Units and Measurement

Definition of Units

A unit is the chosen standard of measurement of a quantity which has essentially the same nature as that of the quantity. It serves as a reference to express the magnitude of physical quantities.

Fundamental Units

Fundamental physical quantities are independent of each other and can represent other physical quantities. Their units are called fundamental units. The seven fundamental physical quantities in the SI system are:

• Mass - kilogram (kg)
• Length - meter (m)
• Time - second (s)
• Temperature - kelvin (K)
• Electric current - ampere (A)
• Luminous intensity - candela (cd)
• Amount of substance - mole (mol)

Derived Units

Derived units are units of physical quantities obtained from fundamental units through mathematical relationships. Examples include velocity (m/s), acceleration (m/s²), pressure (pascal, Pa), and force (newton, N).

Systems of Units

Different systems of units group fundamental and derived units:

• FPS system: Foot, Pound, Second
• CGS system: Centimeter, Gram, Second
• MKS system: Meter, Kilogram, Second
• SI system: Internationally accepted system based on MKS units

Significant Figures

Significant figures are the digits in a number that convey its accuracy. They include all non-zero digits, zeros between non-zero digits, and certain zeros depending on their position relative to the decimal point.

Rules for Significant Figures

• All non-zero digits are significant. Example: 198745 has six significant digits.
• Zeros between non-zero digits are significant. Example: 108.0097 has seven significant digits.
• Zeros to the left of the first non-zero digit are not significant. Example: 0.00798 has three significant digits.
• Zeros to the right of a decimal point and after a non-zero digit are significant. Example: 20.00 has four significant digits.
• Zeros to the right of the last non-zero digit after the decimal point are significant. Example: 0.0079800 has five significant digits.
• Zeros to the right of the last non-zero digit in a measurement are significant. Example: 1090 m has four significant digits.

Additional Concepts

Units should be universally accepted, non-perishable, well-defined, and consistent over time. Dimension formulae express how base quantities represent the dimensions of a physical quantity, for example, force has the dimension [MLT⁻²].

Key Formulas

• 1 AU = 1.496 × 10¹¹ m
• 1 ly = 9.46 × 10¹⁵ m
• 1 parsec = 3.1 × 10¹⁶ m
• 1 Å = 10⁻¹⁰ m; 1 nm = 10⁻⁹ m
• 1 μm = 10⁻⁶ m; 1 mm = 10⁻³ m
• 60 seconds (of arc) = 1 minute (of arc)
• 60 minutes (of arc) = 1 degree (of arc)
• 180 degrees (of arc) = π radians

Magnification

• Linear magnification = Size of image / Size of object
• Linear magnification = √(Areal magnification)

Mnemonic for Fundamental Quantities

At the last moment she luckily caught the train.

• A – Amount of substance
• t – Temperature (thermodynamic)
• l – Length
• m – Mass
• lu – Luminous intensity
• c – Current
• t – Time

Solved Examples

Example 1: Express the velocity of a car moving at 72 km/h in m/s.

Solution:

Given velocity = 72 km/h

1 km = 1000 m, 1 hour = 3600 s

Velocity in m/s = 72 × (1000 m / 1 km) × (1 hour / 3600 s) = 72 × (1000 / 3600) m/s = 20 m/s

Example 2: How many significant figures are there in the number 0.004560?

Solution:

Zeros to the left of the first non-zero digit are not significant.

Digits: 4, 5, 6 are significant.

Trailing zero after decimal point and after non-zero digit is significant.

Total significant figures = 4

Practice Set

• Level 1: Define a fundamental unit and give two examples.
• Level 1: State the rule for significant figures regarding zeros between non-zero digits.
• Level 2: Convert 90 km/h into m/s.

Answer Key

• Level 1: Fundamental units are units of physical quantities that are independent and cannot be derived from other units. Examples: meter (length), kilogram (mass).
• Level 1: All zeros occurring between two non-zero digits are significant. For example, in 108.0097, all digits including zeros are significant.
• Level 2: 90 km/h = 90 × (1000 m / 3600 s) = 25 m/s.

Dimensional Analysis

Definition and Principle

Dimensional analysis studies the relationship between physical quantities and fundamental quantities by expressing physical quantities in terms of their dimensions.

The principle of homogeneity states that the dimensions of each term in a physical equation must be the same on both sides.

Dimensional Equation

A dimensional equation expresses a physical quantity in terms of the fundamental quantities using powers of mass (M), length (L), and time (T).

Conversion of Units

If a physical quantity has magnitude n₁ in one system with dimensions [M₁ᵃ L₁ᵇ T₁ᶜ] and magnitude n₂ in another system with dimensions [M₂ᵃ L₂ᵇ T₂ᶜ], then:

n₁ [M₁ᵃ L₁ᵇ T₁ᶜ] = n₂ [M₂ᵃ L₂ᵇ T₂ᶜ]

Uses of Dimensional Analysis

• Deriving formulas for physical quantities.
• Checking the correctness of physical equations using the principle of homogeneity.

Limitations

• Cannot provide information about dimensional constants.
• Cannot derive formulas involving trigonometric, logarithmic, or exponential functions.
• Cannot determine whether a quantity is scalar or vector.
• Difficult to apply when a quantity depends on more than three variables.
• Cannot find formulas correctly if quantities are not related by multiplication.

Key Terms

• Dimensions of physical quantity: Powers to which fundamental quantities are raised to represent a derived unit.
• Dimensional formula: Expression showing how fundamental quantities represent the dimensions of a physical quantity.
• Dimensional constants: Quantities with constant values and dimensions (e.g., gravitational constant G).
• Dimensional variables: Quantities with variable values and dimensions (e.g., area, volume).
• Dimensionless constants: Constants without dimensions (e.g., π, e).
• Dimensionless variables: Variables without dimensions (e.g., angle, strain).

Key Formula

Conversion of units between two systems:

n₂ = n₁ \times (M₁ / M₂)^a \times (L₁ / L₂)^b \times (T₁ / T₂)^c

Solved Examples

Example 1: Find the dimensional formula of force.

Solution:

Force = mass × acceleration

Mass dimension = [M]

Acceleration = change in velocity / time = [LT^{-1}] / [T] = [LT^{-2}]

Therefore, force dimension = [M] × [LT^{-2}] = [MLT^{-2}]

Practice Set

• Level 1: State the principle of homogeneity.
• Level 1: Define dimensional formula.
• Level 2: Using dimensional analysis, find the dimensional formula of pressure.

Answer Key

• Level 1: The principle of homogeneity states that the dimensions of each term in a physical equation must be the same on both sides.
• Level 1: Dimensional formula is the expression that shows how fundamental quantities represent the dimensions of a physical quantity.
• Level 2: Pressure = Force / Area

Force dimension = [MLT^{-2}]

Area dimension = [L^2]

Pressure dimension = [MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]

Quick Reference Table

• Fundamental Units: Mass (kg), Length (m), Time (s), Temperature (K), Electric current (A), Luminous intensity (cd), Amount of substance (mol).
• Derived Units: Velocity (m/s), Acceleration (m/s8), Force (N), Pressure (Pa), Energy (J).
• Systems of Units: FPS (Foot, Pound, Second), CGS (Centimeter, Gram, Second), MKS (Meter, Kilogram, Second), SI (International System of Units).
• Significant Figures: Rules for determining significant digits in measurements to express precision.
• Dimensional Analysis: Principle of homogeneity, dimensional formulae, and unit conversions.
• Key Formulas: 1 AU = 1.496 d7 10b111 m, 1 ly = 9.46 d7 10b115 m, 1 parsec = 3.1 d7 10b116 m, 1 c5 = 10b1b110 m, 1 nm = 10b1b19 m.
• Conversion Formula: n = n
  (M
  / M) (L
  / L) (T
  / T) for unit conversions between systems.

Common Mistakes and Misconceptions

• Confusing significant zeros: Zeros to the left of the first non-zero digit are not significant, but zeros between non-zero digits are significant.
• Ignoring the principle of homogeneity: Equations must have the same dimensions on both sides to be valid.
• Misapplying dimensional analysis to formulas involving trigonometric or exponential functions, which dimensional analysis cannot handle.
• Assuming dimensional analysis can determine if a quantity is scalar or vector, which it cannot.
• Forgetting to convert units properly when switching between systems, leading to incorrect calculations.

Glossary

• Unit: A standard quantity used to specify measurements.
• Fundamental Quantity: Physical quantities independent of each other, e.g., mass, length, time.
• Derived Quantity: Quantities derived from fundamental quantities, e.g., velocity, force.
• Dimension: The nature of a physical quantity expressed in terms of fundamental quantities.
• Dimensional Formula: Expression showing the powers of fundamental quantities representing a physical quantity.
• Significant Figures: Digits in a number that contribute to its precision.
• Principle of Homogeneity: All terms in a physical equation must have the same dimensions.
• Dimensional Constant: A constant with dimensions, e.g., gravitational constant G.
• Dimensionless Quantity: A quantity without dimensions, e.g., angle, strain.
• System of Units: A set of units used to measure physical quantities consistently.