Units and Measurements

The One Thing

Physical measurements are meaningful only when numerical values are stated with units and an appropriate indication of precision. The SI system standardises units, while significant figures, uncertainty analysis, and dimensional analysis establish how reliably measurements and equations can be interpreted.

Reactions, Processes and Experiments

Key Terms

• Physical quantity: A measurable property of a body or phenomenon, such as length, mass, time, or force.
• Unit: A fixed, agreed standard used to measure a physical quantity.
• SI system: The International System of Units, based on seven fundamental units and used as the standard system in physics.
• Base quantity: A fundamental physical quantity defined independently of other quantities, such as length, mass, and time.
• Derived quantity: A quantity expressed in terms of base quantities, such as velocity, force, and density.
• Accuracy: How close a measured value is to the true or accepted value.
• Precision: How closely repeated measurements agree with one another.
• Least count: The smallest measurement that an instrument can reliably read.
• Significant figures: The meaningful digits in a measured value, including all certain digits and the first uncertain digit.
• Rounding off: The process of reducing digits while retaining the required precision according to specified rules.
• Absolute error: The magnitude of the difference between a measured value and the accepted or mean value.
• Relative error: The ratio of the mean absolute error to the mean measured value.
• Percentage error: Relative error expressed as a percentage.
• Dimensions: The powers of base quantities used to represent a physical quantity.
• Dimensional formula: An expression showing how a physical quantity depends on base quantities, such as [M L T^-2] for force.
• Principle of homogeneity: Every term added or equated in a physical equation must have the same dimensions.
• Dimensional constant: A physical constant that has dimensions, such as the gravitational constant G.
• Dimensionless quantity: A quantity with no dimensions, such as strain, refractive index, and the angle measured in radians.

The seven SI base quantities and units are:

• Length—metre (m)
• Mass—kilogram (kg)
• Time—second (s)
• Electric current—ampere (A)
• Thermodynamic temperature—kelvin (K)
• Amount of substance—mole (mol)
• Luminous intensity—candela (cd)

Easily Confused

• Accuracy and precision: Accuracy concerns closeness to the true or accepted value; precision concerns agreement between repeated measurements.
• Base and derived quantities: Base quantities are defined independently; derived quantities are expressed in terms of base quantities.
• Absolute, relative, and percentage error: Absolute error is an error magnitude, relative error is the ratio of mean absolute error to mean measured value, and percentage error is relative error multiplied by 100.
• Significant figures and decimal places: Significant figures count meaningful digits, whereas decimal places count digits after the decimal point; addition and subtraction use decimal places, while multiplication and division use significant figures.
• Dimensional constant and dimensionless quantity: A dimensional constant, such as the gravitational constant G, has dimensions; a dimensionless quantity, such as strain, refractive index, or the angle measured in radians, has none.
• Dimensional consistency and physical validity: A dimensionally correct equation may still be physically incorrect because dimensional consistency is necessary but not sufficient for validity.
• Dimensional analysis and complete derivation: Dimensional analysis can derive relationships only up to a dimensionless constant and cannot determine numerical constants such as 2 or pi.

What Gets Asked

• Identify the seven SI base quantities and their units. Marks are lost by confusing the base units with derived units such as the newton, joule, watt, and pascal.
• Express derived quantities in SI units or give their dimensional formulae. The required forms include force: newton (N) = kg m s^-2 and force: [M L T^-2], as well as the corresponding expressions for velocity, acceleration, momentum, work, power, pressure, and density.
• Calculate mean value, mean absolute error, relative error, or percentage error from repeated readings. The relevant distinction is between Δx_mean, Δx_mean / x_mean, and (Δx_mean / x_mean) × 100.
• Apply significant-figure and rounding rules. Addition and subtraction are controlled by the fewest decimal places, whereas multiplication and division are controlled by the fewest significant figures.
• Propagate errors through sums, differences, products, quotients, and powers. Absolute errors are added for sums or differences, fractional errors for products or quotients, and Δz/z = |n| Δx/x for z = x^n.
• Use dimensional analysis to test or derive an equation and identify its limitations. It cannot distinguish between quantities with the same dimensions, determine numerical constants such as 2 or pi, or verify equations involving trigonometric, exponential, or logarithmic functions by dimensions alone.