# Questions about newton metre units

I have a few values in Newton metres and I need to make sure each value is an SI unit for calculations. However when I try to search for the units I am given, I can only find similar, but not quite the same, units.

Can someone help clear up the confusion?

• I'm given the unit Nm$$^{-1}$$, the unit of a spring constant. As far as I can tell this is already an SI unit but I'm confused as I can only find the unit Nm online. Are Nm and Nm$$^{-1}$$ the same? I would say that Nm is a newton metre whereas Nm$$^{-1}$$ is newtons per metre, which is why I'm confused about this
• Similarly, I'm given Nsm$$^{-1}$$ but I can only find Nsm or Nsm$$^{2}$$ online, leading to more confusion over whether or not I have the correct units.
• $Nm$ is vastly more common than $Nm^{-1}$ because it's a popular unit of torque and quite prevalent in automotive, motorsports communities etc, enthusiasts comparing torque curves of their favorite cars. Plus completely independently it happens to be equal to 1 Joule, and is sometimes used as unit of work to distinguish it from energy. Additionally, in case of a torsional spring (twisting, not squeezed) the torque may appear as momentary value of torque exerted by/on the spring, while the constant will be something like N/radian. For linear spring, $Nm^{-1}$ is right as the constant.
– SF.
Jul 20 '20 at 12:55
• SI units named after a person have their symbols capitalised but are lowercase when spelled out. 'V' for volt, 'A' for ampere, 'K' for kelvin, 'Ω' (capital omega) for ohm, etc. Your first sentence should read "newton metres". Jul 21 '20 at 8:04

N/m or Nm$$^{-1}$$ is the correct unit of a spring constant and it's already in SI units. Nm or N*m on the other hand is the unit of a torque (or moment) and is also in SI units. So no, Nm and Nm$$^{-1}$$ are not the same at all, they measure very different things.

Similarly, Nsm$$^{−1}$$, Nsm and Nsm$$^2$$ are all different units of different dimensions to measure different things, and are all in SI units.

There are seven fundamental SI units (seconds (time), metres (length), kilograms (mass), ampere (current), kelvin (temperature), mole (amount of substance) and candela (luminous intensity) - https://en.wikipedia.org/wiki/International_System_of_Units.

There are a bunch of derived units and acceptable prefixes (for orders of magnitude) (see that Wikipedia page). When you see a measurement expressed purely in terms of one or more of those units (like Nm or N/m), then it is an acceptable SI unit. But, if you mix and match prefixes in a unit (gram per cubic centimeter, for example), then you've fallen off the SI truck. As the Wiki page puts it: When prefixes are used to form multiples and submultiples of SI base and derived units, the resulting units are no longer coherent.

One thing that is important to note is the importance of dimensional analysis. As others have pointed out, Nm and N/m are very different, as are $$Nsm^{−1}$$, Nsm and $$Nsm^2$$. When you multiply or divide two quantities, carry the corresponding units around and do a similar multiplication or division of the units to figure out what the resuling unit should be. For example, the specific heat of water at standard conditions is about $$4180\ Jkg^{−1}K^{−1}$$ (joules per kilogram per kelvin). It I want to raise the temperature of 10 kg of water 2 kelvin, I will need $$10kg\ \cdot \ 2K\ \cdot \ 4180\ Jkg^{−1}K^{−1}$$.

Multiplying the numbers is easy (10 x 2 x 4180 = 83,600). But, then you need to consider the units of the result. Multiplying the units out yields: $$kg \cdot K \cdot J \cdot kg^{−1}K^{−1}$$ Then go through and delete units that "cancel themselves out" ($$kg \cdot kg^{-1}$$ and $$K \cdot K^{−1}$$). What's left is Joules. So the answer is 83,600 J. Since our expectation is that some amount of energy is needed to raise the temperature of water, we can be more confident in the answer.