Units, Measurement & Physical Quantities

Measurement and Units: The Basics

• Measurement compares an unknown amount with a fixed, agreed-upon standard.
• A physical quantity is anything that can be measured: length, time, mass.
• Every measurement has two parts: a number (how many) and a unit (the standard).
• A unit is a fixed reference amount of a quantity that everyone accepts.
• A number without a unit is meaningless (“the rope is 7”), so the world needs one common system: a measurement made in India must mean exactly the same in Japan or Brazil.

The SI System and 7 Base Units

• SI = Système International d'Unités, adopted in 1960 as the world standard.
• Base (fundamental) quantities are independent building blocks, not defined by any other quantity. SI has exactly seven base quantities and seven base units.
• Derived quantities combine base quantities, e.g. speed = length ÷ time.
• Older systems: CGS (cm-g-s), FPS (foot-pound-second) and MKS (m-kg-s). SI is essentially the upgraded, expanded version of MKS.
• The 7 SI Base Units

Derived Units and Special Names

• Every unit with a special name can always be rewritten purely in base units.
• Units behave like algebra in multiplication and division; identical units cancel.
• Force = mass × acceleration; unit newton: 1 N = 1 kg·m/s².
• Pressure = force ÷ area; unit pascal: 1 Pa = 1 N/m².
• Power = energy ÷ time (the rate of doing work); unit watt: 1 W = 1 J/s.
• Magnetic flux density: 1 tesla = 1 N/(A·m).
• Derived Units in Base Units

Units of Length and Special Big/Small Units

• The SI base unit of length is the metre (m); every true length reduces to metres.
• A light-year is a distance, not a time: how far light travels in one year, about 9.46 × 10¹⁵ m.
• An astronomical unit (AU) is the average Earth–Sun distance; 1 parsec is about 3.26 light-years.
• The angstrom, Å = 10⁻¹⁰ m, is used for atomic sizes.

• Length Conversions

The Electricity 'Unit' and Energy in Joules

• One “unit” on a home electricity bill is one kilowatt-hour (kWh), which is not an SI unit.
• It is the energy used by a 1000 W appliance running for 1 hour.
• 1 unit = 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J; the 3600 is 60 minutes × 60 seconds.
• Example: 250 units = 250 × 3.6 × 10⁶ = 9 × 10⁸ J.

Scalar and Vector Quantities

• A scalar has only magnitude: mass, time, speed, distance, temperature, energy.
• A vector has magnitude and direction: displacement, velocity, acceleration, force, momentum.
• Pairs to remember: distance (scalar) and displacement (vector); speed and velocity.
• A unit vector (î, ĵ, k̂) has magnitude 1 and no dimension and no unit; it shows direction only.

• Scalars vs Vectors

Dimensions and Dimensional Formula

• A dimension shows which base ingredients (M, L, T) make up a quantity, ignoring numbers.
• Base dimension symbols: [M] mass, [L] length, [T] time, [A] current, [K], [mol], [cd].
• A dimensional formula shows the powers of the base dimensions combined: speed = length ÷ time → [M⁰L¹T⁻¹]; force = mass × acceleration → [M¹L¹T⁻²].
• Dimensionless quantities (no dimensions or units): angle, strain, refractive index.
• Dimensional Formulae

Uses of Dimensional Analysis

• Principle of homogeneity: every term in a valid equation must have the same dimensions.
• Three uses: checking whether an equation is correct, converting units between systems, and deriving relations between quantities.

Measuring Instruments You Must Know

• Odometer: the distance travelled by a vehicle. Speedometer: its instantaneous speed.
• Pyrheliometer: the intensity of solar radiation (direct sunlight).
• Barometer: atmospheric pressure. Manometer: gas pressure.
• Vernier caliper and screw gauge: small lengths and thicknesses, accurately.
• Ammeter: current. Voltmeter: potential difference.
• Instruments and What They Measure

Other Common Units and Pairs

• Power of a lens: the dioptre (D). Heat and energy: the joule (J).
• Momentum ÷ velocity gives mass (momentum = mass × velocity).
• Temperature scales: kelvin (SI), Celsius and Fahrenheit.
• Non-SI units of pressure: atmosphere, bar and torr.
• Radioactivity (activity of a radioactive substance): curie and becquerel.
• Other Common Units

Significant Figures, Accuracy and Errors

• Significant figures: the reliable digits in a measurement plus one uncertain digit.
• Accuracy is how close a reading is to the true value; precision is how finely it is measured.
• Systematic errors have a fixed cause (such as a faulty instrument) and a consistent direction; random errors vary unpredictably and are reduced by taking many readings.
• Absolute error: the difference between a reading and the true value. Least count: the smallest value an instrument can measure.