How to Calculate Torque: Formulas, Units, and Real-World Applications

Master torque calculations for engineering applications. Covers torque formulas, moment arms, units conversion, motor torque, bolt torque, and the relationship between torque, speed, and power.

What Is Torque?

Torque is a measure of the rotational force applied to an object about an axis. It is the rotational equivalent of linear force: while a force pushes or pulls an object in a straight line, torque twists or rotates it. Mathematically, torque (T or tau) equals force (F) multiplied by the perpendicular distance from the axis of rotation to the line of action of the force, known as the moment arm (r): T = F times r. The SI unit of torque is the newton-meter (N-m), and in imperial systems it is the pound-foot (lb-ft) or pound-inch (lb-in). Torque is a vector quantity with a direction determined by the right-hand rule, pointing along the axis of rotation.

Torque, Power, and Speed Relationship

The relationship between torque, power, and rotational speed is fundamental to mechanical engineering. Power (P) equals torque (T) multiplied by angular velocity (omega): P = T times omega. Since angular velocity in radians per second relates to RPM by omega = 2 pi N / 60, the formula becomes P = T times 2 pi N / 60, or equivalently T = 9549 times P / N when P is in kilowatts, T is in N-m, and N is in RPM. This relationship means that for a given power output, torque and speed are inversely proportional. A low-speed motor produces high torque, and a high-speed motor produces low torque, which is why gearboxes are used to match motor characteristics to load requirements.

Motor Torque Calculations

Electric motors are specified by their rated power and speed, from which the rated torque can be calculated using T = 9549 P / N. For example, a 15 kW motor running at 1450 RPM produces T = 9549 times 15 / 1450 = 98.8 N-m. Motors also have a starting torque (typically 150% to 300% of rated torque for induction motors) and a breakdown torque (the maximum torque before the motor stalls). When selecting a motor, the starting torque must exceed the load's breakaway torque, and the rated torque must exceed the continuous load torque with an adequate margin. Variable frequency drives (VFDs) can adjust motor speed and torque characteristics to match varying load demands efficiently.

Bolt Torque and Preload

In bolted joints, torque is applied to the bolt head or nut to develop a clamping force (preload) that holds the joint together. The relationship is T = K times F times d, where T is the tightening torque, F is the desired preload force, d is the nominal bolt diameter, and K is the torque coefficient (also called the nut factor), which typically ranges from 0.15 to 0.25 depending on lubrication and surface conditions. Only about 10% to 15% of the applied torque actually goes into stretching the bolt; the rest is consumed by friction under the nut face and in the threads. Proper bolt torque is critical: too little allows the joint to loosen under vibration, and too much risks yielding the bolt or stripping the threads.

Torque in Rotating Shafts

Shafts transmit torque from a power source (such as a motor or engine) to a driven load (such as a pump, fan, or wheel). The shear stress in a solid circular shaft due to torque is tau = T times r / J, where r is the radial distance from the center and J is the polar moment of inertia (J = pi d^4 / 32 for a solid shaft of diameter d). The maximum shear stress occurs at the outer surface. Engineers size shafts so that this maximum stress remains below the allowable shear stress of the material, typically the yield shear strength divided by a safety factor. Hollow shafts are more material-efficient because they place material where stress is highest (at the outer radius) and remove unstressed material from the center.

Torque Measurement Methods

Torque can be measured using several methods depending on the application. A torque wrench measures the torque applied to a fastener and is essential for precision assembly. Strain-gauge-based torque transducers (also called torque cells or torque sensors) measure the torsional strain in a shaft to determine the transmitted torque in real time, and are used in engine dynamometers, drivetrain testing, and industrial process monitoring. Reaction torque sensors measure the torque by sensing the reaction force on a stationary housing. Prony brakes and dynamometers absorb the output power while measuring the torque and speed, allowing engineers to characterize motor or engine performance curves.

Unit Conversions and Common Pitfalls

Torque unit conversions are a frequent source of errors. One N-m equals approximately 0.7376 lb-ft, and 1 lb-ft equals 1.3558 N-m. One lb-ft equals 12 lb-in. In automotive contexts, engine torque is often quoted in lb-ft in the US and N-m elsewhere. A critical pitfall is confusing torque (N-m) with energy (also measured in N-m or joules): they have the same units but are fundamentally different quantities. Another common mistake is forgetting to account for the moment arm direction. Only the component of force perpendicular to the lever arm contributes to torque; the formula with an angle is T = F times r times sin(theta), where theta is the angle between the force vector and the lever arm.

Real-World Applications

Torque calculations arise in an enormous range of engineering contexts. In automotive engineering, engine torque curves and gear ratios determine acceleration performance. In wind turbine design, aerodynamic torque on the blades must be matched to the generator's torque-speed characteristic. In robotics, joint torques determine how much weight a robotic arm can lift and how quickly it can move. In manufacturing, cutting torque dictates spindle motor sizing for CNC machines. In civil construction, torque wrenches ensure that structural bolts achieve the correct preload. Wherever rotation or tightening is involved, torque is the governing quantity, making it one of the most universally important concepts in mechanical and structural engineering.

Try These Calculators

Put what you learned into practice with these free calculators.

Shaft Torque Calculator Motor Torque Calculator Bolt Torque Calculator Shaft Power Calculator