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In textbooks, we define force as the rate of change of momentum. However, when we look at the mathematical formulation of Newton’s Second Law, we express force as being directly proportional to the rate of change of momentum. This seems to imply that force is a separate quantity that depends on the rate of change of momentum rather than being identical to it.

If force is not exactly the rate of change of momentum but rather something proportional to it, what is force actually? How should we truly understand its nature?

Qmechanic
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2 Answers2

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Whether or not there is any distinction between “force is the rate of change of momentum” and “force is proportional to the rate of change of momentum” is simply a question of your units.

In a system where “force is the rate of change of momentum”, like SI units, then $$\frac{d\vec p}{dt}=\Sigma \vec F$$ is dimensionally consistent.

In a system where “force is proportional to the rate of change of momentum then that will not be dimensionally consistent and must be changed to $$\frac{d\vec p}{dt}=k\ \Sigma \vec F$$ where $k$ is a dimensionful constant that converts the units on the right to the units on the left.

An example of such units is using feet ($\mathrm{ft}$) for distances, seconds ($\mathrm{s}$) for time, pounds ($\mathrm{lbm}$) for mass, and pounds ($\mathrm{lbf}$) for force. In those units the left hand side of equation 1 is in units of $\mathrm{lbm \ ft \ s^{-2}}$ and the right hand side is in units of $\mathrm{lbf}$. This is incompatible, so we have to use equation 2 with $k=32 \mathrm{\ lbm \ ft \ s^{-2} \ lbf^{-1}}$

Dale
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Nothing personal, but I fiercely disagree with the answer by Dale. It's not just matter of units, it's about the concept itself: a force may or may not change the state of a system, depending on constraints. I'd say that force and momentum belong to different topics in classical mechanics: force is a model of actions - whatever they are, and that's what the answer is about - while momentum is a dynamical quantity (qualitativley involving product of inertia and kinematic quantities). Newton's 2nd principle of dynamics - that branch of classical mechanics that "puts everything together", kinematics, inertia, actions,...) - summarizes the experimental evidence that the sum of external forces equals the time derivative of momentum of a system.

What's a force? It could be a philosophical question about the existence of things or our knowledge.

IMHO in Physics we should focus on measurements and interactions with other physical quantities (other "things" that can be measured) to define "things".

Limiting the discussion to classical physics, I'm introducing other "things" before force, to describe what I meant before:

  • what's space? Space is something that you can measure with rulers and squares, for distances and angles. And it's possible to "tune" measurement instruments so that different observers agree on the measures of the "thing", the physical quantity object of measurement. These measurements helps you in comparing angles, distances (the qualitative distinction close or far), and build other geometrical concepts, like those of Eulcidean geometry, that could describe a model of space that is good enough for a lot of applications in classical mechanics;

  • what's time? Time is something you can measure with time-keeping devices, like clocks. Again, locally absolute time "flowing with the same speed" - or with absolute nature, so that different observer agree on time measurement - is a model that is good enough applications governed by classical mechanics. Start thinking about time and measurements, that instruments "don't measure time", or put it in another way instruments count oscillations, that are stable enough to tune instruments that agree on measurements of what we call time. Time is qualitatively associated with evolution, and sequence of events, where - if any - causes come before consequences (causality).

  • now, what's a force? A force could be defined as something that can be measured with instruments called dynamometers, or balances. Rough models of these instruments may involve springs or other elastic materials, whose behavior is stable and predictable (if you don't exceed yield loads) enough to build instruments that agree on measurements about these things we call force. Some measurements of dynamometers may depend on the motion of different observers: these contributions are due to the non-inertial motion of the observer/instruments that see "fictitious" forces/actions, not due to one of the fundamental interactions of physics (the magic 4 interactions, just 2 of them - gravitation and electromagnetic - manifest themselves in classical mechanics); from this observation, the definition of inertial reference frames and Galileian relativity follow.

    From a more-mathematical point of view, a force - that thing/interaction that can be measured with those instruments, as dynamometers, balances or other "advanced springs" - may or may not produce changes in momentum of a system, depending on the constraints on the system:

    • As an a basic example, it's possible to study statics of systems that are loaded with many forces, moments, and other actions: despite forces act on a system, in static conditions nothing moves, since constraints prevent motion though other forces - sometimes called reactions.

    • on the other hand, Newton's 2nd principle of dynamics tells us that net external force acting on a closed mechanical system is proportional to the time derivative of the momentum of the system,

      $$\dfrac{d \vec{Q}}{d t} = \vec{R}^e \ .$$

      where $\vec{Q}$ is the momentum of the system and $\vec{R}^e$ is (defined to be) the sum of all the external forces - those kind of physical quantities bla bla you should already know that tale about dynamometers now - acting on the system

    $$\vec{R}^e := \sum_{i} F^e_i \ .$$

basics
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