Force That ClicksEpisode 1 / So you won't be fooled by the word "force"

A follow-up to the sister series "Cosmology That Clicks" ── this time, peeling back the true nature of "force," one layer at a time

F=ma: Law or Definition? Everyone knows $F=ma$. But look closely and this equation turns out to be a loop in which force and mass define each other.
So where is the actual "content" of this equation? Peel it back, and force reveals itself to be not a "thing" but a "relationship."

Tools you'll need: division, ratios, $F=ma$ What an interaction gives you is only the "ratio" of masses

Say "force," and we picture pushing, pulling, heaviness ── that feeling transmitted to the hand. Physics captures it all in the single line $F=ma$. It's so famous that we hardly think to doubt it. But this time, as our very first step, let's doubt this equation. Is $F=ma$ a law that nature obeys, or is it merely a definition of the word "force"? Push on this point and the ground beneath the concept of force starts to wobble ── opening the door to the question this whole series will chase: isn't "force" not a real thing, but the name of a relationship?

01Breaking F=ma down, symbol by symbol

Let's interrogate the three symbols in $F=ma$, one by one, asking "what is this, really?"

$a$ (acceleration): This one is clear. It's position differentiated twice with respect to time. With a ruler and a clock, you can measure it. No complaints.
$m$ (mass): What is it? "How hard something is to accelerate when you apply the same force." Written as an equation, $m=F/a$. ── Wait, force just showed up.
$F$ (force): What is it? "Mass × acceleration," that is, $F=ma$. ── Wait, mass just showed up.

The crux this time ── the loop

To know force you need mass ($F=ma$). To know mass you need force ($m=F/a$).
Force and mass define each other using each other. The only thing you can measure is the acceleration $a$; from this single equation alone, neither $F$ nor $m$ is determined on its own.

So naively, $F=ma$ looks less like a "law of nature" and more like a convention (a definition): let's agree to call this quantity "force." In fact, this very point is a venerable sore spot, argued over again and again ever since the physicist Mach. So then ── how do we break this loop?

02Breaking the loop ── an interaction gives you the "ratio" of masses

The key is to stop thinking about force in isolation. Let two bodies interact. Squeeze a spring between them, collide them ── anything works. By Newton's third law (action and reaction), the forces the two feel are equal in magnitude and opposite in direction. Therefore,

You get the ratio of masses without knowing the value of the force

The forces the two bodies feel have the same magnitude $F$

$$m_1 a_1 = F = m_2 a_2$$

Eliminating the force F

$$\frac{a_1}{a_2}=\frac{m_2}{m_1}$$

You never need to know how big $F$ is. Measure the two accelerations and take their ratio, and the mass ratio $m_2/m_1$ drops right out. The lighter one accelerates more, and how much more is exactly the mass ratio.

This is a big deal. Without ever going through the "undetermined" absolute value of $F$, just by letting them interact and looking at the ratio of accelerations, you can measure the mass ratio ── a number with the units gone. The refrain repeated throughout the sister series "Cosmology That Clicks" ── absolute values are conventions, ratios are physics ── is already at work from the very first step of mechanics. In the figure below, change the mass ratio and watch the acceleration ratio move in the opposite direction.

Figure: Two carts pushing off each other via a spring. Let go, and the lighter one accelerates more. Even if the value of the force F is unknown, the acceleration ratio a₁:a₂ is fixed by the mass ratio m₂:m₁

Mass ratio m₂ / m₁ (the weight of the right cart. The left one is the reference m₁=1)1.0

Cart 1 (reference m₁=1) Cart 2 (m₂)

A connecting voice ── so where did the kg come from? An interaction gives you only the ratio of masses. The absolute value "1 kg" is fixed only once humans agree on a single standard to compare against (once a metal prototype; since 2019, defined by fixing Planck's constant $h$). This is the same story as Bonus ② of the sister series, "Why did we fix the speed of light" ── absolute values are the ledger (a convention); physics lives on the side of ratios. Mechanics, it turns out, is no exception.

03So where is the "content" of F=ma?

Hearing only the loop story, you might think, "Is $F=ma$ an empty definition, then?" But no. The empirical content lives outside $F=ma$ ── on the side of the individual "force laws."

The content of force is carried by the "force laws"

Universal gravitation $F=\dfrac{G m_1 m_2}{r^2}$, the Coulomb force $F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}$, a spring $F=kx$ …
These independently determine "for a given arrangement, how much force appears." $F=ma$ is the patch panel that translates that force into "motion." The content (the physics) is the force laws; $F=ma$ is the framework of translation.

In other words, $F=ma$ by itself is close to a definition, and the physical content resides in "what force laws exist." And when you get down to it, there are only four kinds of force laws in nature ── the recurring cast of characters that will appear again and again throughout this series.

The cast behind the "true nature" of force (the four forces)

Gravity (attracts everything; the weakest, yet it rules the cosmos) / the electromagnetic force (electricity and magnetism; atoms, molecules, light) / the strong force (binds atomic nuclei) / the weak force (causes radioactive decay).

And ── the forces you feel in daily life (pushing, friction, tension, normal force) are almost all the electromagnetic force wearing different makeup. In Episodes 2 and 3 we'll peel back their true nature. The "many names of forces" are, in fact, just a few faces.

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04What the peeling revealed ── force is the name of a "relationship"

Here is the conclusion of Episode 1. Breaking down $F=ma$ revealed that "force" is not a thing a body possesses on its own. Force appears as a relationship in which two things interact. The third law is emblematic: force always appears in pairs; a lone force does not exist. The reason the lone absolute value couldn't be pinned down is precisely that force is, at its heart, a "relationship" ── and a relationship exists only between two parties.

The backbone of this series

"Force" is not a noun (a thing) but the name of a relationship.
The true nature of "A exerts a force on B" will, in the episodes ahead, be replaced one layer at a time by "relationships" ── fields, geometry, the relabeling of coordinates, the localization of symmetry. The final destination of this peeling is the curvature of a gauge field we saw in the sister series (Episode 9).

Think of force as "a thing a body holds," and you'll try to measure its lone value and fall into the loop of $F=ma$. Think of it as "a relationship between two parties," and everything runs on measurable things alone ── ratios, interactions, fields. Starting from the everyday push, this series will carry this replacement into relationships all the way to the end, for all four forces.

An honest line ── calling it "just a definition" also overshoots

Declaring $F=ma$ an "empty definition" goes too far. This equation contains genuine empirical content: that a special coordinate system (an inertial frame) exists in which $F=ma$ holds in its simple form, that forces add together as vectors, that the third law holds ── and so on. So more precisely, "$F=ma$ is a framework where definition and law are mixed together." The claim this time is one narrow point: the meaning of the absolute value of $F$ is loop-like, and the physics lives on the side of ratios and force laws.

Also, as we'll see in Episode 4, once the premise of an inertial frame breaks down, an "inertial force" ── a "force that isn't a force" ── wells up. The very footing of $F=ma$ actually depends on your choice (of coordinates) ── but that story is for then.

Practice problems (solvable with just this episode's equations)

Cart 1 ($m_1=1$) and Cart 2 push off each other via a spring, and the accelerations come out as $a_1:a_2 = 3:1$. What is $m_2$?
See the answer
From $a_1/a_2=m_2/m_1$, we get $m_2/m_1=3$. Since $m_1=1$, $m_2=3$. The one with the smaller acceleration (Cart 2) is heavier. You get the answer without knowing the value of the force F.
What does it mean to say "$F=ma$ is loop-like"? In one sentence.
See the answer
It means that defining force requires mass ($F=ma$) and defining mass requires force ($m=F/a$), so from this single equation alone neither $F$ nor $m$ is determined on its own.
Give one piece of empirical content showing that $F=ma$ is not an "empty definition."
See the answer
Examples: that a special coordinate system called an inertial frame exists / that forces superpose as vectors / that action and reaction (the third law) hold. These are real claims about nature, not definitions.

SummaryForce turned out to be a relationship, not a thing

$F=ma$ contains a loop in which force and mass define each other. The only measurable thing is acceleration. But let two bodies interact and use action and reaction, and without knowing the value of $F$ you can measure $a_1/a_2=m_2/m_1$ ── the ratio of masses. The absolute value is a convention (the kg is defined via Planck's constant); physics is on the side of ratios. The sister series' "absolute values are the ledger, ratios are physics" was already at work from the opening move of mechanics.

And the empirical content of $F=ma$ resides in the "force laws" outside the equation (gravity, electromagnetism, strong, weak). Everyday forces are almost all electromagnetism in different makeup. What the peeling revealed is that ── force is not a thing a body holds, but the name of a relationship between two parties. There is no lone force; it always comes in pairs. This series will carry this "replacement into relationships" all the way to the end, for all four forces.

Next time ── Episode 2 "The true nature of 'pushing' is all electromagnetism." Push a desk, push a wall ── that resistance you feel is the electromagnetic repulsion between the electron clouds of atoms. Your hand isn't even touching the desk. "Contact," that most everyday of sensations, is the very first thing to be peeled away. We'll see how thoroughly one of the four forces ── electromagnetism ── dominates daily life.

This document is Episode 1 of the "Force That Clicks" series, a reading for physics-loving high-schoolers and undergraduates. The circularity in the definitions of force and (inertial) mass in $F=ma$ is a real issue discussed repeatedly since Mach, and obtaining the mass ratio independent of units from an interaction (action-reaction and conservation of momentum), $a_1/a_2=m_2/m_1$, is a standard operational definition. The absolute unit of mass, the kg, has been fixed since the 2019 SI revision by fixing Planck's constant $h$ to a defined value. $F=ma$ contains empirical content ── the existence of inertial frames, the vectorial superposition of forces, the third law ── and it is more accurate to describe it as "a framework where definition and law are mixed." The fundamental interactions in nature are the four: gravity, electromagnetism, strong, and weak; everyday contact forces, friction, tension, and normal force are all essentially derived from the electromagnetic force. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider shows the inverse relationship between the mass ratio and the acceleration ratio. Click "See the answer" to open a solution.

F=ma: Law or Definition? Everyone knows \(F=ma\). But look closely and this equation turns out to be a loop in which force and mass define each other.So where is the actual "content" of this equation? Peel it back, and force reveals itself to be not a "thing" but a "relationship."

01Breaking F=ma down, symbol by symbol

02Breaking the loop ── an interaction gives you the "ratio" of masses

03So where is the "content" of F=ma?

04What the peeling revealed ── force is the name of a "relationship"

SummaryForce turned out to be a relationship, not a thing

F=ma: Law or Definition? Everyone knows \(F=ma\). But look closely and this equation turns out to be a loop in which force and mass define each other.
So where is the actual "content" of this equation? Peel it back, and force reveals itself to be not a "thing" but a "relationship."