# In a force model, $F = ma$, how to understand the units?

For instance, in an aerodynamic force model, the force terms can be lift terms, drag terms, both of which have translational velocities as factors in their models; but, velocities are in units of, say, meters/sec.

Meanwhile, there are gravity force terms to add in to the model, but the gravitational acceleration, g, is in units of meters/seconds-squared.

So, how do we resolve this apparent issue of adding force terms with different units?

So, how do we resolve this apparent issue of adding force terms with different units?

They do not have different units.

Put all the units of each side down :

A force is in Newtons ($N$) and you already now what mass and acceleration are in.

$$Force = mass \times acceleration$$

gives us :

$$N = kg\,m\,s^{-2}$$

Which is exactly what it should be because that's what a Newton's units are.

All forces will work out that way and if they don't you have either gotten the units of something wrong or done the maths wrong (a useful check).

For instance, in an aerodynamic force model, the force terms can be lift terms, drag terms, both of which have translational velocities as factors in their models; but, velocities are in units of, say, meters/sec.

Forces like this will have some constant term which has the right units to balance everything out.

Sometimes aerodynamic drag is modeled as :

$$F = kv^2$$

The units of $k$ will be what is needed to give a unit of force to both sides.

You have proportionality coefficients (which may or may not be constant) which themselves have units; in the end the product has the dimensions of force. For example if you have linear drag, then that's given by a force of $-\gamma v$ where the dimensions of the viscosity coefficient $\gamma$ are $M L T^{-1}$. If you have gravity, then the coefficient of $g$ is of course the mass itself. And so on.

Sometimes people don't want to deal with these coefficients so they will be set equal to some value (usually $1$). This can create seemingly contradictory equations, like $F=-v$. This equation isn't really contradictory because the viscosity $\gamma$ is built into the system of units in which the equation is stated. Removing dimensional quantities by choosing a problem-specific system of units in this way is called nondimensionalization.

You might be interested in the Buckingham Pi Theorem, which essentially defines the maximum extent to which a problem can be nondimensionalized in terms of some simple linear algebra. Since you said you are interested in aerodynamics, this will help you understand where things like Reynolds numbers and such came from: they are the dimensionless quantities which are allowed to contribute nonlinearly to the problem according to the Buckingham Pi Theorem.