Units and Measurements

Physical quantities are classified as fundamental (or base) quantities, which are defined independently of one another, and derived quantities, which are expressed in terms of the fundamental ones. The SI system fixes seven base quantities and their 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). Two further units — the radian (rad) for plane angle and the steradian (sr) for solid angle — are used alongside these; both are dimensionless ratios of lengths or areas.

The dimensions of a derived quantity are the powers to which the base quantities must be raised to represent it, written in square brackets. Force, for instance, has the dimensional formula $[ML{T}^{-2}]$, since $F=ma$ combines mass with acceleration ($L{T}^{-2}$). This gives every physical quantity a dimensional formula built from the base dimensions $M$, $L$, $T$, $A$, $K$, $\mathrm{mol}$, and $\mathrm{cd}$. The principle of homogeneity of dimensions requires that only quantities of the same dimensions may be added, subtracted, or equated — this is what makes an equation like $s=ut+\frac{1}{2}a{t}^2$ checkable term by term, since each term must reduce to $[L]$.

Dimensional analysis exploits this principle in three ways:

• Checking whether an equation is dimensionally consistent (a necessary, not sufficient, test of correctness).
• Deriving the form of a relationship between quantities, such as showing that a simple pendulum's period must take the form $T=k\sqrt{l/g}$.
• Converting a physical quantity's numerical value from one system of units to another.

Its limits matter just as much for NEET: it cannot fix dimensionless constants such as the $2\pi$ in the pendulum formula, it fails wherever trigonometric, exponential, or logarithmic functions appear, and with only a handful of base dimensions available it cannot resolve a relation among more than three unknown quantities.

A frequent trap is assuming that identical dimensional formulas mean identical physical quantities. Work and torque both reduce to $[M{L}^2{T}^{-2}]$, yet work is a scalar (force dotted with displacement) while torque is a vector (force crossed with position) — dimensional analysis cannot tell them apart, so the physical definition, not the formula, must decide.

Every measurement carries some uncertainty. Systematic errors are one-sided, consistently making a reading too high or too low, and stem from instrumental defects (a zero error in a vernier or screw gauge), flawed technique, or observer bias; correcting the instrument or method reduces them. Random errors come from unpredictable fluctuations — in temperature, supply voltage, or judgement — and scatter readings both ways; averaging many observations reduces their effect. Gross errors are outright mistakes, such as misreading a scale, avoidable only through care. Every instrument also carries a least count error set by its smallest readable division.

For $n$ measurements of a quantity with mean value $a_{mean}$, the absolute error in one reading is $\Delta a_i=|a_{mean}-a_i|$, and the mean absolute error $\Delta a_{mean}$ is the average of these over all $n$ readings. The ratio $\Delta a_{mean}/a_{mean}$ is the relative (fractional) error, and $100$ times this is the percentage error.

Errors combine differently depending on the operation:

• For a sum or difference, $Z=A\pm B$, absolute errors always add: $\Delta Z=\Delta A+\Delta B$.
• For a product or quotient, $Z=AB$ or $A/B$, relative errors add: $\Delta Z/Z=\Delta A/A+\Delta B/B$.
• For $Z=A^p{B}^q/C^r$, relative errors add after weighting by the powers: $\Delta Z/Z=p\Delta A/A+q\Delta B/B+r\Delta C/C$.

Significant figures are the digits known with certainty plus one uncertain digit. All non-zero digits count; zeros between non-zero digits count; leading zeros before the first non-zero digit never count; and trailing zeros count only when a decimal point is present — which is why an unqualified number like 12300 is ambiguous and is better written in scientific notation to fix its precision. In calculations, a product or quotient is rounded to the least number of significant figures among the inputs, while a sum or difference is rounded instead to the least number of decimal places.

A frequent NEET trap is assuming subtraction reduces uncertainty: for $Z=A-B$, the temptation is to subtract the errors instead of adding them. Errors add for every sum or difference, never cancel, so the difference of two large, nearly equal, similarly uncertain quantities can carry a large percentage error even though each quantity alone was measured precisely.