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Class XI · Chapter 01

Units and Measurement

The ground rules of all physics: how quantities are defined, how precisely they can be measured, and how dimensions catch a wrong equation before you've even solved it.

● Easy ⏱ 24 min read 🎯 28 practice questions 📊 Not yet revised
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1The International System of Units

Every measurement is a comparison against an agreed standard. Physics uses the International System of Units (SI), built from seven base units, chosen so that every other physical quantity can be expressed as some combination of them.

  • 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 units — newtons, joules, volts, and so on — are derived units, built by combining these seven algebraically.

2Measuring Length, Mass and Time

Ordinary rulers and balances handle everyday scales, but physics regularly deals with distances far too large or far too small to measure directly.

Parallax Method (for large distances, e.g. to a star)
D = b / θ
b = baseline (distance between two observation points), θ = parallax angle (in radians) — the apparent shift in the object's position when viewed from the two ends of the baseline. Used for everything from surveying to measuring astronomical distances.

Very short time intervals (atomic and nuclear processes) are measured using the extremely regular periodic vibrations of atoms — this is exactly what defines the modern second, and what makes atomic clocks so precise.

3Accuracy, Precision and Errors

Accuracy is how close a measurement is to the true value. Precision is how consistently repeated measurements agree with each other — a measurement can be precise without being accurate (consistently wrong) or accurate without being especially precise.

  • Systematic errors: consistent, one-directional errors from a flawed instrument, technique, or environmental factor — they can, in principle, be identified and corrected.
  • Random errors: unpredictable fluctuations from measurement to measurement, with no consistent direction — reduced (not eliminated) by taking many readings and averaging.
Absolute, Relative and Percentage Error
Δa = amean − ai, relative error = Δamean/amean, % error = relative error × 100
Absolute error has the same units as the quantity itself; relative and percentage error are unitless, which is exactly why they're used to compare the quality of very different measurements fairly.

4Combination of Errors

When a result is calculated from several measured quantities, its error depends on how those quantities are combined.

Sum or Difference
Δ(A ± B) = ΔA + ΔB
Absolute errors always add — even for a subtraction, since both original measurements could each be off in the worst-case direction.
Product or Quotient
ΔZ/Z = ΔA/A + ΔB/B (for Z = AB or Z = A/B)
Relative errors add.
Power
ΔZ/Z = n (ΔA/A), for Z = Aⁿ
A quantity raised to a power amplifies its relative error by that power — which is exactly why measuring radius carefully matters so much for anything involving r² or r³.

5Significant Figures

Significant figures communicate how precisely a number is actually known — reporting more digits than your measurement supports is misleading, not more accurate.

  • All non-zero digits are significant.
  • Zeros between non-zero digits are significant (e.g. 405 has 3 significant figures).
  • Leading zeros are never significant (0.0042 has 2 significant figures).
  • Trailing zeros after a decimal point are significant (4.500 has 4 significant figures).
  • Trailing zeros in a whole number with no decimal point are ambiguous — scientific notation removes the ambiguity (4500 is unclear; 4.500×10³ is unambiguously 4 significant figures).

6Significant Figures in Calculations

  • Multiplication and division: the result keeps the same number of significant figures as the least-precise factor involved.
  • Addition and subtraction: the result keeps the same number of decimal places as the term with the fewest decimal places — not the fewest significant figures.
Common Slip Students often apply the multiplication rule (matching significant figures) to addition and subtraction too. The rules are genuinely different — addition/subtraction cares about decimal places, not total significant figure count.

7Dimensions of Physical Quantities

Every physical quantity can be expressed in terms of powers of the base quantities: Mass [M], Length [L], Time [T], Current [A], Temperature [K], Amount of substance [mol], and Luminous intensity [cd]. This is a quantity's dimensional formula.

Examples
Velocity: [LT⁻¹] Force: [MLT⁻²] Energy: [ML²T⁻²]
Dimensions deliberately say nothing about the numerical value or unit system used — only how a quantity is built from the base quantities.

8Dimensional Analysis and Its Uses

Principle of Homogeneity
Every term on both sides of a valid physical equation must have identical dimensions.
This gives a fast, powerful check: if the dimensions don't match, the equation is definitely wrong (though matching dimensions alone doesn't prove an equation is fully correct — a missing dimensionless constant, like ½ or π, wouldn't be caught).
  • Checking equations: verify dimensional consistency before trusting a derived formula.
  • Deriving relationships: figure out how a quantity depends on others, when you know (or guess) which variables are involved.
  • Converting units between different unit systems.
Limitations Dimensional analysis can't determine dimensionless constants, can't handle equations involving sums of terms with different powers (like s = ut + ½at²), and can't handle trigonometric, exponential, or logarithmic relationships at all — it only works cleanly for single-term, power-law relationships.

Formula Summary

Parallax Method
D = b/θ
Percentage Error
(Δa_mean/a_mean) × 100
Sum/Difference Error
Δ(A±B) = ΔA+ΔB
Product/Quotient Error
ΔZ/Z = ΔA/A + ΔB/B
Power Error
ΔZ/Z = n(ΔA/A)
Force Dimensions
[MLT⁻²]
Energy Dimensions
[ML²T⁻²]

Solved Examples

Example 1 · Dimensional Consistency

Check whether the equation v = u + at is dimensionally consistent.

Solution: [v] = [LT⁻¹], [u] = [LT⁻¹], [at] = [LT⁻²][T] = [LT⁻¹].

All three terms share the same dimension [LT⁻¹], so the equation is dimensionally consistent.

Example 2 · Combination of Errors

The length and breadth of a rectangular sheet are measured as (16.2 ± 0.1) cm and (10.1 ± 0.1) cm. Find the area, with its error.

Solution: Area = l × b = 16.2 × 10.1 = 163.62 cm².

ΔA/A = Δl/l + Δb/b = 0.1/16.2 + 0.1/10.1 ≈ 0.0062 + 0.0099 ≈ 0.0161.

ΔA ≈ 0.0161 × 163.62 ≈ 2.6 cm². So the area is (164 ± 3) cm², rounded to appropriate significant figures.

Quick Check

1. The number of significant figures in 0.00620 is:
6
5
✓ 3 (correct)
2
2. The dimensional formula of force is:
[MLT⁻¹]
✓ [MLT⁻²] (correct)
[ML²T⁻²]
[MLT²]
Up Next · Chapter 02

Motion in a Straight Line

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