Newton’s 2nd Law
\[F=ma\]
Thanks for tuning in.
But seriously…
This is the typical starting point for anyone writing down the 2nd law of motion. But it’s just not what it is.
The version written above is a special case. Firstly, acceleration \(a\) is a vector, so we should writhe \(\vec{a}\) instead of \(a\). If our analysis is one dimensional - then we can reduce \(\vec{a}\) to \(a\), but in general it is a vector quantity as a vector can be one dimensions - or however many dimensions you like.
Secondly, Newton wasn’t thinking about acceleration of a fixed mass - rather he was thinking about how momentum changes with time. His original formulation was: \[\vec{F}=\frac{d\vec{p}}{dt}\]
Where \(\vec{p}\) is momentum, defined as: \[\vec{p}=m\vec{v}\]
and \(\vec{F}\) is the total external force acting upon a system. So a total force \(\vec{F}\) causes a change in momentum \(\vec{p}\) with respect to time \(t\) - and an observed change in momentum can be related to a total force \(\vec{F}\) acting on the system.
In the absence of external forces, \(\vec{F}=0\), momentum is conserved, i.e. \(\frac{d\vec{p}}{dt}=0\) - i.e. it does not change with time.
A conserved quantity in physics is gold - as it allows us to relate two states of a system without knowing the details of how the system got from one state to another, since (in this case) the momentum is the same at both states. Another key conserved quantity is energy \(E\).
Since mass \(m\) is a scalar, and velocity \(\vec{v}\) is a vector, the momentum is a scaled velocity vector, i.e. it has the same direction as velocity, but its magnitude is scaled by mass.
The general form
Carrying out the differentiation using the product rule gives: \[\vec{F}=\frac{d(m\vec{v})}{dt}=m\frac{d\vec{v}}{dt}+\vec{v}\frac{dm}{dt}\]
What we typically do is consider mass to be constant, so the second term vanishes to \(dm/dt=0\), leaving us with: \[\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}\] where \(\frac{d\vec{v}}{dt}=\vec{a}\) is acceleration, the change in velocity with time. This is our familiar form of the 2nd law of motion.
So our typical form is just a special case of the more general form.
Because this is a vector equation, it encodes quite alot of information. If our problem is one dimensions, this reduces to a scalar equation (one equation). For two dimensions, the vector equation encodes two scalar equations (one for each component), and in three dimensions it encodes three scalar equations.
Why does this matter?
In many practical applications, the mass of the system is not constant with respect to time.
Anyone who has studied fluid mechanics or rocket propulsion will be familiar with the more general form, where mass is not constant with respect to time. In rockets, mass is expelled (the propellant) and in fluid dynamics mass flow in a control volume is variable with time. Starting out with a scalar \(F=ma\) in those fields will not get you very far.
Embracing first principles
I’ve always tried to keep in mind the first principles of physics. Newton’s laws are exactly that - first principles. They are not just equations to be memorized, but rather they are the foundation upon which we build our understanding of mechanics.
When we start with the more general form of Newton’s 2nd law, we are reminded that forces cause changes in momentum, and that momentum can change not only due to changes in velocity but also due to changes in mass. This broader perspective is crucial for understanding a wide range of physical phenomena.
Instead of starting half way down the road with \(F=ma\), starting with \(\vec{F}=d\vec{p}/dt\) and then performing acceptable adjustments or simplifacations to the problem at hand keeps us grounded in the fundamental concepts of physics and encourages us to think more deeply about the systems we are analyzing. It does not cost anything extra to remember the more general form, and it pays off in the long run.
A tangent - mass in other contexts
Let’s venture off on a tangent from this topic and consider mass in other contexts.
In simple terms, mass is a measure of inertia. Inertia is the resistance of an object to change in its state of motion. The greater the mass, the greater the inertia, and the more force is required to change its velocity.
Contrast this to a large corporation. A large corporation has a lot of inertia - it is resistant to change. Changing the direction of a large corporation requires significant effort and resources, much like changing the velocity of a massive object requires a large force. It is much easier to change the direction of a small startup company (low mass) than a large multinational corporation (high mass).
However, the mass of a corporation is not fixed. Mergers, acquisitions, layoffs, and growth can all change the “mass” of a corporation over time. This is analogous to the variable mass scenarios in physics where the mass of a system changes with time. However, in this context, these changes with time are not continuous, so the function \(m(t)\) is not differentiable everywhere.
Parkinson’s Law - the mass of bureaucracy
Parkinson’s Law states that “work expands to fill the time available for its completion.” In the context of a corporation, this can be interpreted as the tendency for bureaucratic processes to grow in complexity and size over time, regardless of the actual workload. This growth in bureaucracy can be seen as an increase in the “mass” of the corporation, making it more resistant to change and less agile.
This means that the change in mass of government - and thus of its inertia - is always increasing with time, i.e. \(dm/dt>0\) always. As such, considering it at moving at constant velocity at a given time (\(d\vec{v}/dt=0\)), the magnitude of the force needed to make any change also increases with time \(\left|{d\vec{F}/dt}\right|>0\). In other words, it just gets harder and harder with time to notice any impact of changes made, or a lot more effort has to be expelled for a unit change to be observed. This is an interesting contrast to physical systems where mass can both increase and decrease with time.
However, I have worked in companies that display a chronic beurocratic bloat, where the mass seems to increase without bound. In such cases, the corporation becomes so massive and inertial that it is nearly impossible to change its direction or adapt to new circumstances. So this is not limited to just government…