Units, Dimensions & Measurement

A physical quantity is anything we can measure and express as a number together with a unit. Saying a tablet weighs "500" means nothing until we add milligram — the unit gives the number its meaning. For NEET, the habit to build early is simple: every numerical answer carries a unit, and a missing or wrong unit usually means a lost mark.

Quantities are of two kinds. Fundamental (base) quantities are chosen by convention and defined independently of one another — length, mass and time are the familiar three. Derived quantities are built from these by multiplication or division; speed is length divided by time, density is mass divided by volume. There is nothing absolute about which quantities are "base" — it is an agreed choice, and the modern agreement is the SI system (Système International).

SI fixes seven base units: the 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. Every other unit in physics — newton, joule, pascal, volt — is a combination of these seven. Two dimensionless supplementary units, the radian (plane angle) and steradian (solid angle), complete the set.

Because measured values span an enormous range — from the radius of a nucleus near $10^{-15}$ m to galactic distances near $10^{21}$ m — SI uses prefixes as shorthand: nano ($10^{-9}$), micro ($10^{-6}$), milli ($10^{-3}$), kilo ($10^{3}$), mega ($10^{6}$), giga ($10^{9}$). Writing $5\ \mu\text{m}$ is cleaner than $0.000005$ m and far less error-prone. A common NEET trap is mishandling these prefixes during conversion, so always convert to base SI units before substituting into a formula.

The dimensions of a physical quantity tell us how it is built from the base quantities mass [M], length [L] and time [T] (and where needed current [A], temperature [K]). Writing speed as $[\text{M}^{0}\text{L}^{1}\text{T}^{-1}]$ says: speed is independent of mass, varies as the first power of length and inversely with time. This compact statement is the dimensional formula, and the equation that links a quantity to its formula is the dimensional equation.

The single most useful rule is the principle of homogeneity: every term that is added or equated in a physical equation must have the same dimensions. You cannot add a length to a time any more than you can add rupees to litres. This gives a fast way to check an equation — if the two sides do not match dimensionally, the equation is certainly wrong. For example, in $v = u + at$, each term reduces to $[\text{LT}^{-1}]$, so the equation is dimensionally consistent.

Dimensional analysis does three jobs in NEET problems. First, it checks the correctness of a formula. Second, it converts a quantity from one system of units to another by comparing the number of base units. Third, it lets you derive a relation between quantities when you already know which quantities are involved — for instance, guessing that the time period of a simple pendulum depends on length and $g$ leads to $T \propto \sqrt{L/g}$.

Be clear about its limits, because NEET likes to test them. Dimensional analysis cannot find pure numbers or dimensionless constants (the $2\pi$ in the pendulum formula is invisible to it). It fails when a quantity depends on more than three others, and it cannot handle equations with trigonometric, exponential or logarithmic functions — whose arguments must themselves be dimensionless. So treat it as a powerful consistency tool, not a complete derivation method.

No measurement is exact. The difference between a measured value and the true value is the error, and a good experimenter does not pretend errors away but estimates and reports them. NEET distinguishes accuracy (how close a reading is to the true value) from precision (how closely repeated readings agree with one another). A clock that is fast by exactly five minutes is precise but not accurate; scattered readings around the true value are accurate on average but not precise.

Errors come in three families. Systematic errors push readings consistently in one direction — a zero error in a screw gauge, an un-calibrated scale, or a thermometer reading high. They can in principle be found and corrected. Random errors arise from unpredictable fluctuations and scatter readings both ways; we reduce them by taking many readings and averaging. Gross errors are plain mistakes by the observer, such as misreading a scale, and are removed only by care and repetition.

To quantify error we use three measures. The absolute error of a reading is $|a_{\text{mean}} - a_i|$, the size of its deviation from the mean. The mean absolute error is the average of these deviations and is reported with the result as $a = a_{\text{mean}} \pm \Delta a$. The relative error is $\Delta a / a_{\text{mean}}$, and multiplying by 100 gives the percentage error, the form NEET asks for most often.

When a result is calculated from several measured quantities, the errors combine. For a sum or difference, absolute errors add. For a product or quotient, relative (fractional) errors add. When a quantity is raised to a power, its relative error is multiplied by that power — so in $T = 2\pi\sqrt{L/g}$, the relative error in $g$ is twice the relative error in $T$ plus the relative error in $L$. Recognising which rule applies is exactly the skill NEET error questions test.

Significant figures are the digits in a measurement that are known reliably, plus one last digit that is estimated. They communicate the precision of a reading: writing a length as $2.50$ m claims more precision than $2.5$ m, because the trailing zero is a measured digit, not decoration. NEET expects you to count significant figures correctly and to carry the right number through a calculation.

The counting rules are worth memorising. All non-zero digits are significant. Zeros between non-zero digits are significant ($1005$ has four). Leading zeros are never significant — they only fix the decimal point, so $0.0032$ has two significant figures. Trailing zeros after a decimal point are significant ($2.300$ has four), while trailing zeros in a whole number with no decimal point are ambiguous and best resolved using scientific notation. Indeed, scientific notation removes all ambiguity: $4.500 \times 10^{3}$ unmistakably shows four significant figures.

When measurements are combined, the result cannot be more precise than the data. In multiplication and division, the result keeps as many significant figures as the quantity with the fewest. In addition and subtraction, the result keeps as many decimal places as the term with the fewest decimal places. Rounding follows the usual convention — digits below 5 are dropped, 5 or above rounds the previous digit up — and you round only the final answer, never the intermediate steps.

A closely related idea is the order of magnitude: the power of ten nearest to a quantity, found by writing it as $a \times 10^{b}$ with $1 \le a < 10$ and rounding $a$ to the nearest power of ten. The order of magnitude of the speed of light, $3 \times 10^{8}$ m/s, is $10^{8}$. NEET uses order-of-magnitude estimates to test whether you can judge the plausibility of an answer at a glance.