Torque Formula
Calculate rotational force (torque) from force and lever arm distance.
Essential for mechanical and automotive engineering.
The Formula
τ = F × r × sin(θ)
Torque is the rotational equivalent of force. It depends on the force applied, the distance from the pivot point, and the angle of application.
Variables
| Symbol | Meaning |
|---|---|
| τ | Torque (Newton-meters, N⋅m) |
| F | Applied force (Newtons) |
| r | Distance from the axis of rotation to the force (meters) |
| θ | Angle between the force vector and the lever arm |
Example 1
You apply 50 N to a wrench 0.3 m from the bolt, perpendicular to the wrench
θ = 90°, sin(90°) = 1
τ = 50 × 0.3 × 1
τ = 15 N⋅m
Example 2
Same wrench but force applied at 60° angle
τ = 50 × 0.3 × sin(60°)
τ = 50 × 0.3 × 0.866
τ = 13.0 N⋅m (less torque because force is not perpendicular)
When to Use It
Use the torque formula when:
- Tightening bolts to specification (torque wrenches)
- Calculating engine and motor output
- Designing levers, cranks, and rotating machinery
- Analyzing the turning effect of forces in structures
Key Notes
- Maximum torque occurs at θ = 90° (force perpendicular to the arm, sin 90° = 1) — pushing straight down on a wrench handle produces more torque than pushing at an angle
- Torque and energy share the unit N⋅m (= joule) but are different quantities — torque is a vector with a rotational direction; work is a scalar with no direction
- In automotive specs, engine torque (N⋅m or lb⋅ft) and power (kW or hp) are linked: Power (W) = Torque (N⋅m) × Angular velocity (rad/s)
Key Notes
- Formula: τ = r × F × sinθ: r is the moment arm (distance from pivot to point of force application), F is the force magnitude, and θ is the angle between the force vector and the moment arm. Maximum torque occurs at θ = 90° (force perpendicular to the arm).
- Newton's law for rotation: τ = Iα: Torque causes angular acceleration α (rad/s²) in proportion to the moment of inertia I (kg·m²). This is the rotational analogue of F = ma. A larger I (mass farther from axis) requires more torque for the same angular acceleration.
- Power: P = τ × ω: Rotational power equals torque times angular velocity (in rad/s). Engine power ratings: 1 hp = 745.7 W. At a given power, higher torque comes with lower RPM. Diesel engines produce high torque at low RPM; gasoline engines produce peak power at higher RPM.
- Torque units: N·m, not joules: Both are dimensionally identical (force × distance) but torque is a moment (rotational tendency), not energy. Never convert between them — they are conceptually distinct physical quantities.
- Applications: Torque calculations govern fastener preload (bolt torque specs), gear and pulley system design, motor and engine selection, structural moment analysis, wheel nut tightening standards, and the design of torque wrenches and dynamometers.