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Torque

Turning effort is force times radius - how to calculate it, which unit belongs to it, and why the order of the two words is not decoration.

Torque is force applied at a radius; Watch out

Torque is force applied at a radius

A force pushes. A torque turns. The same force produces more torque the further it acts from the axis of rotation, which is why a longer wrench loosens a tighter nut with the same pull.

τ = rF  —  torque (τ, tau) equals radius times force.

Torque on a shaft A circle represents a pulley on a shaft, with a dot at its centre. A dashed line runs from the centre out to the rim, labelled r for radius. At the rim, an arrow points along the tangent, at right angles to that line, labelled F for force. The turning effort produced is the radius multiplied by the force. r F τ = rF
The force acts at right angles to the radius. Everything below assumes that; a force pointing partly along the radius contributes only the part of it that is perpendicular.

r is the radius, measured from the axis of rotation to where the force acts. Pulleys, rollers, and wheels are specified by diameter, so halve the number on the drawing before it goes into this formula.

Note: when a turning effort is described about an axis rather than about a shaft, it is often called a moment and given the symbol M. It is the same calculation.

The unit, and the word order that identifies it

The unit, and the word order that identifies it

Torque units and rounded conversion factors between them.
SystemUnitSymbolConversion
SInewton metreN·m1 N·m = 0.7375621 lbf·ft
Foot–slug–secondpound-force footlbf·ft1 lbf·ft = 1.355818 N·m

Case matters, and this one bites. The SI torque unit is N·m with a lower-case m. Written N·M it is not a valid SI torque unit: M is the prefix for 106, but a prefix cannot stand alone as a unit. The same trap catches Kg for kg and S for s. See Measurement Language.

Here torque is written lbf·ft; energy is written ft·lbf. The same two units, the same numerical factor, written in the opposite order — and NIST prints them that way round on purpose: pound-force foot converts to the newton metre, foot pound-force converts to the joule. They are different physical quantities that happen to share a unit combination. Word order is a convention, not a mathematical distinction; use the quantity and context to tell them apart. SI avoids the ambiguity by giving energy its own name, the joule; a torque is never written in joules. See Work and Energy.

Worked example: can the motor hold the arm up?

Worked example: can the motor hold the arm up?

A robot-arm joint motor produces 15 N·m. The arm segment is 0.35 m long and carries a 25 N load at its end. Will the motor hold the arm horizontal?

  1. The load acts straight down, at right angles to a horizontal arm, at the full 0.35 m: τ = 0.35 m × 25 N = 8.75 N·m.
  2. Compare with what the motor produces: 8.75 N·m < 15 N·m.

Yes, with 6.25 N·m to spare — for the load alone. The weight of the arm segment itself also produces a torque about the joint, and no mass was given for it, so this answer is a partial one. In a real selection you would add the arm weight acting at its own centre of mass, then keep margin on top of that.

This is the shape of most torque questions: work out what the load demands, then check it against what the drive can supply, and be explicit about what you have left out.

Worked example: torque from a tangential force

Worked example: torque from a tangential force

A wheel 250 mm in diameter needs 3.5 kN of tangential force to overcome resistance. What torque must the motor supply to the axle?

  1. r = 125 mm = 0.125 m.
  2. 3.5 kN = 3,500 N.
  3. τ = 0.125 m × 3,500 N = 437.5 N·m.

The same formula, rearranged

When the torque is what you know and the force is what you want, F = τ/r. That single rearrangement explains a family of everyday machines: a longer wrench, a bigger steering wheel, and a larger pulley all trade radius against force. Double the radius and the same torque delivers half the force — and, at the same rotational speed, the rim travels twice as far, which is why Power and Efficiency comes out unchanged.

Sources

Sources

Where these values were checked

NIST SP 811, Appendix B lists pound-force foot (lbf·ft) to newton meter as 1.355 818 E+00, and separately lists foot pound-force (ft·lbf) to joule as the same 1.355 818 E+00 — the two entries that establish the word-order convention used above. The reciprocal, 0.7375621, gives newton metres to pound-force feet.

That M is the prefix symbol for 106 and m the symbol for the metre is from the prefix and base-unit tables in NIST SP 811, Chapter 4.

Appears in
DET 403 DET Rotation and Torque