Physics And Measurement

Almost every numerical problem you will solve in physics ends with a number and a unit, and a surprising number of NEET questions test nothing more than whether you can handle those units, their dimensions, and the uncertainty attached to them. This opening topic is cheap marks: dimensional formulae, significant figures and error combination questions appear almost every year and need no lengthy calculation.

Physical Quantities, Units and Standards

A physical quantity is anything that can be measured and expressed as a number multiplied by a unit — length, time, charge, magnetic flux. The number alone is meaningless: "5" tells you nothing until you say 5 metres or 5 seconds. So every measurement is written as

$$Q = n \times u$$

where $n$ is the numerical value and $u$ the chosen unit. Because the product $Q$ is fixed by nature, $n$ and $u$ are inversely related: choose a bigger unit and the number shrinks. This gives the very useful conversion relation $n_1u_1 = n_2u_2$. A length of 2 m becomes 200 cm because the centimetre is 100 times smaller.

Quantities are of two kinds. Base (fundamental) quantities are chosen by convention as independent — no one derives length from mass. Derived quantities are built from base ones by multiplication and division: speed = length/time, density = mass/volume, force = mass × acceleration. Correspondingly we have base units and derived units. A system of units is a complete set of base units plus the derived units built on them; historically CGS (centimetre, gram, second), FPS (foot, pound, second) and MKS were used, but the internationally agreed system today is SI.

A good standard unit should be well-defined, invariant with time and place, reproducible in any laboratory, and of a convenient size. This is why modern definitions are tied to atomic and fundamental constants rather than to metal artefacts kept in a vault.

The SI System: Base Units, Derived Units and Prefixes

SI (Système International d'Unités) uses seven base units:

Two supplementary units are also used: the radian (rad) for plane angle and the steradian (sr) for solid angle. Both are ratios of lengths (or areas), so they are dimensionless.

The definitions have been sharpened over time. The second is fixed by 9,192,631,770 oscillations of the radiation from a specified hyperfine transition of the caesium-133 atom. The metre follows from fixing the speed of light in vacuum at exactly 299,792,458 m/s — it is the distance light travels in 1/299792458 of a second. Since 2019 the kilogram is defined by fixing Planck's constant $h = 6.62607015 \times 10^{-34}\ \text{J s}$, replacing the old platinum–iridium prototype cylinder. The kelvin is fixed via Boltzmann's constant and the mole by the Avogadro number $6.02214076 \times 10^{23}$.

Derived units are combinations; many have their own names — newton (kg m s⁻²), joule (kg m² s⁻²), watt, pascal, coulomb, volt, ohm, farad, tesla, weber, henry. It is worth being able to unpack any of these into base units, because dimensional questions often disguise themselves as "the unit of X is equivalent to...".

Because physics spans from nuclear radii to galactic distances, standard prefixes are attached to units: deca (10¹), hecto (10²), kilo (10³), mega (10⁶), giga (10⁹), tera (10¹²), and going down deci (10⁻¹), centi (10⁻²), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), pico (10⁻¹²), femto (10⁻¹⁵). A few practical non-SI units also recur: 1 angstrom = 10⁻¹⁰ m, 1 fermi = 10⁻¹⁵ m, 1 light year ≈ 9.46 × 10¹⁵ m, 1 parsec ≈ 3.08 × 10¹⁶ m, 1 astronomical unit ≈ 1.496 × 10¹¹ m, 1 atomic mass unit ≈ 1.66 × 10⁻²⁷ kg, 1 barn = 10⁻²⁸ m².

Large distances are measured by parallax: an object viewed from two points separated by a baseline $b$ shifts through an angle $\theta$ (in radians), so its distance is $D = b/\theta$. Small sizes (a molecule, the thickness of a hair) are estimated by indirect tricks such as spreading a known volume of oil into a monolayer.

Dimensions and Dimensional Analysis

The dimensions of a physical quantity record which base quantities enter it and to what powers. Using the symbols M, L, T, A, K, mol, cd, the dimensional formula of force is $[\text{M L T}^{-2}]$ and of pressure $[\text{M L}^{-1}\text{T}^{-2}]$. Some to memorise:

• Velocity $[\text{LT}^{-1}]$, acceleration $[\text{LT}^{-2}]$, momentum and impulse $[\text{MLT}^{-1}]$
• Work, energy, heat, torque $[\text{ML}^2\text{T}^{-2}]$; power $[\text{ML}^2\text{T}^{-3}]$
• Pressure, stress, modulus of elasticity, energy density $[\text{ML}^{-1}\text{T}^{-2}]$
• Surface tension, spring constant, force gradient $[\text{MT}^{-2}]$
• Coefficient of viscosity $[\text{ML}^{-1}\text{T}^{-1}]$; frequency and angular velocity $[\text{T}^{-1}]$
• Gravitational constant G $[\text{M}^{-1}\text{L}^3\text{T}^{-2}]$; Planck's constant $[\text{ML}^2\text{T}^{-1}]$ (same as angular momentum)
• Dimensionless: angle, strain, refractive index, relative density, Poisson's ratio, coefficient of friction, all trigonometric and exponential arguments

Dimensional analysis rests on the principle of homogeneity: every term added or equated in a physical equation must carry identical dimensions. From this follow three standard uses:

1. Checking an equation. If the dimensions of the two sides disagree, the equation is certainly wrong. For example, in $v^2 = u^2 + 2as$ every term is $[\text{L}^2\text{T}^{-2}]$, so the relation passes the test.
2. Converting between systems. Write the quantity's dimensional formula and use $n_1[\text{M}_1^a\text{L}_1^b\text{T}_1^c] = n_2[\text{M}_2^a\text{L}_2^b\text{T}_2^c]$. This is how one shows 1 joule = 10⁷ erg.
3. Deducing a relation. If you suspect the time period of a simple pendulum depends on length, mass and $g$, write $T \propto l^x m^y g^z$, match dimensions, and you get $x = 1/2$, $y = 0$, $z = -1/2$, i.e. $T = k\sqrt{l/g}$.

The limitations matter as much as the uses:

• Dimensionless constants (like the $2\pi$ in the pendulum formula) cannot be found.
• A dimensionally correct equation may still be physically wrong (e.g. $s = ut + at^2$ passes the dimension test but the coefficient is wrong).
• The method fails when a quantity depends on more than three base quantities, or when the relation is a sum of several terms with different powers.
• Trigonometric, logarithmic and exponential relationships cannot be derived this way.

Errors in Measurement

No measurement is exact. Error is the difference between the measured and true value; accuracy describes how close a reading is to the true value, while precision describes the resolution of the instrument — a stopwatch reading 3.14 s is more precise than one reading 3.1 s, but need not be more accurate.

Errors are classified as:

• Systematic errors — one-sided, with an identifiable cause: instrumental defects (zero error in a vernier or screw gauge, a wrongly calibrated thermometer), imperfect experimental technique (not correcting for buoyancy or heat loss), and personal bias (parallax while reading a scale). These can in principle be removed or corrected for.
• Random errors — irregular, fluctuating in sign and size, arising from uncontrollable variations. They are reduced by repeating the measurement many times and averaging.
• Least count error — the resolution limit of the instrument; it is the smallest value the instrument can read (0.1 mm for a typical vernier calliper, 0.01 mm for a screw gauge). It is both systematic and random in character.
• Gross errors — outright mistakes by the observer (misrecording a reading); no theory can handle these.

For $n$ readings $a_1, a_2, \dots a_n$:

• Mean (best estimate): $\bar{a} = \dfrac{1}{n}\sum a_i$
• Absolute error of a reading: $|\Delta a_i| = |\bar{a} - a_i|$
• Mean absolute error: $\overline{\Delta a} = \dfrac{1}{n}\sum |\Delta a_i|$
• Relative (fractional) error: $\overline{\Delta a}/\bar{a}$; multiply by 100 for percentage error
• Result reported as $a = \bar{a} \pm \overline{\Delta a}$

Combination of errors — the rules to memorise:

1. Sum or difference ($Z = A + B$ or $A - B$): absolute errors add, $\Delta Z = \Delta A + \Delta B$. Note that for a difference the absolute error still adds while the value shrinks, so the relative error can become huge — avoid experiments that depend on small differences of large numbers.

2. Product or quotient ($Z = AB$ or $A/B$): relative errors add, $\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$

3. Power ($Z = A^p B^q / C^r$): each relative error is multiplied by the magnitude of its power before adding, $\dfrac{\Delta Z}{Z} = p\dfrac{\Delta A}{A} + q\dfrac{\Delta B}{B} + r\dfrac{\Delta C}{C}$. This is why a quantity that enters as a square or cube (radius in a volume, or in $g = 4\pi^2 L/T^2$) contributes its percentage error multiplied by that power — a 1% error in a measured radius becomes a 3% error in a computed volume.

Significant Figures

A measurement can never be reported with more digits than the instrument justifies. The significant figures in a number are all the digits known with certainty plus one estimated (uncertain) digit. Rules for counting them:

• All non-zero digits are significant.
• Zeros between non-zero digits are always significant (e.g. 1008 has 4 significant figures).
• Leading zeros (to the left of the first non-zero digit) are never significant — they only fix the position of the decimal point (0.0025 has 2 significant figures).
• Trailing zeros after a decimal point are significant (2.500 has 4 significant figures).
• Trailing zeros in a number without a decimal point are ambiguous by convention, which is exactly why scientific notation is preferred: writing $2.5\times10^3$ makes it unambiguous that there are only 2 significant figures, whereas "2500" on its own is not clear.

Changing the unit must never change the number of significant figures — this is the operational test for whether a zero is significant. $4.700$ m has 4 significant figures whether rewritten as $470.0$ cm or $4700\times10^{-3}$ m, because scientific notation preserves the count.

Arithmetic with significant figures:

• 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.
• Intermediate results should carry one extra guard digit, with only the final answer rounded, to avoid compounding rounding errors.

Rounding off follows the standard rule: if the digit to be dropped is more than 5, round up; less than 5, round down; if it is exactly 5, round to make the preceding digit even (2.45 rounds to 2.4, and 2.35 also rounds to 2.4).

Common Mistakes and Exam Traps

• Forgetting that percentage error in a computed quantity scales with the power of each measured quantity. A radius error of 1% becomes a 2% error in area and a 3% error in volume — dropping the exponent when combining errors is one of the most frequent numerical slips.
• Counting leading zeros as significant. 0.00420 has 3 significant figures (4, 2, 0), not 6 — leading zeros only locate the decimal point.
• Applying multiplication rules (matching the fewest significant figures) to an addition problem, or vice versa. The two operations use different rules — decimal places for +/-, significant figures for ×/÷ — and swapping them is an easy but wrong shortcut.
• Treating a dimensionally correct formula as necessarily true. Dimensional analysis cannot catch a missing dimensionless constant (like $2\pi$) or verify an equation built from a sum of terms with different powers, so it should be used to check plausibility, not to prove correctness.