AkCalculators

Torque — definition, physical meaning, equations and practical applications

Torque is a fundamental concept in mechanics describing the tendency of a force to rotate an object about an axis, pivot or fulcrum. In simple terms, torque measures how effective a force is at producing rotational motion. The scalar magnitude of torque τ for a point force acting at a distance from an axis is given by the cross-product relation:

τ = r × F = r · F · sinθ

Here r is the lever arm (distance from the axis to the line of action of the force), F is the force magnitude, and θ is the angle between the lever arm and the force vector. When the force is perpendicular to the lever arm (θ = 90°), the formula reduces to τ = r·F, which is often the practical case for wrenches, levers and simple torque-transfer scenarios.

Units and physical interpretation

The SI unit of torque is the newton-metre (N·m). Although the unit is dimensionally equivalent to joules (N·m = J), torque and energy represent different physical quantities — torque is a vector (or a pseudo-vector in 3D) representing rotational effect, while energy is a scalar. In some regions and industries, torque is also expressed in pound-foot (lb·ft) or kilogram-force metre (kgf·m). Converting between units requires multiplying by the appropriate conversion factor.

Why the sine of the angle?

The sinθ factor appears because only the component of the force perpendicular to the lever arm produces rotation. If the force acts along the lever arm (θ = 0° or 180°), sinθ = 0 and no torque is produced. If the force is perpendicular, sinθ = 1 and torque is maximal for that lever arm and force magnitude.

Torque and moment of force — vector form

In vector notation, torque is the cross product of position vector r and force vector F: τ = r × F. The direction of the torque vector follows the right-hand rule and indicates the axis of rotation. For planar problems we often work with scalar magnitudes and sign conventions (e.g., counterclockwise positive).

Design, safety and real-world examples

Torque matters in many practical contexts: tightening bolts with a torque wrench, designing gearboxes, sizing motors, calculating stresses on shafts, and analysing braking systems. For example, a mechanic uses a torque wrench to apply a specified torque to a bolt; applying too little torque can allow the joint to loosen, while too much can strip threads or overstress components.

Consider a wrench 0.25 m long where you apply 80 N of force perpendicular to the handle. The torque applied to the nut is τ = 0.25·80 = 20 N·m. Doubling the lever arm yields the same torque with half the force — this is the mechanical advantage of a lever.

Torque and power — the rotational link

Torque and angular speed together determine mechanical power in rotating systems. The instantaneous mechanical power delivered by a torque is the product of torque and angular velocity:

P = τ · ω

where ω is angular speed in radians per second (rad/s). If rotational speed is specified in revolutions per minute (RPM), convert to radians per second using ω = 2π·RPM / 60. Thus, if τ is given in N·m and RPM in rev/min, power in watts is:

P (W) = τ (N·m) × 2π × RPM / 60

This relation is essential when converting motor torque ratings and speed to deliver mechanical power — or conversely when computing required torque from desired power at a given rotational speed. For instance, a motor rated at 3 kW operating at 1500 RPM produces torque τ = (P·60) / (2π·RPM) ≈ (3000·60) / (2π·1500) ≈ 19.1 N·m.

Computing unknowns — common problem types

Typical tasks include:

• Compute torque from force, lever arm and angle.
• Given torque and lever arm, compute the force required.
• Compute torque from power and RPM (or compute power from torque and RPM).
• Convert torque between units (N·m ↔ lb·ft ↔ kgf·m).

These problems are algebraically straightforward but it is crucial to maintain consistent units, and convert angular measures to radians when using P = τ·ω. Also check whether force is given as a weight (mass × g) or as a direct force in newtons — improperly treating kg as newtons is a common source of error.

Units, conversions and common pitfalls

Useful conversion factors (approximate):

• 1 N·m = 0.73756 lb·ft
• 1 lb·ft = 1.35582 N·m
• 1 kgf·m ≈ 9.80665 N·m
• Power: 1 kW = 1000 W, 1 HP (mechanical) ≈ 745.7 W

Common pitfalls:

• Mixing mass (kg) and force (N). Convert mass into force using F = m·g (g ≈ 9.80665 m/s²) when appropriate.
• For non-perpendicular forces, include sinθ.
• When using RPM, always convert to rad/s for P = τ·ω, or use the direct RPM formula P = τ·2π·RPM/60.

Worked example

Suppose a gear requires 500 W of mechanical power at 1200 RPM. What torque must the driving shaft supply?

Use τ = P / ω, with ω = 2π·RPM/60 = 2π·1200/60 = 40π ≈ 125.66 rad/s. Thus τ = 500 / 125.66 ≈ 3.979 N·m. In lb·ft this is ≈ 3.979 × 0.73756 ≈ 2.93 lb·ft.

Practical measurement and instrumentation

Torque is measured using transducers such as strain-gauge-based torque sensors, reaction torque sensors, and calibrated torque wrenches. For rotating shafts, rotary torque transducers measure torque while allowing rotation through slip rings or wireless telemetry. Calibration and appropriate measurement range are important to ensure accuracy.

Design considerations

When designing shafts and couplings, engineers use torque together with material strength and safety factors to size components (torsional shear stresses, angle of twist). Powertrain design also combines torque-speed curves of engines or motors with gear ratios to meet application requirements.

In summary, torque is a simple but powerful quantity that links force to rotational effect and connects directly to power when combined with rotational speed. Accurate unit management, clear sign conventions and awareness of angular measures will avoid common mistakes in calculations.