# What is an intuitive explanation of units like $N \cdot m$? [duplicate]

Every quantity in Physics has a unit. These units can be in 3 forms:

1. A single unit. An example is kg or m or J
2. A division of two or more units. An example is $\frac{m}{s}$ or $\frac{m}{s^2}$
3. A product of two or more units. An example is $N \cdot m$ or $J \cdot s$

The first case is simple. A person whose mass is 70 kg simply has a mass of 70 kg, simple enough.

The second case is also very simple. A speed of 1 $\frac{m}{s}$ means that 1m of distance is covered per second/every second.

The third case, however, has eluded me for a long time. What does it mean to have a unit like $N \cdot m$? What does the multiplication sign between the newton and meter mean? I tried stripping down the unit to the fundamental parts and I got $\frac{kg \cdot m^2}{s^2}$, which does not help.

Can someone please provide me with an understanding of a unit made of a product of two or more other units?

• – Farcher Dec 6 '16 at 15:51

The structure of the units is directly tied to the equations they come from. We are used to units of speed, hence their appearance of being less mysterious. However quite logical interpretations can be given to the one you mention:

$Nm$ can be units of work. In this case, it really means pushing with a force in Newtons $N$ over a distance in meters $m$. An unmoving force does not imply a transfer of energy, but a force applied during a displacement do imply a transfer of energy which is the work. In other words, an integral of force over distance.

$Nm$ can also be units of torque and, although a bit more perplexing, it means pushing with a force in Newton $N$ with a lever of a certain length in meters $m$. As vector quantities (the length is a directed length) their product is given by the cross product, yielding the torque, but the magnitude of their product is the product of their magnitudes times a unitless angular factor. Hence, the $Nm$. It has the same units as work while still being more of a "force" because the associated angular displacement is unitless.

$Js$ is units for angular momentum. Since a torque has the same units as energy (because of the dimensionless angular displacement) angular momentum Is the cumulative rotational effect of a torque over time. In other words, the integral of the torque over time.

$Js$ is a unit of action. This would be the most esoteric, because action is not a quantity present in daily life. It amounts to the integral of energy over time. An analogy with angular momentum can be made where in action/angle variables of a classical system, the action is the "angular momentum" that keeps the angle variable "spinning". In any case, it is still an integral over time of energy.

Maybe you start seeing the pattern here. Alot of these "product" units are naturally associated to integrals of a quantity over another one. As such, they are related to cumulative quantities or effets over another dimension.

This is a unit for a physical quantity called work. Work is defined as the product of an applied force (N) and the distance applied (m).

There are 2 kinds of Newton meters. A vector force (newton) dotted into a vector distance (meter) results in a scalar Joule: work is force times distance. If you take the cross product, it is a force times a lever arm, which results in a torque, which is a (pseudo)vector. Traditionally the Newton and meter are NOT converted to a Joule, even though mathematics allows it. In the lab, when you tighten a bolt on your space craft and it says "10 Nm", you can have 100 Newton an a 0.1 meter wrench, or a 1 meter wrench with 10 Newtons. If you ask your tech for 10 Joules of torque, it won't happen.