Units and Measurements

Physics is a science of measurement. Every law you will study this year, from Newton's second law to Coulomb's law, is a relation between measured quantities, and it only means something once you have agreed on what a metre, a second and a kilogram actually are. Physical quantities are split into two kinds. A fundamental (base) quantity is one that is measured directly and independently of others: length, mass, time, electric current, temperature, amount of substance and luminous intensity. A derived quantity is built from these by multiplication or division: speed = length/time, force = mass × acceleration, and so on. This chapter fixes the units for the base quantities, tells you how those units are realised in a laboratory, and gives you the tools, significant figures, error analysis and dimensional analysis, to handle any measurement honestly and to check whether an equation can possibly be correct before you even test it.

Before SI, different countries used different systems: CGS (centimetre, gram, second), FPS (foot, pound, second) and MKS (metre, kilogram, second). To remove this confusion, the General Conference on Weights and Measures adopted the SI (Système International d'Unités), built on seven base units: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance and candela (cd) for luminous intensity. Two supplementary units are added for angles: radian (rad) for plane angle and steradian (sr) for solid angle. Every other physical unit (newton, joule, volt, ohm...) is derived from these seven by combining them according to the quantity's defining equation. SI also fixes rules for writing units correctly: unit symbols are never pluralised (5 kg, not 5 kgs), no full stop follows a symbol unless it ends a sentence, and a space separates the number from the unit (10 m, not 10m). Since 2019, all seven base units are defined by fixing the numerical value of a fundamental constant (e.g. the kilogram is now defined by fixing the Planck constant h), rather than by a physical artefact.

For everyday lengths we use metre scales, vernier callipers and screw gauges. For very large distances, such as the distance to the moon or a nearby star, direct measurement is impossible, so we use the parallax method. Parallax is the apparent shift in the position of an object against a distant background when viewed from two different points. To find the distance D of a distant object, observe it from two points separated by a known baseline b, and measure the parallax angle θ (in radians) it subtends. Since θ is small, the arc length b ≈ Dθ, giving D = b/θ. The same idea, using the angular size of the object as seen from Earth, gives the diameter of a distant star once its distance is known: if α is the angular diameter, the star's diameter d = Dα. For distances within the solar system, the echo (radar) method is used: a radio pulse is sent, its reflection is timed, and distance = (speed of pulse × time)/2. For very small lengths (atomic and molecular size, of the order 10⁻¹⁰ m), electron microscopes and X-ray diffraction are used; a rough estimate can also be made from Avogadro's number and the molar volume of a substance.

Mass is measured against the SI kilogram using a common balance for ordinary masses and a sensitive spring or inertial balance in the absence of gravity (as on a spacecraft). For atomic and subatomic masses, a common unit is the unified atomic mass unit (u), defined as 1/12 of the mass of a carbon-12 atom: 1 u = 1.66 × 10⁻²⁷ kg. Atomic and molecular masses are measured very precisely using a mass spectrograph, in which charged particles are deflected by electric and magnetic fields and their radius of curvature is used to find their mass-to-charge ratio. The range of masses in nature is enormous: from about 10⁻³⁰ kg for an electron to about 10⁵⁵ kg for the known universe, which is why NCERT wants you to appreciate the scale as much as the method.

Time is measured using a clock; the SI second was earlier defined from Earth's rotation, but that rotation is not perfectly uniform, so the modern definition uses atomic vibrations instead. One SI second is defined as the time taken by 9,192,631,770 periods of the radiation emitted during a specific transition of the caesium-133 atom. Devices built on this principle, atomic clocks, are accurate to about 1 second in many thousands of years, and India's National Physical Laboratory (NPL), New Delhi maintains the atomic clock that provides Indian Standard Time.

Accuracy is how close a measured value is to the true value; precision is how close repeated measurements are to each other, regardless of the true value — a clock can be precise (giving the same wrong time every day) without being accurate. Every measurement has some uncertainty, called an error. Errors are of three kinds. Systematic errors are one-sided (always positive or always negative) and arise from instrumental defects (zero error), imperfect technique, or an external condition; they can be reduced by correcting the instrument or the method. Random errors arise from unpredictable, uncontrollable causes and cause a scatter of readings around the true value; they are reduced by taking many readings and averaging. Gross errors are human mistakes in reading or recording.

If a₁, a₂, ..., a_n are n measurements of a quantity, the best estimate is their arithmetic mean a_mean. The absolute error in each reading is Δa_i = |a_mean − a_i|, and the mean absolute error is the average of these. The relative (fractional) error is Δa_mean/a_mean, and multiplying it by 100 gives the percentage error. When quantities are combined, their errors combine by definite rules: for a sum or difference the absolute errors add; for a product or quotient the relative errors add; and for a quantity raised to a power, the relative error is multiplied by that power.

The significant figures in a measured number are all the digits that are reliably known plus one estimated (uncertain) digit; they tell you how precisely a quantity was actually measured. The rules: all non-zero digits are significant; zeros between two non-zero digits are significant (1005 has 4 sig. figs.); leading zeros (before the first non-zero digit) are never significant, they only fix the decimal point (0.0025 has 2 sig. figs.); trailing zeros after a decimal point are significant (2.500 has 4 sig. figs.); trailing zeros in a number without a decimal point are ambiguous unless written in scientific notation (2500 could have 2 to 4 sig. figs., but 2.500 × 10³ has exactly 4). Scientific (power-of-ten) notation is preferred exactly because it removes this ambiguity.

When numbers are combined, the result cannot be more precise than the least precise input. In addition or subtraction, the result is rounded to the same number of decimal places as the term with the fewest decimal places. In multiplication or division, the result is rounded to the same number of significant figures as the factor with the fewest significant figures. Rounding off follows the standard rule: if the digit to be dropped is more than 5, the preceding digit is raised by 1; if less than 5, it is left unchanged; if exactly 5, the preceding digit is raised by 1 if it is odd and left unchanged if it is even (this keeps rounding unbiased on average).

The dimensions of a physical quantity are the powers to which the base quantities must be raised to represent it. Physics uses seven dimensional symbols: [M] for mass, [L] for length, [T] for time, [A] for electric current, [K] for temperature, [mol] for amount of substance and [cd] for luminous intensity. For example, speed = distance/time has the dimensional formula [M⁰L¹T⁻¹], because it involves length to the first power and time to the power −1, and no mass at all. Writing a quantity's dimensional formula this way, in square brackets with the powers of M, L, T (and other base quantities where needed), is the language the rest of the chapter is built on.

A dimensional equation is the equation obtained when a physical quantity is equated to its dimensional formula, for example [F] = [MLT⁻²] for force. To write the dimensional formula of any derived quantity, start from its defining relation in terms of quantities you already know the dimensions of, and substitute. For instance, force = mass × acceleration = mass × (velocity/time) = mass × (length/time)/time, giving [F] = [MLT⁻²]; and since pressure = force/area, [P] = [MLT⁻²]/[L²] = [ML⁻¹T⁻²]. Building up dimensional formulae step by step from familiar definitions, rather than memorising a long list, is both faster and less error-prone in the exam.

Dimensional analysis rests on the principle of homogeneity: in a dimensionally correct equation, every term on both sides must have the same dimensions, because you can only add, subtract or equate quantities of the same kind. This gives three uses. First, checking correctness: if the dimensions on the two sides of an equation do not match, the equation must be wrong (though matching dimensions do not guarantee the equation is numerically correct — a dimensionless constant could still be wrong or missing). Second, deriving a relation: if you can guess which quantities a result depends on, you can assume a power relation between them, match dimensions on both sides, solve for the unknown powers, and arrive at the form of the formula (up to an undetermined dimensionless constant). For example, assuming the time period T of a simple pendulum depends on its length l, the mass m of the bob and g, writing T = k l^ag^bm^c and matching dimensions gives a = ½, b = −½, c = 0, so T ∝ √(l/g) — mass drops out entirely, which the method itself reveals. Third, converting units from one system to another using n₁u₁ = n₂u₂. The method has real limits: it cannot fix the dimensionless constant k, it fails if a quantity depends on more than three unknown powers (only three equations are available, one each for M, L, T), and it cannot be applied if the relation involves a sum of terms like s = ut + ½at² or a trigonometric/exponential/logarithmic function, since dimensional analysis cannot tell you the numerical coefficients or that such functions must be dimensionless in their argument.