Units, Measurements & Errors

Physics begins with measurement, and every measurement requires a standardised system of units. The International System of Units (SI) defines seven base quantities: 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). All other physical quantities are derived from these base units.
For instance, force is measured in newtons (N = kg m $s^{-2}$), energy in joules (J = kg $m^{2} \; s^{-2}$), pressure in pascals (Pa = kg $m^{-1} \; s^{-2}$), and power in watts (W = kg $m^{2} \; s^{-3}$).

Every physical quantity can be expressed as a combination of the base dimensions M (mass), L (length), and T (time). The dimensional formula of velocity is [$M^{0} \; L^{1} \; T^{-1}$], acceleration is [$M^{0} \; L^{1} \; T^{-2}$], force is [$M^{1} \; L^{1} \; T^{-2}$], energy is [$M^{1} \; L^{2} \; T^{-2}$], pressure is [$M^{1} \; L^{-1} \; T^{-2}$], power is [$M^{1} \; L^{2} \; T^{-3}$], and angular momentum is [$M^{1} \; L^{2} \; T^{-1}$]. Dimensional analysis serves three purposes: (1) checking the correctness of a physical equation by verifying that dimensions on both sides match, (2) deriving relationships between quantities when the exact form is unknown, and (3) converting a physical quantity from one unit system to another.
However, dimensional analysis has important limitations: it cannot determine dimensionless constants (such as the factor of $\frac{1}{2}$ in KE = $\frac{1}{2}$ $mv^{2}$), and it cannot distinguish between two different formulas that share the same dimensions (both v = u + at and v = u + 2at have identical dimensions [$M^{0} \; L^{1} \; T^{-1}$] on each side).

Significant figures reflect the precision of a measurement. Leading zeros are never significant (0.00450 has 3 significant figures). Trailing zeros after a decimal point are significant (2.30 has 3 significant figures). In arithmetic operations, for addition and subtraction the result retains as many decimal places as the term with the fewest decimal places. For multiplication and division, the result retains as many significant figures as the factor with the fewest significant figures.

Errors in measurement are classified as systematic (consistent offset due to instrument calibration, personal bias, or environmental conditions) and random (unpredictable fluctuations). The absolute error $\Delta$-a is the magnitude of the difference between the measured value and the true value. The relative error is $\Delta$-a/a (dimensionless), and the percentage error is ($\Delta$-a/a) x 100%.
For error propagation: when quantities are added or subtracted, absolute errors add ($\Delta$-Z = $\Delta$-A + $\Delta$-B); when multiplied or divided, relative errors add ($\Delta$-Z/Z = $\Delta$-A/A + $\Delta$-B/B); for a quantity raised to the power n, the relative error is multiplied by n ($\Delta$-Z/Z = n x $\Delta$-A/A).

Solved Numerical 1: Convert G = 6.67 x $10^{-11}$ N $m^{2} \; kg^{-2}$ from SI to CGS. The dimensional formula of G is [$M^{-1} \; L^{3} \; T^{-2}$].
In SI: M = 1 kg, L = 1 m, T = 1 s.
In CGS: M = 1 g, L = 1 cm, T = 1 s.
$G_{CGS}$ = $G_{SI}$ x (1 kg / 1 g)^-1 x (1 m / 1 cm)³ x (1 s / 1 s)^-2 = 6.67 x $10^{-11}$ x ($10^{3}$)^-1 x ($10^{2}$)³ x 1 = 6.67 x $10^{-11}$ x $10^{-3}$ x $10^{6}$ = 6.67 x $10^{-8}$ dyne $cm^{2} \; g^{-2}$.

Solved Numerical 2: The period T of a pendulum depends on length L [$M^{0} \; L^{1} \; T^{0}$] and gravity g [$M^{0} \; L^{1} \; T^{-2}$].
Let T = k L^a g^b.
Dimensions: [$T^{1}$] = [L^a] [L^b $T^{-2}$b] = [L^(a+b) T^(-2b)].
Equating: a + b = 0 and -2b = 1, giving b = -$\frac{1}{2}$, a = $\frac{1}{2}$.
Therefore T = k $\sqrt{L/g}$, where k = $2\pi$ (dimensionless constant, undetermined by dimensional analysis).

Solved Numerical 3: Given a = P x $Q^{2}$ / R, with P = 4.0 +/- 0.2, Q = 3.0 +/- 0.1, R = 2.0 +/- 0.1.
Percentage error in a: ($\Delta$-a/a) x 100 = ($\Delta$-P/P + 2 x $\Delta$-Q/Q + $\Delta$-R/R) x 100 = (0.$\frac{2}{4}$.0 + 2 x 0.$\frac{1}{3}$.0 + 0.$\frac{1}{2}$.0) x 100 = (0.05 + 0.0667 + 0.05) x 100 = 16.67%.
The value of a = 4.0 x 9.0 / 2.0 = 18.0, so a = 18.0 +/- 16.67%.

The key testable concept is dimensional analysis for verifying equations, converting units, and error propagation using the power rule (relative error multiplied by the exponent).