Units, Measurement and Instruments

Suppose you tell a shopkeeper, "Give me five of rice." Five what: kilograms, grams, bags? A number alone means nothing; it needs a unit. Physics is built on measurement, and every measurement has two parts: a number (magnitude) and a unit. Anything that can be measured is a physical quantity: length, mass, time, speed, force, temperature and so on. Feelings such as love or anger are not physical quantities because we cannot measure them with a standard.

Physical quantities are of two kinds. Fundamental (base) quantities are independent of every other quantity: length, mass, time, electric current, temperature, amount of substance and luminous intensity. Derived quantities are made by combining base quantities: speed = length / time, force = mass × acceleration, pressure = force / area. The units follow the same split: fundamental units for base quantities and derived units for the rest.

Over history, different countries used different yardsticks (foot, cubit, seer, maund). To end the confusion, scientists agreed on systems of units. The four you must know are: CGS (centimetre–gram–second, the 'Gaussian' system), MKS (metre–kilogram–second), FPS (foot–pound–second, the British system) and the modern SI (Système International d'Unités), which was adopted in 1960 and is now used almost everywhere. SI is an extension of MKS. India adopted the metric system through the Standards of Weights and Measures Act, 1956.

The SI has exactly seven base quantities, and each has one base unit. This list is the single most asked fact of the chapter, so study the table carefully. Since 20 May 2019 all seven units are defined from fixed values of natural constants (for example, the metre from the speed of light, the second from the caesium-133 atom's vibration, and the kilogram from the Planck constant), instead of a metal bar or block kept in a laboratory. The old platinum–iridium kilogram kept near Paris was retired.

Once you know the seven base units, every other unit is just a recipe made from them. Speed is metre per second (m/s) because speed = distance / time. Force is mass × acceleration, so its unit is kg m/s², and this long unit was given a short name, the newton. Many derived units are named after scientists, which is why exams ask 'named after whom?' as often as 'unit of what?'. Learn the table below as pairs: quantity, unit, and the one-line recipe.

Very big and very small numbers are written with prefixes. Moving up from kilo, every named prefix is 1000 times the previous one (kilo, mega, giga, tera, peta, exa); moving down from milli, every prefix is 1/1000 of the previous one (milli, micro, nano, pico, femto, atto). The in-between prefixes hecto, deca, deci and centi are the odd ones that do not follow the step of three.

Besides SI, a number of special and practical units keep appearing in exams. Group them by what they measure and you will never confuse them: first the ones for distance (astronomy and sea), then those for mass, energy and pressure.

Ask a friend, "How far is your house?" and he says "5 km." That is enough. Now ask, "How do I reach there?" and he must add a direction: "5 km towards the north." Some quantities are complete with just a number and a unit; others need a direction as well. That is the whole difference between scalars and vectors.

How do vectors combine? Two forces pulling in the same direction give the sum; pulling in opposite directions give the difference; at right angles they give the hypotenuse. In general, for two vectors of magnitudes A and B, the resultant R lies between (A − B) and (A + B). The maximum is at 0° (same direction) and the minimum at 180° (opposite). At 90°, R = √(A² + B²).

Vectors can also be multiplied in two ways. The dot (scalar) product A · B gives a scalar: work = F · d and power = F · v. The cross (vector) product A × B gives a vector: torque = r × F and angular momentum = r × p. This is why work (a scalar) and torque (a vector) can have the same unit and same dimensions, and yet are completely different things.

The dimensions of a physical quantity show how it is built from the base quantities. We write mass as [M], length as [L] and time as [T]. Speed = length / time, so its dimensional formula is [L T⁻¹]. Think of it as the 'DNA' of a quantity: it ignores the system of units and the numerical value, and keeps only the recipe.

Some quantities have no dimensions at all: they are pure numbers or ratios of two similar quantities. Examples: angle (radian), strain, refractive index, relative density (specific gravity), coefficient of friction, magnification, Poisson's ratio and Mach number. They have no unit either, or at most a 'name' like radian.

Every measurement carries some doubt. Significant figures (sig figs) are the digits that we actually trust. Four quick concepts decide the count. (1) All non-zero digits are significant: 1.73 has 3. (2) Zeros between non-zero digits are significant: 1007 has 4. (3) Leading zeros are not significant: 0.006 has 1. (4) Trailing zeros after a decimal point are significant: 2.50 has 3.

While multiplying or dividing, the answer keeps as many sig figs as the number with the fewest sig figs. While adding or subtracting, the answer keeps as many decimal places as the number with the fewest decimal places. Now come errors. Accuracy means how close a measurement is to the true value. Precision means how close repeated measurements are to one another. A rifle that always hits the same spot, but far from the bull's eye, is precise but not accurate.

Every instrument was invented to answer one question: 'how much?' How much pressure, how humid, how fast, how hot, how pure? The names look scary, but they are built in a pattern. A word ending in -meter measures something, -graph records it (a continuous record on paper), -scope is for viewing or detecting, and -phone is for sound. If you know the front part of the name, the job is almost obvious.

Weather and atmosphere. These instruments sit in a weather station.

Liquids, density and flow. Mostly based on floating and pressure ideas that you will meet again in the fluids chapter.

Motion, direction, length and navigation.

Heat and light. The thermometer family is the most examined group, so learn it in detail below.

Electricity and magnetism.

Sound, earth and body.

A thermometer works because liquids expand evenly when heated. Mercury is used because it expands uniformly, is a good conductor of heat, does not wet the glass, and has a wide liquid range (−39 °C to 357 °C). Since mercury freezes at −39 °C, alcohol thermometers (alcohol freezes near −115 °C) are used in very cold regions. The clinical thermometer has a small kink (constriction) near the bulb so that the mercury does not fall back by itself; we shake it down after use. Fixed points: Celsius 0 and 100; Fahrenheit 32 and 212; Kelvin 273 and 373.

Conversion in one line: C / 5 = (F − 32) / 9 = (K − 273) / 5. Check with body temperature: 37 °C gives (37 × 9 / 5) + 32 = 98.6 °F. The only temperature where °C and °F show the same number is −40.