Problem — Fundamental or derived

Problem

Fundamental or derived? Distinguish fundamental quantities from derived ones, and state which of the two categories the following belong to: time, velocity, mass, density, temperature. Justify your answer by recalling the operational definition of each.

Solution

Step 1 — The criterion. A quantity is fundamental if the SI assigns it a base unit, defined by an autonomous measurement procedure that does not require first measuring other quantities. It is derived if its unit (and its operational definition) is built from other quantities via a relation.

Step 2 — Classification of the required quantities.

• Time — fundamental. Unit: the second (s). Operational definition: count the number of periods of a reference periodic phenomenon (today the hyperfine transition of caesium-133). No other measurement is needed.
• Mass — fundamental. Unit: the kilogram (kg). Operational definition: comparison on a balance with a standard (today fixed via the Planck constant). It is a primitive property of the body.
• Temperature — fundamental. Unit: the kelvin (K). Operational definition: read a thermometer calibrated on fixed points (today fixed via the Boltzmann constant). It is one of the seven SI base quantities.
• Velocity — derived. Unit: $\text{m/s}$. Operational definition: a distance and a time are measured separately and their ratio taken, $v = \Delta s/\Delta t$. It presupposes length and time.
• Density — derived. Unit: $\text{kg/m}^3$. Operational definition: mass and volume are measured separately and their ratio taken, $\rho = m/V$. It presupposes mass and length.

Step 3 — Conclusion.

$\ev{\text{fundamental: time, mass, temperature}\quad\text{derived: velocity, density}}$

The discriminating factor is always the operational definition: fundamental quantities are measured “on their own” with a standard, derived ones are calculated by combining measurements of more basic quantities.

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Topics: Grandezze fisiche e misura