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Units and Measurement – Class 11 Physics Chapter 1 Notes & NCERT PDF

Class 11PhysicsChapter 1NCERT book: Physics Part I - Class 11

Notes and a simple summary of Chapter 1 of the NCERT Class 11 Physics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 1: Units and Measurement

Introduction

Physics is a quantitative science — every law we study is built on measurement. Measuring a physical quantity means comparing it with an internationally accepted reference standard called a unit, and the result is always written as a number + unit. This chapter builds the toolkit you will use throughout Classes 11 and 12: the SI system, significant figures (how to report precision honestly) and dimensions (how to check and even derive equations).

A measurement result = numerical measure × unit (e.g. 1.62 s, 287.5 cm).

Key Concepts

1.1 Fundamental and derived units

  • Base (fundamental) quantities have base units; all other quantities are expressed as combinations of them — these are derived units.
  • A complete set of base and derived units is a system of units.

    • 1.2 The International System of Units (SI)

  • Older systems: CGS (centimetre, gram, second), FPS (foot, pound, second), MKS (metre, kilogram, second).
  • SI (Système International d'Unités), developed by BIPM in 1971 and revised in November 2018, is the accepted system today. It uses the decimal system, so conversions are easy.
  • Seven base quantities and units: length – metre (m); mass – kilogram (kg); time – second (s); electric current – ampere (A); thermodynamic temperature – kelvin (K); amount of substance – mole (mol); luminous intensity – candela (cd).
  • Two dimensionless supplementary units: plane angle dθ = ds/r, unit radian (rad); solid angle dΩ = dA/r², unit steradian (sr).
  • The modern definitions fix constants of nature: e.g. the metre fixes c = 299 792 458 m s⁻¹; the kilogram fixes Planck's constant h = 6.626 070 15 × 10⁻³⁴ J s; one mole contains exactly 6.022 140 76 × 10²³ entities (Avogadro number). The book notes these values need not be memorised.

    • 1.3 Significant figures

    Reliable digits plus the first uncertain digit are the significant figures. In 1.62 s, 1 and 6 are certain and 2 is uncertain → 3 significant figures.

    Rules

  • All non-zero digits are significant.
  • Zeros between non-zero digits are significant (2.308 cm → 4 s.f.).
  • For a number less than 1, zeros after the decimal point but before the first non-zero digit are not significant (0.002308 → 4 s.f.).
  • Trailing zeros in a number without a decimal point are not significant (123 m = 12300 cm → 3 s.f.).
  • Trailing zeros in a number with a decimal point are significant (3.500 or 0.06900 → 4 s.f.).
  • Changing units never changes the number of significant figures — so report measurements in scientific notation a × 10ᵇ (1 ≤ a < 10): 4.700 m = 4.700 × 10² cm = 4.700 × 10³ mm (4 s.f. each).
  • Order of magnitude: round a to 1 (if a ≤ 5) or 10 (if 5 < a ≤ 10); the power b is the order. Earth's diameter 1.28 × 10⁷ m (order 7); hydrogen atom 1.06 × 10⁻¹⁰ m (order −10) → Earth is 17 orders of magnitude larger.
  • Exact numbers (like the 2 in s = 2πr, or n in T = t/n) have infinitely many significant figures.

    • 1.3.1 Arithmetic with significant figures

  • Multiplication/division: keep as many s.f. as the quantity with the fewest s.f. Density = 4.237 g ÷ 2.51 cm³ = 1.69 g cm⁻³ (3 s.f.). Light year = 3.00 × 10⁸ m s⁻¹ × 3.1557 × 10⁷ s = 9.47 × 10¹⁵ m.
  • Addition/subtraction: keep as many decimal places as the quantity with the fewest decimal places. 436.32 g + 227.2 g + 0.301 g = 663.821 g → 663.8 g. 0.307 m − 0.304 m = 0.003 m = 3 × 10⁻³ m.

    • 1.3.2 Rounding off

  • Drop digit > 5 → raise the preceding digit (2.746 → 2.75); < 5 → leave it (1.743 → 1.74).
  • Drop digit exactly 5 → raise the preceding digit only if it is odd: 2.745 → 2.74, 2.735 → 2.74.
  • In multi-step work keep one extra digit in intermediate steps: 1/9.58 = 0.1044 → reciprocal gives 9.58 back, but 0.104 → 9.62.

    • 1.3.3 Uncertainty in results

  • l = 16.2 ± 0.1 cm (±0.6 %), b = 10.1 ± 0.1 cm (±1 %) → l × b = 164 ± 3 cm².
  • Relative error depends on the number itself: ±0.01 g in 1.02 g is ±1 %, but in 9.89 g it is ±0.1 %.

    • 1.4–1.5 Dimensions and dimensional formulae

  • The dimensions of a quantity are the powers to which base quantities are raised: [L], [M], [T], [A], [K], [cd], [mol].
  • Volume [M⁰ L³ T⁰]; speed [M⁰ L T⁻¹]; acceleration [M⁰ L T⁻²]; force [M L T⁻²]; density [M L⁻³ T⁰].
  • A dimensional equation equates a quantity to its dimensional formula, e.g. [F] = [M L T⁻²].

    • 1.6 Dimensional analysis and its applications

  • Principle of homogeneity: only quantities with the same dimensions can be added, subtracted or equated.
  • Checking equations: x = x₀ + v₀t + ½at² — every term has dimension [L], so it is dimensionally correct. ½mv² = mgh: both sides [M L² T⁻²].
  • Limits: a dimensionally correct equation need not be correct (dimensionless constants cannot be found), but a dimensionally wrong equation is definitely wrong. Arguments of sin, log, exp must be dimensionless.
  • Deducing relations: assuming T = k lˣ gʸ mᶻ for a pendulum gives x = ½, y = −½, z = 0 → T = k√(l/g); experiment gives k = 2π.

    • Worked Examples

  • Cube of side 7.203 m. Surface area = 6(7.203)² = 311.299254 m² → 311.3 m²; volume = (7.203)³ = 373.714754 m³ → 373.7 m³ (4 s.f.).
  • Density with limited precision. 5.74 g occupies 1.2 cm³ → 5.74/1.2 = 4.78 → 4.8 g cm⁻³ (volume has only 2 s.f.).
  • Which kinetic-energy formulae are impossible? K must be [M L² T⁻²]. m²v³ → [M² L³ T⁻³] ✗; ma → [M L T⁻²] ✗; ½mv² + ma adds unlike quantities ✗. Both ½mv² and (3/16)mv² pass the test — dimensions cannot choose between them; the definition gives ½mv².
  • Metal sheet 4.234 m × 1.005 m × 2.01 cm. Area = 2(lb + bh + hl) = 8.72 m² (3 s.f.); volume = 0.0855 m³.
  • Density of the Sun (M = 2.0 × 10³⁰ kg, R = 7.0 × 10⁸ m): ρ = M / (4/3 πR³) ≈ 1.4 × 10³ kg m⁻³ — in the range of solids and liquids, not gases.

    • Formula / Key-Facts Sheet

  • dθ = ds/r (rad); dΩ = dA/r² (sr)
  • Multiplication/division → least number of significant figures
  • Addition/subtraction → least number of decimal places
  • Order of magnitude: a × 10ᵇ ≈ 10ᵇ (a ≤ 5) or 10ᵇ⁺¹ (a > 5)
  • [F] = [M L T⁻²]; [Energy] = [M L² T⁻²]; [ρ] = [M L⁻³]; [v] = [L T⁻¹]; [a] = [L T⁻²]
  • Pendulum by dimensions: T = k√(l/g), k = 2π

    • Exam Focus

    CBSE questions come straight from the chapter's exercises:

  • Count significant figures (Ex. 1.10): 0.007 m² → 1; 2.64 × 10²⁴ kg → 3; 0.2370 g cm⁻³ → 4; 6.320 J → 4; 6.032 N m⁻² → 4; 0.0006032 m² → 4.
  • Unit conversion using dimensions (Ex. 1.3): a calorie = 4.2 J = 4.2 kg m² s⁻²; in units α kg, β m, γ s it becomes 4.2 α⁻¹ β⁻² γ². Outline: write 1 J in new units, substitute each base unit.
  • Speed of light as unit length (Ex. 1.5): light takes 8 min 20 s = 500 s, so the Sun–Earth distance is 500 new units.
  • Guess the missing constant (Ex. 1.13): (1 − v²)^½ must be dimensionless → m = m₀ / (1 − v²/c²)^½.
  • Magnified hair (Ex. 1.7): 3.5 mm ÷ 100 = 0.035 mm.

    • Key Terms

  • Unit (मात्रक)
  • Base / fundamental quantity (मूल राशि)
  • Derived unit (व्युत्पन्न मात्रक)
  • SI units (अंतर्राष्ट्रीय मात्रक पद्धति)
  • Radian (रेडियन)
  • Steradian (स्टेरेडियन)
  • Significant figures (सार्थक अंक)
  • Order of magnitude (परिमाण की कोटि)
  • Rounding off (पूर्णांकन)
  • Dimensions (विमाएँ)
  • Dimensional formula (विमीय सूत्र)
  • Dimensional equation (विमीय समीकरण)
  • Principle of homogeneity (विमीय समांगता का सिद्धांत)
  • Dimensionless quantity (विमाहीन राशि)
  • Relative error (आपेक्षिक त्रुटि)

    • Common Mistakes

  • Counting leading zeros in 0.0006032 as significant (it has 4 s.f., not 7).
  • Writing 4700 mm and claiming 2 s.f. after converting 4.700 m — use 4.700 × 10³ mm.
  • Using the s.f. rule for addition: 436.32 + 227.2 + 0.301 is 663.8 g (decimal places), not 664 g.
  • Rounding 2.745 to 2.75 — with 5 dropped and an even preceding digit, it stays 2.74.
  • Thinking a dimensionally correct equation is proved correct; it is only not proved wrong.
  • Forgetting that the method of dimensions cannot give constants like 2π or ½.

    • 💡 Key Learning Points

    • ✓SI has seven base units (m, kg, s, A, K, mol, cd) plus radian and steradian for plane and solid angle; SI was revised in November 2018
    • ✓Significant figures = reliable digits + first uncertain digit; changing units never changes their number, so use scientific notation
    • ✓Multiply/divide: keep the least number of significant figures; add/subtract: keep the least number of decimal places
    • ✓Dimensions express a quantity in powers of base quantities, e.g. [F] = [M L T⁻²]; only like dimensions can be added or equated
    • ✓Dimensional analysis checks equations and deduces relations like T = k√(l/g), but cannot find dimensionless constants

    👨‍🏫 Teaching Tips

    • →Give pairs a set of measured values (0.0200, 4700, 3.500 × 10²) and have them classify significant figures before revealing the rules
    • →Convert 4.700 m into cm, mm and km on the board to show why scientific notation removes trailing-zero ambiguity
    • →Run a density calculation (4.237 g / 2.51 cm³) on a calculator and let students argue why 1.68804780876 is absurd
    • →Use the 2.745 / 2.735 rounding pair as a quick quiz to fix the even–odd rule
    • →Let students derive the pendulum formula by dimensions, then time a real pendulum to find k ≈ 2π
    • →Show the five kinetic-energy options from Example 1.4 and have groups eliminate them using dimensions alone
    • →Connect orders of magnitude to real objects: Earth (10⁷ m) vs hydrogen atom (10⁻¹⁰ m) — 17 orders apart

    📋 Assessment Questions

    1. State the number of significant figures in 0.2370 g cm⁻³ and 0.0006032 m², giving the rule used
    2. Report 436.32 g + 227.2 g + 0.301 g correctly and justify
    3. Write the dimensional formulae of force, energy and density
    4. Check the dimensional correctness of ½mv² = mgh
    5. Use the method of dimensions to show that the period of a simple pendulum is proportional to √(l/g)

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    Keep learning

    All chapters in Physics Part I - Class 11

    1. 1. Units and Measurement
    2. 2. Motion in a Straight Line
    3. 3. Motion in a Plane
    4. 4. Laws of Motion
    5. 5. Work, Energy and Power
    6. 6. System of Particles and Rotational Motion
    7. 7. Gravitation