What this chapter is about
Physics rests on measurement. Before you can describe motion, forces or energy, you must know how to express and compare quantities. This chapter introduces the International System of Units (SI), the standard language scientists everywhere use so that a measurement made in Chennai means the same thing in Geneva or Tokyo. You will learn the seven base units, how derived units are built from them, and how to write any physical quantity in terms of its dimensions.
The chapter also deals with the limits of measurement. Every instrument has a smallest reading it can give, and every observer introduces some error. Understanding significant figures, absolute error, relative error and the rules for combining errors lets you report a result honestly—neither claiming false precision nor throwing away information you actually have. These skills matter in every experiment you will do in the laboratory.
By the end you should be able to convert units confidently, check whether an equation is dimensionally consistent, estimate an unknown quantity by dimensional analysis, and state the uncertainty in a measured or calculated result.
Key ideas
- A physical quantity equals a numerical value multiplied by a unit; changing the unit changes the number but not the quantity itself.
- SI defines seven base units: metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (thermodynamic temperature), mole (amount of substance) and candela (luminous intensity).
- Every other unit (newton, joule, pascal, hertz, etc.) is a derived unit, expressible as a product or quotient of powers of base units.
- The dimensions of a quantity show how it depends on the base quantities; for example, velocity has dimensions [L T⁻¹] because it is length divided by time.
- Dimensional analysis can verify equations, convert units and sometimes derive relations, but it cannot find pure numbers or distinguish quantities with the same dimensions.
- Errors are classified as systematic (consistent bias) and random (unpredictable scatter); accuracy refers to closeness to the true value, precision to the spread of repeated readings.
- Significant figures are the digits in a measurement that carry real information; rules for rounding after arithmetic keep the final answer from pretending to be more precise than the data allow.
- When quantities are added or subtracted, absolute errors add; when they are multiplied or divided, relative (percentage) errors add.
Formulas and facts to remember
1. Dimensional formula of a derived quantity: write the quantity in terms of base quantities using [M], [L], [T], [A], [K], [mol], [cd]. Example: Force = mass × acceleration → [M L T⁻²].
2. Dimensional consistency check: both sides of any valid equation must have the same dimensions.
3. Conversion of units: if a quantity has dimensions [Mᵃ Lᵇ Tᶜ], and the unit changes from system 1 to system 2, then n₂ = n₁ × (M₁/M₂)ᵃ × (L₁/L₂)ᵇ × (T₁/T₂)ᶜ.
4. Absolute error in a single measurement: Δa = |measured value − true value|.
5. Mean absolute error of n readings: Δa_mean = (|Δa₁| + |Δa₂| + … + |Δaₙ|) / n.
6. Relative (fractional) error = Δa / a; percentage error = (Δa / a) × 100 %.
7. Propagation rules (for small errors):
- Sum or difference Z = A ± B: ΔZ = ΔA + ΔB.
- Product or quotient Z = A × B or A / B: ΔZ/Z = ΔA/A + ΔB/B.
- Power Z = Aⁿ: ΔZ/Z = |n| × (ΔA/A).
8. Significant figures: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are not significant; trailing zeros after the decimal point are significant.
Worked examples
### Example 1: Converting a derived unit
A car's kinetic energy is 45 000 J in SI. Express this in CGS units (erg).
Step 1. Energy has dimensions [M L² T⁻²].
Step 2. Ratios of base units from SI to CGS:
- Mass: 1 kg = 1000 g, so M₁/M₂ = 1000.
- Length: 1 m = 100 cm, so L₁/L₂ = 100.
- Time: 1 s = 1 s, so T₁/T₂ = 1.
Step 3. Apply the conversion formula: n₂ = n₁ × (1000)¹ × (100)² × (1)⁻² = 45 000 × 1000 × 10 000 × 1 = 4.5 × 10¹¹.
Answer: 45 000 J = 4.5 × 10¹¹ erg.
### Example 2: Dimensional analysis to check an equation
A student writes the time period of a simple pendulum as T = 2π √(g/l). Is this dimensionally correct?
Step 1. Dimensions of T (time period): [T].
Step 2. Dimensions of g (acceleration due to gravity): [L T⁻²].
Step 3. Dimensions of l (length): [L].
Step 4. Dimensions of √(g/l) = √([L T⁻²]/[L]) = √[T⁻²] = [T⁻¹].
Step 5. Right-hand side has dimensions [T⁻¹], not [T].
Conclusion: The equation is dimensionally wrong; the correct form is T = 2π √(l/g).
### Example 3: Combining errors in a calculation
A student measures the diameter of a wire with a screw gauge as d = 1.25 mm with absolute error Δd = 0.01 mm. The length of the wire is L = 80.0 cm with ΔL = 0.1 cm. She uses resistance R = ρL/A, where A = πd²/4. Find the percentage error in A.
Step 1. A depends on d² so the rule for powers applies: ΔA/A = 2 × (Δd/d).
Step 2. Δd/d = 0.01/1.25 = 0.008.
Step 3. Percentage error in A = 2 × 0.008 × 100 % = 1.6 %.
Answer: The percentage error in the cross-sectional area is 1.6 %.
Common mistakes
Writing 2500 m as having four significant figures when trailing zeros without a decimal point are ambiguous → write 2.500 × 10³ m to show four significant figures clearly.
Adding percentage errors when quantities are added → use absolute errors for addition; percentage errors are added only for multiplication and division.
Assuming dimensional correctness proves an equation is correct → it only shows the equation could be correct; numerical factors and terms with the same dimensions cannot be found this way.
Confusing precision with accuracy → a balance may give readings that cluster tightly (high precision) yet all differ from the true mass (low accuracy) because of a systematic error.
Dropping powers of ten during unit conversion → track each base-unit ratio raised to its dimensional power carefully.
Quick revision
- SI has seven base units; all others are derived from these.
- Dimensions show the dependence on base quantities and must match on both sides of any equation.
- Systematic errors shift all readings one way; random errors scatter them.
- Add absolute errors for sums and differences; add relative errors for products, quotients and powers.
- Significant figures communicate how precise a measurement really is.
- Dimensional analysis checks equations but cannot find pure numbers.