Units and Measurements: NEET notes

1. Units: base, derived and systems

NCERT §1.1

• To measure a physical quantity is to compare it with a reference standard that has been chosen arbitrarily and agreed internationally; that standard is the unit.
• A measured result always has two parts, a numerical value and a unit: a pendulum string is 0.994 m long, never just 0.994.
• There are very many physical quantities, but they are linked to each other by equations, so a small set of units is enough to express all of them.
• The quantities chosen as the starting set are the fundamental or base quantities, and their units are the base units (for example metre, kilogram, second).
• Every other unit is built from the base units by multiplying and dividing them; these are derived units, such as m s⁻² for acceleration or kg m s⁻² (the newton) for force.
• The base units together with all the derived units made from them form a system of units.

2. The SI system

NCERT §1.2

• Three older systems were widely used: CGS (centimetre, gram, second), FPS or British (foot, pound, second) and MKS (metre, kilogram, second).
• Today's internationally accepted standard is SI, short for the French Système Internationale d'Unités. The BIPM (International Bureau of Weights and Measures) set up its scheme of units and symbols in 1971, and the General Conference on Weights and Measures revised it in November 2018.
• SI has seven base quantities and units: 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).
• Since the 2018 revision each base unit is defined by fixing the numerical value of a constant of nature: the caesium-133 hyperfine frequency for the second, the speed of light c for the metre, the Planck constant h for the kilogram, the elementary charge e for the ampere, the Boltzmann constant k for the kelvin, the Avogadro constant N_A for the mole, and a luminous efficacy of 683 lm W⁻¹ at 540 × 10¹² Hz for the candela.
• The exact defining values are shown to indicate how precisely the units are fixed; NCERT states they need not be memorised.
• When the mole is used, the elementary entities must be specified: atoms, molecules, ions, electrons or other particles.
• Two supplementary units are defined as ratios: the plane angle dθ = ds/r in radian (rad), and the solid angle dΩ = dA/r² in steradian (sr). Both are dimensionless.
• SI is a decimal system, so converting between multiples and sub-multiples only shifts powers of 10; some derived units carry special names, such as newton, joule and watt.

3. Significant figures

NCERT §1.3

• A reported measurement should contain every digit known reliably plus the first uncertain digit; together these are the significant figures, and their number shows how precise the measurement was.
• The precision depends on the least count of the instrument: 1.62 s has three significant figures (1 and 6 certain, 2 uncertain), 287.5 cm has four.
• Changing the unit never changes the number of significant figures: 2.308 cm = 0.02308 m = 23.08 mm all have four.
• All non-zero digits are significant, and zeros between two non-zero digits are significant wherever the decimal point is.
• For a number less than 1, zeros after the decimal point but before the first non-zero digit are not significant; the zero written before the decimal point (as in 0.1250) is never significant.
• Trailing zeros are significant in a number with a decimal point (3.500 and 0.06900 have four each) but not in a number without one (123 m = 12300 cm has three).
• Scientific notation a × 10ᵇ with 1 ≤ a < 10 removes the ambiguity: 4.700 m = 4.700 × 10³ mm keeps its four significant figures, and every zero in a is significant.
• For the order of magnitude, round a to 1 if a ≤ 5 and to 10 if 5 < a ≤ 10; the resulting power of ten is the order. The earth's diameter, 1.28 × 10⁷ m, is of order 10⁷ m; a hydrogen atom, 1.06 × 10⁻¹⁰ m, is of order 10⁻¹⁰ m, 17 orders smaller.
• Exact numbers in a formula, such as the 2 in s = 2πr or a counted number of oscillations, have an unlimited number of significant figures.

4. Arithmetic with significant figures

NCERT §1.3.1

• A result calculated from measured values cannot be more precise than the data it came from, so it should not carry more significant figures than the data justify.
• Multiplication and division: keep as many significant figures as the input with the fewest significant figures. 4.237 g ÷ 2.51 cm³ gives 1.69 g cm⁻³, not 1.68804780876.
• Addition and subtraction: keep as many decimal places as the input with the fewest decimal places. 436.32 g + 227.2 g + 0.301 g = 663.821 g is reported as 663.8 g.
• Subtraction can cut the number of figures: the difference of 0.307 m and 0.304 m is just 3 × 10⁻³ m, one significant figure, although each input has three.
• Do not mix the two rules: the sum above is not 664 g, and the difference is not 3.00 × 10⁻³ m.
• With c = 3.00 × 10⁸ m s⁻¹ (three significant figures) and one year = 3.1557 × 10⁷ s (five), the light year comes out as 9.47 × 10¹⁵ m (three).

5. Rounding off

NCERT §1.3.2

• When a result carries more than one uncertain digit, round it to the right number of significant figures.
• If the digit being dropped is more than 5, raise the preceding digit by 1 (2.746 → 2.75); if it is less than 5, leave the preceding digit alone (1.743 → 1.74).
• If the digit being dropped is exactly 5, look at the preceding digit: if it is even, simply drop the 5 (2.745 → 2.74); if it is odd, raise it by 1 (2.735 → 2.74).
• In a multi-step calculation, carry one digit more than the final answer needs through the intermediate steps and round only at the end, or rounding errors build up.
• Rounding too early can shift an answer: 1/9.58 rounded to 0.104 and inverted gives 9.62, but keeping 0.1044 gives back 9.58.
• Constants known to many figures are rounded to what the problem needs: c = 2.99792458 × 10⁸ m s⁻¹ is often used as 3 × 10⁸ m s⁻¹, and π as 3.14 or 3.142.
• A cube of side 7.203 m has surface area 6 × (7.203)² = 311.3 m² and volume (7.203)³ = 373.7 m³, both to four significant figures; 5.74 g in 1.2 cm³ gives a density of 4.8 g cm⁻³, to two.

6. Uncertainty in calculated results

NCERT §1.3.3

• A reading from an instrument carries an uncertainty of about its least count: a length of 16.2 cm read on a metre scale is 16.2 ± 0.1 cm.
• The relative (percentage) error is the uncertainty divided by the value: ±0.1 cm on 16.2 cm is about ±0.6%, and ±0.1 cm on 10.1 cm is about ±1%.
• When measured values are multiplied or divided, their percentage errors add: the area l × b = 163.62 cm² carries ±1.6%, about ±2.6 cm², so it is quoted as 164 ± 3 cm².
• The same absolute uncertainty is a larger relative error on a smaller value: ±0.01 g is ±1% of 1.02 g but only ±0.1% of 9.89 g.
• Uncertainties in a sum or difference follow decimal places, not significant figures, so a difference can lose figures: 12.9 g − 7.06 g is 5.8 g, not 5.84 g.
• If the data are good to n significant figures, a result obtained by combining them is valid to about n significant figures, fewer if data are subtracted.

7. Dimensions and dimensional formulae

NCERT §1.4, §1.5

• The dimensions of a quantity are the powers to which the base quantities are raised to express it; they describe its nature, not its size.
• Each base quantity is one dimension, written in square brackets: [M] for mass, [L] for length, [T] for time, [A] for electric current, [K] for thermodynamic temperature, [mol] for amount of substance and [cd] for luminous intensity.
• In mechanics every quantity can be written with [M], [L] and [T] alone. Volume is [L³]; force = mass × acceleration is [M L T⁻²].
• Magnitudes do not enter: speed, initial velocity, final velocity and change in velocity all have the same dimensions, [L T⁻¹].
• The dimensional formula shows which base quantities appear and with what powers, zero powers included: volume [M⁰ L³ T⁰], velocity [M⁰ L T⁻¹], acceleration [M⁰ L T⁻²], density [M L⁻³ T⁰].
• A dimensional equation sets a quantity equal to its dimensional formula, for example [F] = [M L T⁻²] or [ρ] = [M L⁻³ T⁰].
• A dimensional formula can be read off from any defining equation: momentum = mass × velocity gives [M L T⁻¹]; work = force × distance gives [M L² T⁻²].

8. Checking dimensional consistency

NCERT §1.6.1

• Only quantities with the same dimensions can be added, subtracted or equated; this principle of homogeneity of dimensions is a quick test of an equation.
• In a correct equation every term on both sides has the same dimensions. In x = x₀ + v₀t + ½at², each term, x, x₀, v₀t and ½at², has dimension [L].
• Units and dimensions are handled like algebraic symbols: identical ones in the numerator and denominator cancel.
• The argument of a trigonometric, logarithmic or exponential function must be dimensionless, and pure numbers and ratios of like quantities (an angle, refractive index) have no dimensions.
• An equation that fails the test is certainly wrong; one that passes is not thereby proved right, because dimensionless factors cannot be checked.
• Of the candidates K = m²v³, ½mv², ma, (3/16)mv² and ½mv² + ma for kinetic energy, dimensions rule out m²v³, ma and the mixed sum, but cannot choose between ½mv² and (3/16)mv².
• ½mv² = mgh passes: both sides are [M L² T⁻²].

9. Deducing relations by dimensions

NCERT §1.6.2

• If you know which quantities (up to three independent ones) a result depends on, assume it is a product of their powers and match dimensions to find the powers.
• For a simple pendulum take T = k lˣ gʸ mᶻ. Matching [T] = [L]ˣ [L T⁻²]ʸ [M]ᶻ gives x + y = 0, −2y = 1 and z = 0.
• So x = ½, y = −½, z = 0 and T = k√(l/g): the period does not depend on the mass of the bob.
• The dimensionless constant k cannot come from dimensions; experiment or theory gives k = 2π, so T = 2π√(l/g).
• The method fails to give dimensionless constants, cannot tell apart quantities with the same dimensions (work and torque, for example), and cannot handle sums of terms or trigonometric, logarithmic and exponential dependence.

Must-know facts

1. Seven SI base units: metre, kilogram, second, ampere, kelvin, mole, candela.
2. Radian (plane angle, ds/r) and steradian (solid angle, dA/r²) are dimensionless.
3. CGS: cm, g, s. FPS: foot, pound, second. MKS: m, kg, s.
4. Significant figures = reliable digits + the first uncertain digit.
5. Changing units does not change the number of significant figures.
6. Leading zeros are never significant; trailing zeros count only if there is a decimal point.
7. Order of magnitude: round a in a × 10ᵇ to 1 if a ≤ 5, to 10 if a > 5.
8. Product or quotient: fewest significant figures. Sum or difference: fewest decimal places.
9. Rounding a final 5: even preceding digit stays, odd preceding digit goes up (2.745 → 2.74, 2.735 → 2.74).
10. Carry one extra digit through intermediate steps; round at the end.
11. In a product or quotient, percentage errors add.
12. Exact numbers (2 in 2πr, a count) have unlimited significant figures.
13. Force [M L T⁻²], velocity [L T⁻¹], acceleration [L T⁻²], density [M L⁻³], energy and work [M L² T⁻²].
14. Only like dimensions can be added; arguments of sin, log and exp are dimensionless.
15. Dimensionally wrong means wrong; dimensionally right does not mean right.
16. Pendulum by dimensions: T ∝ √(l/g), independent of mass; k = 2π comes from theory.

Common traps

Formulas

Key terms

Unit

An agreed reference standard with which a physical quantity is compared.

Base quantity

One of the seven quantities chosen as independent starting points, such as length, mass and time.

Derived unit

A unit built from base units, such as m s⁻¹ or kg m s⁻².

SI

The internationally accepted system of units, with seven base units defined by fixed constants of nature.

Radian

Unit of plane angle: arc length divided by radius.

Steradian

Unit of solid angle: intercepted area on a sphere divided by the square of its radius.

Significant figures

The reliable digits of a measurement together with the first uncertain one.

Least count

The smallest value an instrument can read directly; it limits the precision.

Order of magnitude

The power of ten nearest a quantity, found by rounding a in a × 10ᵇ to 1 or 10.

Relative error

The uncertainty in a value divided by the value, often given as a percentage.

Dimensions

The powers to which base quantities are raised to express a physical quantity.

Dimensional formula

An expression such as [M L T⁻²] showing the base quantities in a quantity and their powers.

Principle of homogeneity

Every term of a physically meaningful equation has the same dimensions.