Units & Measurements — Concept Notes (Class 11 Physics)

UNITS & MEASUREMENTS — CONCEPT NOTES (Class 11 · JEE/NEET) 1. PHYSICAL QUANTITIES & UNITS A physical quantity is anything we can measure. Measuring it needs a unit (a chosen standard) and a number (how many units). Fundamental quantities are independent; derived quantities are built from them (e.g. speed = length/time). 2. THE SI SYSTEM — 7 BASE UNITS Length — metre (m); Mass — kilogram (kg); Time — second (s); Electric current — ampere (A); Temperature — kelvin (K); Amount of substance — mole (mol); Luminous intensity — candela (cd). Every derived unit comes from these, e.g. $1\ \text{N} = 1\ \text{kg}\,\text{m}\,\text{s}^{-2}$. 3. DIMENSIONS & DIMENSIONAL FORMULAE The dimension of a quantity shows how it is built from mass $[M]$, length $[L]$ and time $[T]$, written $[M^a\,L^b\,T^c]$. • Velocity $[L\,T^{-1}]$, acceleration $[L\,T^{-2}]$ • Force $[M\,L\,T^{-2}]$, momentum & impulse $[M\,L\,T^{-1}]$ • Work / energy $[M\,L^{2}\,T^{-2}]$, power $[M\,L^{2}\,T^{-3}]$ • Pressure / stress $[M\,L^{-1}\,T^{-2}]$, density $[M\,L^{-3}]$ A quantity with no $M,L,T$ left over is DIMENSIONLESS (strain, angle, refractive index, relative density). 4. USES OF DIMENSIONAL ANALYSIS (a) Check an equation — every term must have the SAME dimensions (principle of homogeneity). (b) Convert units between systems. (c) Derive a relation between quantities (up to a dimensionless constant). LIMITATIONS: it cannot find dimensionless constants (like $\tfrac12$ or $2\pi$), cannot handle sums/differences of numbers, and fails for trig/log/exponential functions. 5. SIGNIFICANT FIGURES • All non-zero digits are significant. • Zeros between non-zero digits count ($2.003 \to 4$). • Leading zeros do NOT count ($0.0034 \to 2$). • Trailing zeros AFTER a decimal DO count ($2.300 \to 4$). • Add/subtract: keep the least number of DECIMAL PLACES. Multiply/divide: keep the least number of SIGNIFICANT FIGURES. 6. ERRORS IN MEASUREMENT Systematic errors are a consistent bias (fix by calibration); random errors scatter both ways (reduce by averaging). Absolute error $=|a_i - \bar a|$, mean absolute error $\overline{\Delta a}=\dfrac{1}{n}\sum |a_i-\bar a|$, relative error $=\dfrac{\overline{\Delta a}}{\bar a}$, percentage error $=\dfrac{\overline{\Delta a}}{\bar a}\times 100\%$. 7. COMBINATION OF ERRORS • Sum or difference $Z = A \pm B \Rightarrow \Delta Z = \Delta A + \Delta B$. • Product or quotient $Z = AB$ or $\tfrac{A}{B} \Rightarrow \dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$. • Powers $Z = \dfrac{A^{p} B^{q}}{C^{r}} \Rightarrow \dfrac{\Delta Z}{Z} = p\,\dfrac{\Delta A}{A} + q\,\dfrac{\Delta B}{B} + r\,\dfrac{\Delta C}{C}$. EXAM TIP: the quantity raised to the highest power usually contributes the largest error. 8. ACCURACY vs PRECISION Accuracy = closeness to the true value. Precision = closeness of repeated readings to each other. A zero-error instrument can be precise but not accurate. 9. LEAST COUNT (LC) The smallest value an instrument can read. $$\text{Vernier callipers:}\quad \text{LC} = 1\,\text{MSD} - 1\,\text{VSD}$$ $$\text{Screw gauge:}\quad \text{LC} = \dfrac{\text{pitch}}{\text{no. of circular-scale divisions}}$$ A smaller least count means a more precise instrument.