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Class 11 · Chapter 2

Physical World & Measurement

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Physical World & Measurement Formula Sheet

11 formulas across 3 topics in Physical World & Measurement.

2 min read

Updated 2026-07-08 · v1.0.0

Units & Dimensional Analysis

Dimensions are the DNA of a physical quantity — every valid physics equation must have matching dimensions on both sides, and this single check catches a huge fraction of algebra mistakes.

[LHS] = [RHS] for any dimensionally correct equation

Principle of homogeneity of dimensions

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Valid when

  • A necessary but NOT sufficient condition for an equation to be correct — dimensionally consistent equations can still be numerically wrong (missing a dimensionless constant like 2 or π)

Common mistakes

  • Dimensional correctness does not prove an equation right, only that it's not obviously wrong — purely numeric constants (2, ½, π) are invisible to dimensional analysis
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n₁u₁ = n₂u₂ ⟹ n₂ = n₁ [M₁/M₂]ᵃ [L₁/L₂]ᵇ [T₁/T₂]ᶜ

Converting a physical quantity between unit systems using dimensions

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The MAGNITUDE of a physical quantity (n×u) is invariant — only the numerical value n changes when the unit u changes

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
a, b, cpowers of mass, length, time in the quantity's dimensional formuladimensionless[M⁰L⁰T⁰]

Valid when

  • Requires knowing the correct dimensional formula [MᵃLᵇTᶜ] of the quantity being converted
Read the concept →

Worth remembering

  • Dimensional analysis can also derive the FORM of a relation (e.g. T = 2π√(l/g) for a pendulum) up to an unknown dimensionless constant — but it can never find that constant
  • Purely dimensionless quantities (angle, strain, refractive index) cannot be checked or derived by dimensional analysis at all

Significant Figures & Errors in Measurement

No measurement is exact. Knowing how small errors in individual readings combine — adding, multiplying, or raised to powers — decides how much you can trust a calculated final answer.

% error = (Δa / a) × 100

Percentage error in a single measured quantity

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
Δamean absolute error in the measurementsame as asame as a
amean (true) value of the quantitycontext-dependentcontext-dependent
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Δ(a ± b) = Δa + Δb (maximum possible error)

Error propagation in sum or difference

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Worst-case scenario: both individual errors act in the same unfavourable direction simultaneously

Valid when

  • Errors ADD even when the quantities are SUBTRACTED — this is the single most commonly missed point in error analysis

Common mistakes

  • For a difference (a − b), students often (wrongly) subtract the errors too — absolute errors always add, regardless of whether the quantities add or subtract
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Δ(ab)/(ab) = Δa/a + Δb/b (same rule for a/b)

Error propagation in product or quotient

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Taking logarithms of Z = ab (or a/b) turns multiplication into addition, then differentiating

Valid when

  • RELATIVE (fractional) errors add for both multiplication AND division — errors never partially cancel

Common mistakes

  • For a quotient a/b, the relative errors still ADD — they do not subtract even though the operation is division
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For Z = AᵖBᵍ/Cʳ: ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C)

Error propagation with powers

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Generalisation of the product rule — an exponent multiplies through the relative error term

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
p, q, rpowers to which A, B, C are raised in the formula for Zdimensionless[M⁰L⁰T⁰]

Valid when

  • A quantity that appears SQUARED contributes DOUBLE its own relative error — this is the classic source of dominant error in a calculation

Common mistakes

  • Forgetting to multiply by the power — e.g. for g = 4π²l/T², the error in T contributes TWICE (since T² appears), making timing the dominant source of error in a pendulum experiment
Read the concept →

Boards and JEE Main both love pendulum-experiment error questions (g = 4π²l/T²) — the T² term means a small timing error dominates the final error in g. Recognise this pattern instantly.

Vernier Calipers & Screw Gauge

Both instruments work on the same idea — a fixed main scale plus a sliding/rotating scale that lets you read a fraction of the smallest main-scale division.

LC = 1 MSD − 1 VSD

Least count of a Vernier caliper

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If n vernier divisions span (n−1) main-scale divisions, each VSD is slightly smaller than each MSD by exactly this amount

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
MSDmain scale divisiontypically mm[L]
VSDvernier scale divisiontypically mm[L]

Valid when

  • Standard vernier: 1 MSD = 1 mm, 10 VSD = 9 MSD, giving LC = 0.1 mm = 0.01 cm
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True reading = MSR + (VSC × LC) ± zero error correction

Total reading of a Vernier caliper

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
MSRmain scale reading (before the vernier zero)cm or mm[L]
VSCvernier scale coincidence — which vernier division exactly lines up with a main scale divisiondimensionless (a count)[M⁰L⁰T⁰]

Valid when

  • Zero error must be SUBTRACTED if positive, ADDED if negative — sign convention is a frequent source of mistakes

Common mistakes

  • Positive zero error means the instrument OVER-reads, so it must be subtracted from the observed reading — many students add it by mistake
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Pitch = distance moved on main scale / number of full rotations of the screw

Pitch of a screw gauge

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Valid when

  • Typical pitch values: 0.5 mm or 1 mm per full rotation
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LC = Pitch / (number of divisions on the circular scale)

Least count of a screw gauge

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Valid when

  • Typical: pitch 1 mm, 100 circular divisions gives LC = 0.01 mm — ten times finer than a standard Vernier caliper

Common mistakes

  • A screw gauge is finer than a Vernier caliper (LC ~0.01 mm vs ~0.01 cm) — a factor-of-10 difference that's easy to misremember under pressure
Read the concept →

Reading = MSR + (CSR × LC) − zero error correction

Total reading of a screw gauge

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
CSRcircular scale reading — division on the circular scale coinciding with the reference linedimensionless (a count)[M⁰L⁰T⁰]

Valid when

  • Same zero-error sign convention as the Vernier caliper: positive zero error is subtracted, negative is added
Read the concept →

Worth remembering

  • Backlash error (screw gauge) — always turn the screw in the SAME direction when approaching a measurement to avoid this instrumental error

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