# 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 $(\mathrm{N} = \mathrm{kg} \, \mathrm{m} / \mathrm{s}^2)$. For example, a force of $4\, \mathrm{N}$ accelerates $4\, \mathrm{kg}$ of mass by $1\, \mathrm{m/s}^2$, or alternatively $1\, \mathrm{kg}$ of mass by $4\, \mathrm{m/s}^2$ and so on.

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

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

In summary:

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

### 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 $\mathrm{N} \, \mathrm{m}$ as the unit. One might now think that this means that torque and work are the same thing, since work is also measured in $\mathrm{N}\,\mathrm{m}$ after all. However, this is not the case, and it would in fact be invalid to use $\mathrm{J}$ instead of $\mathrm{N}\,\mathrm{m}$ for torque! The reason for this confusion is that the unit of torque is actually $\mathrm{N} \, \mathrm{m}/\mathrm{rad}$, but since the radian part is dimensionless, it is usually omitted.

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

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

## Example

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

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

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