Motor
5 kW motor at 3000 RPM
⚙️ Flywheel Acceleration
I = 2.5 kg·m², α = 4 rad/s²
🎮 Steering Wheel
50 N at 0.18m radius
🌬️ Wind Turbine
2 MW at 15 RPM
🪛 Screwdriver
15 N tangential on 2cm handle
🏭 Industrial Motor
I = 50 kg·m², needs α = 2 rad/s²
Enter Your Values
⚠️For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
τ = r × F; rotational analog of F = ma.
— Mechanics
τ = P/ω; power and angular velocity.
— Motors
1 N·m = 0.738 lb·ft; 1 N·m = 0.102 kgf·m.
— Units
τ = Iα; moment of inertia × angular acceleration.
— Rotation
What is Torque?
Torque (τ) is the rotational equivalent of linear force. It measures the tendency of a force to cause rotational motion about an axis. Think of turning a wrench, opening a door, or tightening a bolt - all involve torque. It depends on the force applied, the distance from the pivot point, and the angle of application.
Basic Definition
Torque is force times perpendicular distance from the axis of rotation. It's a vector pointing along the rotation axis.
Formula:
τ = r × F = rF sin(θ)
Common Units
SI unit is Newton-meter (N·m). Other common units include lb·ft and kgf·m.
Conversions:
1 N·m = 0.738 lb·ft
1 N·m = 0.102 kgf·m
Key Applications
Engine power rating, bolt tightening specs, lever mechanics, and rotational machinery design.
- Automotive engines
- Electric motors
- Fastener specifications
How to Calculate Torque
🧮 Method 1: Force × Distance
F = force applied
r = distance from pivot (lever arm)
θ = angle between F and r
Maximum torque when θ = 90° (force perpendicular to lever arm)
📊 Method 2: Power & Speed
P = power in watts
ω = angular velocity in rad/s
RPM = rotations per minute
Useful for motors and engines: τ (N·m) = 9549 × P (kW) / RPM
When to Use Torque Calculations
🚗 Automotive
Engine specifications, wheel lug torque, transmission design, driveshaft sizing
🔧 Mechanical Fasteners
Bolt tightening specifications, torque wrenches, assembly procedures
⚡ Electric Motors
Motor selection, servo sizing, load calculations, gearbox requirements
🏗️ Structural Engineering
Beam analysis, crane design, structural connections
🤖 Robotics
Joint actuators, gripper force, arm kinematics
🚴 Sports Equipment
Bicycle cranks, golf clubs, racket handles
Complete Torque Formulas
Basic Torque
Newton's 2nd (Rotation)
Power Relationship
Angular Momentum
Work by Torque
Motor Torque
Common Torque Values Reference
| Application | Torque (N·m) | Torque (lb·ft) | Notes |
|---|---|---|---|
| M6 bolt (steel) | 8-10 | 6-7 | Dry threads |
| M10 bolt (steel) | 45-55 | 33-40 | Dry threads |
| Car wheel lug nut | 90-120 | 65-90 | Varies by vehicle |
| Spark plug | 25-35 | 18-26 | 14mm thread |
| Bicycle pedal | 35-45 | 26-33 | Standard pedal |
| Small car engine | 100-150 | 74-110 | 1.5L engine |
| Sports car engine | 400-600 | 295-443 | V8 engine |
| Electric motor (small) | 1-10 | 0.7-7 | NEMA 17/23 |
| Wind turbine (large) | 1,000,000+ | 737,000+ | Multi-MW turbine |
Frequently Asked Questions
What's the difference between torque and force?
Force causes linear motion (F = ma), while torque causes rotational motion (τ = Iα). Torque = Force × Distance from pivot. You need both force AND a lever arm to create torque.
Why do car specs show torque AND power?
Torque determines acceleration and pulling force. Power (P = τω) determines top speed. A diesel truck has high torque for towing; a sports car may have high power for speed. Both matter!
Why use a longer wrench for stuck bolts?
Torque = Force × Distance. A longer wrench increases the lever arm (r), so the same force produces more torque. Double the length = double the torque with same effort!
What happens if I over-torque a bolt?
Over-torquing can strip threads, stretch/break the bolt, or damage the materials being fastened. Always follow manufacturer specifications and use a calibrated torque wrench.
How do I convert between torque units?
Common conversions: 1 N·m = 0.7376 lb·ft = 0.1020 kgf·m. For motor torque: τ (N·m) = 9549 × P (kW) / RPM. Always verify unit conversions to avoid calculation errors.
What is the relationship between torque, power, and RPM?
Power = Torque × Angular Velocity. For rotating systems: P (W) = τ (N·m) × ω (rad/s) = τ × 2π × RPM / 60. Higher torque at lower RPM produces same power as lower torque at higher RPM, but with different applications.
Why is angle important in torque calculations?
Torque depends on the perpendicular component of force: τ = F × r × sin(θ). Maximum torque occurs at 90° (perpendicular). At 0° or 180°, torque is zero because force is parallel to the lever arm. Always apply force perpendicular for maximum effectiveness.
📚 Official Data Sources
⚠️ Disclaimer
Important: This calculator provides theoretical torque calculations based on standard mechanical engineering and physics principles. The results are estimates and should not be used as the sole basis for critical engineering decisions.
- Torque values assume ideal conditions; friction, efficiency losses, and dynamic effects are not accounted for
- For fastener applications, always follow manufacturer torque specifications and consider thread lubrication, surface finish, and material properties
- Motor torque varies with speed; use actual motor curves for accurate performance prediction
- Angle measurements assume perfect geometry; real-world applications may have alignment errors
- Safety factors and design codes (ASME, SAE, ISO) should be applied for critical applications
- For high-torque applications, verify component ratings, material strength, and connection integrity
- Unit conversions are approximate; verify critical calculations with multiple sources
- Consider dynamic loads, shock, vibration, and fatigue in real-world applications
No warranty: The authors and providers of this calculator assume no liability for errors, omissions, or damages resulting from the use of these calculations. Improper torque application can cause equipment failure and safety hazards.
Tips and Common Mistakes
✅ Best Practices
- • Apply force perpendicular to lever arm
- • Use calibrated torque wrenches
- • Consider friction and lubrication
- • Check unit conversions carefully
❌ Common Mistakes
- • Confusing N·m with N/m
- • Ignoring the angle of force application
- • Not accounting for friction losses
- • Mixing up lb·ft and ft·lb (same thing!)
Torque as a Vector
Torque is actually a vector quantity, pointing perpendicular to both the force and position vectors. The direction follows the right-hand rule.
Right-Hand Rule
Point fingers along r (position), curl toward F (force). Thumb points in direction of τ (torque). Counterclockwise = positive (out of page).
Cross Product
|τ| = |r||F|sin(θ)
Practice Problems
Problem 1: Wrench on Bolt
You apply 80 N of force to a wrench 25 cm from the bolt center at a 60° angle. What torque do you apply?
Solution:
τ = F × r × sin(θ) = 80 N × 0.25 m × sin(60°)
τ = 80 × 0.25 × 0.866 = 17.32 N·m
Problem 2: Motor Power Rating
A motor produces 15 kW at 2800 RPM. What is its output torque?
Solution:
τ = P / ω = P × 60 / (2π × RPM)
τ = 15000 × 60 / (2π × 2800) = 51.2 N·m
Or using: τ = 9549 × P(kW) / RPM = 9549 × 15 / 2800 = 51.2 N·m
Problem 3: Rotational Acceleration
A flywheel with I = 5 kg·m² needs to accelerate from rest to 200 rad/s in 10 seconds. What constant torque is required?
Solution:
α = Δω / Δt = (200 - 0) / 10 = 20 rad/s²
τ = I × α = 5 kg·m² × 20 rad/s² = 100 N·m
Torque and Equilibrium
For an object to be in complete equilibrium (not accelerating linearly or rotationally), two conditions must be met:
First Condition
Sum of all forces equals zero (no linear acceleration)
Second Condition
Sum of all torques equals zero (no angular acceleration)
Torque in Different Systems
🔩 Threaded Fasteners
Bolt tension related to torque by coefficient of friction and thread geometry.
K ≈ 0.2 typical
⚙️ Gear Systems
Torque multiplication through gear ratio, inverse of speed ratio.
🔗 Belt/Chain Drives
Torque transmitted through tension difference in belt.
Torque Unit Conversions
| From | To N·m | To lb·ft | To kgf·m |
|---|---|---|---|
| 1 N·m | 1 | 0.7376 | 0.1020 |
| 1 lb·ft | 1.3558 | 1 | 0.1383 |
| 1 kgf·m | 9.8067 | 7.2330 | 1 |
| 1 lb·in | 0.1130 | 0.0833 | 0.0115 |
| 1 oz·in | 0.00706 | 0.00521 | 0.00072 |
Key Insights About Torque
Longer lever arm
= More torque
Same force, less effort
Perpendicular force
= Max torque
θ = 90° is optimal
Power = Torque × Speed
P = τω
Trade-off relationship
Rotation analog
τ = Iα
Like F = ma
Historical Development
The concept of torque (moment of force) has ancient origins in the study of levers by Archimedes around 250 BCE. He famously said "Give me a place to stand, and I shall move the Earth" - understanding that a long enough lever arm could generate enormous torque.
Archimedes (250 BCE)
Established the law of the lever: equilibrium when W₁d₁ = W₂d₂. This was the first mathematical treatment of torque and rotational equilibrium.
Newton & Euler (1700s)
Formalized torque as the rotational equivalent of force in the context of rigid body dynamics and the laws of motion.