N m is not N m

Force, work and power. We often use these words interchangeably, but their meaning is not quite the same.

Force is what causes acceleration of mass, and it's measured in Newton (N=kgm/s2)(\mathrm{N} = \mathrm{kg} \, \mathrm{m} / \mathrm{s}^2). For example, a force of 4N4\, \mathrm{N} accelerates 4kg4\, \mathrm{kg} of mass by 1m/s21\, \mathrm{m/s}^2, or alternatively 1kg1\, \mathrm{kg} of mass by 4m/s24\, \mathrm{m/s}^2 and so on.

Work is measured in Joule (J=Nm)(\mathrm{J} = \mathrm{N} \, \mathrm{m}), which is nothing else than the acceleration of mass by a certain force over a certain distance. For example, 4Nm4\, \mathrm{N}\,\mathrm{m} is the amount of work required to apply a force of 4N4 \, \mathrm{N} over a distance of 1m1\, \mathrm{m}, or alternatively to apply a force of 1N1\, \mathrm{N} over a distance of 4m4 \, \mathrm{m}.

And finally there is power, which is work per time and is measured in Watt (W=J/s=Nm/s)(\mathrm{W} = \mathrm{J}/\mathrm{s} = \mathrm{N}\, \mathrm{m} / \mathrm{s}). For example, doing 1J1\, \mathrm{J} of work per second requires twice as much power as doing 1J1\, \mathrm{J} of work only every two seconds.

In summary:

  • Force — Newton (N)(\mathrm{N})
  • Work — Joule (J=Nm)(\mathrm{J} = \mathrm{N} \, \mathrm{m})
  • Power — Watt (W=J/s=Nm/s)(\mathrm{W} = \mathrm{J}/\mathrm{s} = \mathrm{N} \, \mathrm{m}/ \mathrm{s})

Torque vs Work

Torque is the rotational force resulting from linear force acting on a lever, and is given by the amount of linear force times the length of the lever, leaving us with Nm\mathrm{N} \, \mathrm{m} as the unit. One might now think that this means that torque and work are the same thing, since work is also measured in Nm\mathrm{N}\,\mathrm{m} after all. However, this is not the case, and it would in fact be invalid to use J\mathrm{J} instead of Nm\mathrm{N}\,\mathrm{m} for torque! The reason for this confusion is that the unit of torque is actually Nm/rad\mathrm{N} \, \mathrm{m}/\mathrm{rad}, but since the radian part is dimensionless, it is usually omitted.

NmradJ=Nm\frac{\mathrm{N} \, \mathrm{m}}{\mathrm{rad}} \neq \mathrm{J} = \mathrm{N} \, \mathrm{m}

For example, the work required for a full rotation with 1Nm/rad1 \, \mathrm{N} \, \mathrm{m} / \mathrm{rad} of torque is (2πrad)(1Nm/rad)=2πNm=2πJ(2 \, \pi \, \mathrm{rad}) \cdot (1 \, \mathrm{N} \, \mathrm{m} / \mathrm{rad}) = 2 \, \pi \, \mathrm{N} \, \mathrm{m} = 2 \, \pi \, \mathrm{J}.

Example

The maximum amount of torque that the engine of a Kawasaki Ninja H2R can produce is 165Nm165 \, \mathrm{N} \, \mathrm{m} at 12500rpm12500 \, \mathrm{rpm}. Given that one revolution per minute is 2πrad/60s2 \, \pi \, \mathrm{rad}/60 \, \mathrm{s}, we can infer that the power output at this point is

(165Nm/rad)(125002πrad/60s)216kW,(165 \, \mathrm{N} \, \mathrm{m}/\mathrm{rad}) \cdot (12500 \cdot 2 \, \pi \, \mathrm{rad}/60 \, \mathrm{s}) \approx 216 \, \mathrm{kW},

which is about 290HP290 \, \mathrm{HP}.