Units and Measurements

Chapter 2: Units and Measurements

Measurement is the cornerstone of physics. To describe any physical quantity, we need a number and a unit. Without units, numbers are meaningless — saying a table is "5 long" tells us nothing.

Physical Quantities

A physical quantity is any property of matter or energy that can be measured and expressed as a number with a unit (e.g., mass, length, time, temperature).

Physical quantities are classified as:
Fundamental (Base) Quantities: Cannot be expressed in terms of other quantities. The SI system has 7 base quantities.
Derived Quantities: Defined in terms of base quantities (e.g., velocity = length/time, force = mass × acceleration).

The SI System

The Systeme International d'Unites (SI) is the modern form of the metric system, adopted internationally. Its 7 base units are:

| Quantity | Unit | Symbol |
|---|---|---|
| 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 |

Significant Figures

1.Significant figures (sig figs) convey the precision of a measurement. Rules:
2.All non-zero digits are significant.
3.Zeros between non-zero digits are significant (e.g., 1005 has 4 sig figs).
4.Leading zeros are NOT significant (0.0032 has 2 sig figs).
5.Trailing zeros after a decimal point ARE significant (3.400 has 4 sig figs).

In calculations: For multiplication/division, keep the same number of sig figs as the least precise measurement. For addition/subtraction, keep to the least number of decimal places.

Errors in Measurement

No measurement is perfectly precise. Types of errors:
Systematic error: consistent bias in one direction (e.g., a ruler that starts at 1 mm instead of 0). Reduces accuracy.
Random error: unpredictable fluctuations; can be reduced by repeated measurements. Reduces precision.
Absolute error: |measured value - true value|
Relative (fractional) error: absolute error / true value
Percentage error: relative error × 100%

Error propagation:
Addition/Subtraction: absolute errors ADD.
Multiplication/Division: relative errors ADD.
Power n: relative error is multiplied by n.

Dimensions and Dimensional Analysis

Every physical quantity has dimensions — the powers of base quantities it involves. Notation: [M], [L], [T], [A], [K], etc.

Examples:
Velocity: [L T^-1]
Force: [M L T^-2]
Energy: [M L² T^-2]

1.Dimensional analysis is used to:
2.Check correctness of equations (dimensional homogeneity)
3.Derive relationships between quantities
4.Convert units between systems

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Example 1

Find the dimensions of pressure.
Pressure = Force / Area = [M L T^-2] / [L²] = [M L^-1 T^-2]

Example 2

Check if v = u + at is dimensionally correct.
[LT^-1] = [LT^-1] + [LT^-2][T] = [LT^-1] + [LT^-1]. All terms have same dimension — equation is dimensionally consistent.

Example 3

Express 1 joule in CGS units (ergs).
1 J = 1 kg m² s^-2; in CGS: 1 kg = 1000 g, 1 m = 100 cm.
1 J = 1000 g × (100 cm)² × s^-2 = 1000 × 10000 g cm² s^-2 = 10⁷ erg.

Example 4

A measurement is 5.32 ± 0.05 cm. Find the percentage error.
Percentage error = (0.05 / 5.32) × 100 = 0.94%

Example 5

If T = 2pi × √(l/g), find dimensions of g using dimensional analysis.
T has dimension [T], l has [L]. So [T] = [L/g]^1/2 → [T²] = [L]/[g] → [g] = [L T^-2]. This matches the known dimension of acceleration.

Example 6

Round 0.004563 to 3 significant figures.
Identify the first significant figure: 4 (the leading zeros are not significant). Three sig figs: 4, 5, 6 → 0.00456.

Example 7

A body travels 100.5 m in 10.2 s. Find speed to correct significant figures.
Speed = 100.5 / 10.2 = 9.852... m/s. Both values have 3 sig figs, so answer = 9.85 m/s.

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Common mistakes

Treating leading zeros as significant (0.0032 has only 2 sig figs, not 4).
Adding absolute errors when multiplying quantities — always add relative errors for products.
Forgetting that dimensional analysis cannot detect dimensionless constants (like pi or 2).

Summary

Measurement requires a standard unit. The SI system defines 7 base quantities. Significant figures express precision. Errors are classified as systematic or random. Dimensional analysis provides a powerful tool to verify equations and derive new relationships between physical quantities.