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. To measure it we need a unit (a chosen standard) and a number (how many units). • Fundamental (base) quantities: independent of others. • Derived quantities: built from base quantities (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). All other (derived) units come from these — e.g. newton = kg·m·s^-2. 3. DIMENSIONS & DIMENSIONAL FORMULAE The dimension of a quantity shows how it is built from mass [M], length [L] and time [T] (and [A],[K] etc.). Written as [M^a L^b T^c]. Examples: velocity [M^0 L^1 T^-1], acceleration [L T^-2], force [M L T^-2], work/energy [M L^2 T^-2], power [M L^2 T^-3], momentum/impulse [M L T^-1], pressure/stress [M L^-1 T^-2], density [M L^-3]. A quantity with no [M],[L],[T] left over is DIMENSIONLESS (e.g. 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 from one system to another. (c) Derive a relation between quantities (up to a dimensionless constant). LIMITATIONS: cannot find dimensionless constants (like 1/2 or 2π), cannot check equations with +/-, and fails if a quantity depends on more than 3 base quantities or on trig/log/exponential functions. 5. SIGNIFICANT FIGURES (rules) • All non-zero digits are significant. • Zeros between non-zero digits are significant (2.003 → 4). • Leading zeros are NOT significant (0.0034 → 2). • Trailing zeros AFTER a decimal ARE significant (2.300 → 4). • In addition/subtraction: keep the least number of DECIMAL PLACES. • In multiplication/division: keep the least number of SIGNIFICANT FIGURES. 6. ERRORS IN MEASUREMENT • Systematic errors: consistent bias (faulty instrument, zero error) — reduce by calibration. • Random errors: scatter both ways — reduce by repeating and averaging. Absolute error = |measured − true (or mean)|. Mean absolute error = average of absolute errors. Relative error = mean absolute error / mean value. Percentage error = relative error × 100%. 7. COMBINATION OF ERRORS • Sum or difference Z = A ± B → ΔZ = ΔA + ΔB (absolute errors add). • Product or quotient Z = AB or A/B → ΔZ/Z = ΔA/A + ΔB/B (relative errors add). • Power Z = A^p B^q / C^r → ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C). KEY EXAM POINT: powers multiply the relative error, so the quantity with the highest power usually contributes the most error. 8. ACCURACY vs PRECISION Accuracy = how close to the true value. Precision = how close repeated readings are to each other. You can be precise but not accurate (a zero-error instrument). 9. LEAST COUNT (LC) The smallest value an instrument can read. • Vernier callipers: LC = 1 Main Scale Division − 1 Vernier Scale Division. • Screw gauge / micrometer: LC = pitch ÷ (number of divisions on the circular scale). Smaller least count → more precise instrument.