Physical quantities — fundamental and derived: Visual walkthrough

We only assume three fundamental quantities: length, mass, and time. Everything else is built. Let's build.

Step 1 — The three tools we are allowed to use

WHAT. We lay out exactly three measuring instruments and nothing else: a ruler that measures length, a scale that measures mass, and a stopwatch that measures time.

WHY. You cannot derive a unit unless you first agree on what is not derived. These three are fundamental: each is measured directly against a standard, none is defined using the others. Force is not on this table — that is the whole point. We must construct it.

PICTURE. The three boxes below are our entire toolkit. Each has a symbol (the letter physicists use for its dimension) and an SI unit.

Figure — Physical quantities — fundamental and derived

Step 2 — Speed: our first act of building (length ÷ time)

WHAT. We combine the ruler and the stopwatch to build speed: how much distance is covered per unit of time.

WHY. Force will need acceleration, and acceleration is built on top of speed, which is built on top of length and time. So we climb the ladder one rung at a time. We use division because "per" always means divide: distance per time.

PICTURE. A dot travels a distance $Δ L$ (a change in length, the orange bar) in a time $Δ T$ (the blue bar). Speed is one bar divided by the other.

WHY the $s^{- 1}$ ? Writing $\frac{1}{s}$ as $s^{- 1}$ is pure bookkeeping: a unit in the denominator gets a negative exponent. This lets us multiply and divide units like ordinary algebra later on — the trick that makes the whole derivation work.

Step 3 — Acceleration: how fast the speed itself changes (speed ÷ time)

WHAT. We divide by time a second time to build acceleration: the rate at which speed changes.

WHY. Force acts by changing motion — starting it, stopping it, curving it. "Change of motion per second" is exactly acceleration. So force will be tied to acceleration, and we need its units first. We divide speed by time because acceleration asks "how much speed gained per second".

PICTURE. Two speed arrows one second apart. The green arrow is the extra speed gained. Acceleration is that green gain divided by the time it took.

Step 4 — The physical law that names force: $F = ma$

WHAT. We introduce the one piece of physics that defines force — Newton's second law: force equals mass times acceleration.

WHY. We cannot invent units for force out of thin air; a law must tell us what force is. The law says: to accelerate a heavier object the same amount, you need proportionally more force ( $\times M$ ), and to accelerate it faster you need more force too ( $\times a$ ). Multiplication captures both "more mass" and "more acceleration".

PICTURE. Same push (orange arrow) on a light and a heavy block. The heavy block gains less speed — its acceleration arrow is shorter. Force is the product of the two factors on the right.

Step 5 — Multiply the units like algebra to get the newton

WHAT. We substitute the units of each factor and multiply them together.

WHY. Because the quantities multiply, their units multiply too — units obey the exact same arithmetic as the numbers in front of them. This is the master principle of the whole chapter.

PICTURE. The mass-unit box ( $kg$ ) sits beside the acceleration-unit box ( $m s^{- 2}$ ); we snap them together into one combined block and give the block a nickname: newton.

Step 6 — The degenerate cases (never skip these)

WHAT. We check what the recipe says at the extremes: zero acceleration, zero mass, and negative direction.

WHY. A derivation you trust must survive its edge cases. If the formula gave nonsense when $a = 0$ , we would not believe it when $a \neq = 0$ .

PICTURE. Three mini-scenarios stacked: a block drifting at constant speed ( $a = 0$ ), a hypothetical massless object, and a block pushed the other way (negative force).

Step 7 — Reusing the same machine: energy and pressure in one glance

WHAT. Once force is built, everything downstream is one more multiply or divide.

WHY. To show the newton was not a one-off trick: the same "units multiply like algebra" engine builds the joule and the pascal instantly.

PICTURE. A flow: newton $\times$ metre $\to$ joule; newton $\div$ area $\to$ pascal.

The one-picture summary

The whole page in one image: start with the three tools, divide by time twice to reach acceleration, multiply by mass (Newton's law), and read off $kg m s^{- 2} = N$ — then branch to joule and pascal with one more operation each.

Recall Feynman retelling — say it to a 12-year-old

I only own a ruler, a scale, and a stopwatch. First I use the ruler and stopwatch together: distance divided by time gives me speed — how fast something moves. Then I use the stopwatch again on the speed: how much speed I gain each second is acceleration. Now here's the one law of nature I need — pushing a heavier thing, or pushing it to speed up faster, both take more push, so push = mass × acceleration. When I multiply their labels the same way I multiply the numbers, I get $kg \times m/s/s = kg m s^{- 2}$ , and we nickname that pile "one newton". I never bought a "force meter" — I built force from three cheap tools. And once I have force, one more multiply gives energy (the joule) and one more divide gives pressure (the pascal). The whole of mechanics is LEGO, and length-mass-time are the starter bricks.

Active recall

Recall Rebuild it yourself

Starting from only $L$ , $M$ , $T$ , derive the units of the newton. State each division/multiplication and why.
Why does acceleration carry $s^{- 2}$ and not $s^{- 1}$ ?
What are the base units of $1 N$ , $1 J$ , and $1 Pa$ ?
If $a = 0$ , what is the force and what is its unit?
Does a leftward force have a different unit from a rightward force? Explain.

What is

1 N

in base units?

kg m s^{- 2}

Why is acceleration's exponent

- 2

we divided by time twice (once for speed, once for acceleration)

What is

1 J

in base units?

kg m^{2} s^{- 2}

(force × distance)

What is

1 Pa

in base units?

kg m^{- 1} s^{- 2}

(force ÷ area)

Force when

a = 0

0 N

— the unit stays

kg m s^{- 2}

, only the number is zero

Do units carry direction?

No — direction is in the vector sign; units carry magnitude only

Connections

Physical quantities — fundamental and derived — the parent: the seven base quantities this page draws from.
Newton's Second Law — the law $F = ma$ that names force in Step 4.
Dimensional Analysis — the general algebra of multiplying/dividing units.
SI Units and Prefixes — the standards behind m, kg, s.
Vectors — why force keeps direction in its sign, not its unit.
Errors and Significant Figures — every measured $L$ , $M$ , $T$ carries uncertainty.