Torque Calculator
Last Updated: October 30, 2025
Calculate torque using force and lever arm (τ = F × r) or using moment of inertia and angular acceleration (τ = I × α). Useful for rotational dynamics and mechanical engineering.
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Applied force
Distance from axis to line of action
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Table of Contents
What is a Torque Calculator?
Understanding Rotational Effect of Forces
The Torque Calculator computes the rotational effect of a force about an axis. It supports two common methods: using applied force and perpendicular distance (lever arm), and using moment of inertia with angular acceleration.
This calculator is essential for designing shafts, selecting actuators, sizing motors, and analyzing bolts and joints subjected to turning forces.
Key Concepts
Torque (τ): Rotational effect of force, measured in Newton-meters (N·m). A vector quantity representing the tendency of a force to cause rotation about an axis. Analogous to force in linear motion.
Force (F): Applied force, measured in Newtons (N). The magnitude of the force determines the torque when combined with lever arm length.
Lever Arm (r): Perpendicular distance from the axis of rotation to the line of action of the force, measured in meters (m). Also called the moment arm or torque arm.
Moment of Inertia (I): Resistance to angular acceleration, measured in kilogram-square meters (kg·m²). Depends on mass distribution relative to the axis of rotation.
Angular Acceleration (α): Change rate of angular velocity, measured in radians per second squared (rad/s²). Torque produces angular acceleration through the relationship τ = Iα.
Physical Interpretation
Torque is the rotational equivalent of force. Just as force causes linear acceleration (F = ma), torque causes angular acceleration (τ = Iα). The effectiveness of a force in producing rotation depends not only on the magnitude of the force but also on how far from the axis it's applied and at what angle. This is why longer wrenches make it easier to turn bolts, and why door handles are placed far from the hinge.
The formula τ = F × r shows that torque increases with both force magnitude and lever arm length. The lever arm must be the perpendicular distance - if you push at an angle, only the perpendicular component of the force contributes to torque. This is why "torque wrenches" are essential tools in automotive and mechanical applications, ensuring bolts are tightened to the correct rotational force.
Right-Hand Rule and Direction
Torque is a vector quantity with both magnitude and direction. The direction follows the right-hand rule: curl your fingers in the direction of rotation, and your thumb points in the direction of the torque vector. Torque can be positive (counterclockwise) or negative (clockwise), depending on the convention used.
Relationship to Work and Power
Torque relates to rotational work and power. Rotational work W = τθ (torque times angular displacement), and rotational power P = τω (torque times angular velocity). This is why engines are rated in both horsepower (power) and torque - torque determines acceleration and pulling ability, while power determines top speed and sustained performance.
Historical Development
The concept of torque (or moment) was developed as part of classical mechanics, building upon Archimedes' work on levers and mechanical advantage. The mathematical formulation τ = r × F emerged from vector mechanics developed in the 18th and 19th centuries. Torque is fundamental to understanding rotational dynamics, mechanical advantage, and the design of all rotating machinery from simple tools like wrenches to complex systems like engines and turbines. The relationship τ = Iα (Newton's second law for rotation) is one of the cornerstones of rotational dynamics.
Units and Conversions
SI Unit: N·m (Newton-meter) or J (Joule) - though torque and work have the same units, they represent different physical quantities
Alternative Units: lb·ft (pound-foot), lb·in (pound-inch) - common in US engineering
Conversion: 1 N·m = 0.7376 lb·ft, 1 lb·ft = 1.3558 N·m
Practical Note: In engineering, torque is often specified with force and distance units (e.g., "50 N at 0.3 m" produces 15 N·m torque)
Formulas and Equations
Torque Calculation Methods
1. From Force and Lever Arm
τ = F × r
Where:
- • τ = Torque (N·m)
- • F = Force (N)
- • r = Perpendicular distance (m)
Use case: Calculate torque from force and lever arm distance. This is the most fundamental torque calculation, essential for analyzing wrenches, levers, pulleys, gears, and any mechanical system where force is applied at a distance from a rotation axis. Used extensively in mechanical engineering for bolt tightening, door mechanisms, and calculating mechanical advantage.
2. From Inertia and Angular Acceleration
τ = I × α
Where:
- • τ = Torque (N·m)
- • I = Moment of inertia (kg·m²)
- • α = Angular acceleration (rad/s²)
Use case: Calculate torque from moment of inertia and angular acceleration. This method uses Newton's second law for rotation (τ = Iα) and is essential for motor sizing, actuator selection, and analyzing rotating machinery dynamics. Critical for designing systems that need to accelerate or decelerate rotating masses, such as motors, flywheels, spindles, and robotic joints.
Applications of Torque
Real-World Uses Across Industries
| Industry | Applications | Importance |
|---|---|---|
| Mechanical Engineering | Bolted joints, shaft sizing, gears, levers, actuators | Critical for design and safety |
| Automotive | Lug nuts, engine torque, drivetrain design, brake systems | Vital for performance and safety |
| Industrial | Conveyors, mixers, pumps, valves, robotics joints | Fundamental for machinery |
| Aerospace | Fasteners, control surfaces, actuator sizing, ground support tools | Key for reliability |
| Construction | Torque wrenches, bolting, cranes, hoists, power tools | Essential for safety |
Examples of Torque Calculation
Real-World Applications and Use Cases
Example 1: Wrench on a Bolt
Given:
- • Force: F = 120 N
- • Lever arm: r = 0.25 m
Step-by-step calculation:
Step 1: Apply torque formula
τ = F × r
τ = 120 × 0.25 = 30 N·m
Final Answer
30 N·m
Torque applied to the bolt
Example 2: Flywheel Acceleration
Given:
- • Moment of inertia: I = 0.25 kg·m²
- • Angular acceleration: α = 12 rad/s²
Step-by-step calculation:
Step 1: Apply τ = I × α
τ = 0.25 × 12 = 3.0 N·m
Final Answer
3.0 N·m
Torque required to accelerate the flywheel
Example 3: Motor Torque for Starting Load
Given:
- • Rotor moment of inertia: I = 0.5 kg·m²
- • Required angular acceleration: α = 50 rad/s²
- • Application: Starting a rotating load from rest
Step-by-step calculation:
Step 1: Apply torque formula for angular acceleration
τ = I × α
τ = 0.5 × 50
τ = 25 N·m
Step 2: Convert to alternative units (for reference)
τ = 25 N·m × (0.7376 lb·ft / N·m) = 18.4 lb·ft
Final Answer
25 N·m (18.4 lb·ft)
This is the minimum torque required from the motor to achieve the desired angular acceleration. In practice, additional torque margin is needed to account for friction and load variations.