The International System of Units — Question 1

Question 1

How many base units does an experiment need?

A laboratory database offers the seven SI base units: $$\mathrm{s,\ m,\ kg,\ A,\ K,\ mol,\ cd}.$$ A technician argues: “There are seven base units, so every experiment must use all seven. Also, a named unit such as the newton must be a base unit.”

Consider a cart experiment that records mass $m$, path length $L$, elapsed time $t$, and pulling force $F$. For this task the cart’s acceleration $a$ has unit $\mathrm{m\,s^{-2}}$, and the force model is $F=ma$. No electrical, thermal, chemical, or optical measurement enters the model.

1. Match each of the seven symbols above to its base quantity. Give the unit name as well as the quantity.

2. Find the smallest subset of the listed base units needed to express every quantity in this cart model. Derive the force unit in that subset.

3. Evaluate both of the technician’s claims. Does the absence of temperature from this model establish that temperature can never influence a real cart experiment?

Question 1 - Solution

How many base units does an experiment need?

(a) The seven base quantities and their units.

The seven quantity–unit correspondences are time: second (s); length: metre (m); mass: kilogram (kg); electric current: ampere (A); thermodynamic temperature: kelvin (K); amount of substance: mole (mol); and luminous intensity: candela (cd).

(b) Select the subset from the model.

$$[a]=\mathrm{m\,s^{-2}},\qquad[F]=[m][a]=\mathrm{kg\,m\,s^{-2}}=\mathrm N.$$

Mass, path length, and time require $\mathrm{kg}$, $\mathrm{m}$, and $\mathrm{s}$. The stated force relation gives $$[F]=[m][a]=\mathrm{kg\,m\,s^{-2}}=\mathrm{N}.$$ Thus $\boxed{\{\mathrm{kg,m,s}\}}$ suffices. Each of these is needed for a recorded quantity, and the force introduces no additional base unit. Square brackets here denote the unit structure of a quantity.

(c) Distinguish a system of units from an experimental model.

$$\mathrm N=\mathrm{kg}^{1}\mathrm m^{1}\mathrm s^{-2}\mathrm A^{0}\mathrm K^{0}\mathrm{mol}^{0}\mathrm{cd}^{0}.$$

The SI supplies a common system for many types of measurement; it does not require every experiment to involve all seven base quantities. The newton is a coherent derived unit with a special name, not an eighth base unit. A special name is not a test of base-unit status.

The model excludes temperature as an input; it does not prove thermal effects are physically absent. For example, temperature could alter a wheel’s dimensions or friction. Whether those effects matter requires a more detailed model and evidence at the experiment’s required precision.