Units and Measurements: SI Units, Dimensions and Instruments

Every measurement has two parts, a number and a unit, and the unit decides what the number means. This chapter is one of the most rewarding in general science for competitive exams, because the questions are direct: the unit of a quantity, the instrument that measures it, the dimensional formula, or the meaning of a prefix. The system in use today is the International System of Units, adopted in 1960 by the General Conference on Weights and Measures, and it is built on seven base units.

The seven base units of the SI system

Besides these, the radian for a plane angle and the steradian for a solid angle were long listed as supplementary units. The definitions have changed over time to make them reproducible anywhere: the metre is the distance travelled by light in vacuum in a fixed fraction of a second, the second is fixed by the radiation of the caesium-133 atom, and since the revision that took effect in 2019 the kilogram is defined through the Planck constant instead of the metal cylinder kept near Paris.

Derived units and dimensional formulas

Every other unit is derived from the seven base units, and many derived units carry the name of a scientist. Force is measured in newton, work and energy in joule, power in watt, pressure in pascal, frequency in hertz, electric charge in coulomb, potential difference in volt, resistance in ohm and magnetic flux density in tesla. A dimensional formula shows how a derived quantity is built from mass (M), length (L) and time (T).

• Force: M L T⁻², measured in newton
• Work or energy: M L² T⁻², measured in joule
• Power: M L² T⁻³, measured in watt
• Pressure: M L⁻¹ T⁻², measured in pascal
• Momentum: M L T⁻¹, measured in kilogram metre per second

Dimensional analysis is used to check whether an equation is consistent, to convert a value from one system of units to another, and to work out the form of a relation. It cannot, however, find a dimensionless constant, and it fails for quantities that have the same dimensions, such as work and torque.

Prefixes and practical units

Prefixes save writing long strings of zeros: kilo stands for a thousand, mega for a million, giga for a thousand million and tera for a million million, while milli, micro, nano and pico stand for a thousandth, a millionth, a thousand millionth and a million millionth. For very large and very small measurements, science keeps a set of practical units.

• Light year: the distance light travels in one year, about 9.46 trillion kilometres. It is a unit of distance, not of time.
• Astronomical unit: the mean distance between the Earth and the Sun, about 149.6 million kilometres.
• Parsec: about 3.26 light years, the larger unit used in astronomy.
• Angstrom: one ten thousand millionth of a metre, used for atoms and wavelengths of light.
• Fermi: one thousand million millionth of a metre, used for nuclear sizes.
• Nautical mile: 1,852 metres, used at sea and in the air; one horsepower is about 746 watts; one quintal is 100 kilograms and one tonne is 1,000 kilograms.

Instruments and what they measure

One-line questions on instruments appear in almost every paper, so learn them in pairs.

Scalars, vectors and errors

A scalar quantity has only magnitude, such as mass, time, distance, speed, work, energy and temperature. A vector quantity has both magnitude and direction, such as displacement, velocity, acceleration, force, momentum and weight. Note the pairs that are often confused: distance is a scalar while displacement is a vector, and speed is a scalar while velocity is a vector. In measurement, accuracy means how close a reading is to the true value and precision means how closely repeated readings agree with one another, so a set of readings can be precise without being accurate. The number of significant figures in a result shows how reliable the measurement is, and the least count of an instrument is the smallest value it can read.