SI units, dimensions and dimensional analysis — NEET 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 $[M L T^{- 2}]$ , since $F = ma$ combines mass with acceleration ( $L T^{- 2}$ ). This gives every physical quantity a dimensional formula built from the base dimensions $M$ , $L$ , $T$ , $A$ , $K$ , $mol$ , and $cd$ . 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 $s = u t + \frac{1}{2} a t^{2}$ 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 $T = k l / g$ .
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 $2 π$ 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.