Concept 01 / 08Units and Measurements
The SI System: Base Units, Derived Units and Standards
Measurement as comparison with a standard: the seven SI base units, the 2019 redefinition in terms of fixed constants of nature, derived and practical units, and the rule that numerical value varies inversely with unit size.
Every measurement is a comparison. Saying a rod is 2.5 m long means the rod contains 2.5 copies of an agreed standard called the metre. So a physical quantity is always a pure number multiplied by a unit, . The quantity itself is a fact about nature and does not care which standard we happened to choose, so the product must stay invariant when we switch units:
, that is, .
Choose a bigger unit and the number attached to it shrinks. This single inverse proportionality is the engine behind every unit-conversion problem in this chapter, and it is why a question like "the unit of length is doubled, what happens to the numerical value of a given acceleration" has a sharp answer.
Out of the huge list of measurable quantities, a small set is declared base (fundamental) and everything else is built from them. Which ones are base is a human convention, not a law of nature. SI chooses seven, shown in the diagram above: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), thermodynamic temperature (kelvin, K), amount of substance (mole, mol) and luminous intensity (candela, cd). Plane angle (radian) and solid angle (steradian) were once called supplementary units; since 1995 they are classified as dimensionless derived units, so angle has the dimensional formula . JEE likes this point because it means you may never "cancel radians" the way you cancel metres, yet contributes nothing to a dimensional equation.
A modern subtlety worth knowing. Since 20 May 2019 the SI no longer defines base units using artefacts or the properties of particular objects. Instead seven defining constants are assigned exact numerical values and the units follow as consequences. The caesium hyperfine transition frequency Hz fixes the second; the speed of light m/s then fixes the metre; the Planck constant J s fixes the kilogram (realised with a Kibble balance, replacing the platinum-iridium cylinder); the elementary charge C fixes the ampere; the Boltzmann constant J/K fixes the kelvin; the Avogadro constant per mole fixes the mole; and the luminous efficacy lm/W fixes the candela. Notice the ordering: time first, because frequency is the quantity we can measure most precisely, and everything else is chained onto it.
Derived units are products and quotients of base units, e.g. speed in m/s, density in kg/m. A coherent derived unit has no extra numerical factor; 22 of them carry special names, including newton (N kg m s), joule (J N m), watt (W J/s), pascal (Pa N/m), coulomb (C A s), volt (V J/C), ohm ( V/A), farad (F C/V), tesla (T Wb/m), weber (Wb V s), henry (H Wb/A) and hertz (Hz s).
Prefixes scale units in powers of ten: deca , hecto , kilo , mega , giga , tera , peta , exa ; and deci , centi , milli , micro , nano , pico , femto , atto . A prefix is part of the unit symbol, so km m m, not m - a routine slip in exams.
Practical (allied) units you are expected to recall: fermi m, angstrom m, micron m, astronomical unit (AU) m, light year m, parsec m light year, barn m, atomic mass unit kg, quintal kg, metric tonne kg, shake s, atmosphere Pa, bar Pa, electron volt J, horsepower W.
Finally, the grammar of SI, which carries marks in some board and entrance questions: unit symbols are never pluralised (5 kg, not 5 kgs), never followed by a full stop, and are written in lower case unless named after a person (metre m, but newton N, kelvin K). Write a space between the number and the symbol, and use the degree-free kelvin (300 K, not 300 degrees K).
Key Takeaways:
- and , so ; larger unit gives a smaller number.
- Seven base units: m, kg, s, A, K, mol, cd. Radian and steradian are dimensionless derived units.
- Since 2019 all base units come from seven exactly fixed constants; , , , , , , now have zero uncertainty by definition.
- Pitfall: applying a prefix before raising to a power. m m, not m.
- Pitfall: treating the mole as a "mass" unit; it counts entities, and its dimension symbol is independent of M.
- Pitfall: writing "300 K" as "300 degrees K", or "5 Kg" instead of "5 kg".
Brain-Teaser 1: In a new system of units the unit of length is doubled and the unit of time is halved, while the unit of mass is unchanged. By what factor does the numerical value of a given acceleration change?
Solution: Acceleration has the structure length divided by (time). The new unit of acceleration is . Since , the numerical value becomes . Concretely, take m s with new units 2 m and 0.5 s. One new acceleration unit equals m s, so new units. The trap is to reason that halving the time unit should make the number bigger; it does, but the square makes the time factor 4, and combined with the doubled length unit the net unit is 8 times larger, so the number falls to one eighth.
Brain-Teaser 2: A new system of units takes the unit of mass as kg, the unit of length as m and the unit of time as s. What is the numerical value of 1 joule in this system?
Solution: Energy has the structure . The new unit of energy is in kg m s, i.e. J. Using with and J, . Sanity check with numbers: if the answer is 1, as it must be; if the time unit is made larger, the energy unit gets smaller (because ), so the number should grow, and indeed sits in the numerator.
Brain-Teaser 3: Since 1983 the metre is defined by declaring m/s exactly. Does this mean the speed of light can never again be measured more precisely? What actually carries the uncertainty now?
Solution: It means exactly that, and it is not a loss. Before 1983 the metre was an independent standard (a krypton wavelength), and was an experimental number with a finite uncertainty. Fixing converts the metre into a derived quantity: a length is now realised by timing a light pulse, so what used to be "measuring " becomes "calibrating your length standard". The uncertainty has not vanished, it has moved into the practical realisation of the metre, i.e. into how well you can count caesium oscillations and control your interferometer. The same logic applies after 2019 to and the kilogram: is exact, and the experimental uncertainty now lives in the Kibble balance that realises a mass. The general lesson, which JEE occasionally tests conceptually, is that fixing a constant does not abolish uncertainty, it relocates it to the realisation of the unit.
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