Physical quantities, SI units and dimensional consistency

Say what the number belongs to

A physical quantity is a property that can be expressed by a number and a reference such as a unit. Length, mass and time are different kinds of quantity. “The shelf length is 1.2 m” identifies the particular property, gives its numerical value 1.2, and states the unit metre, symbol m. The shelf does not change when its length is written as 1200 mm; only the numerical value and unit change together.

Symbols save space but need definitions. We might write L = 1.2 m, where L means this shelf’s length. A unit symbol has a fixed meaning: m is metre, s is second and kg is kilogram. Leave a space between number and unit. Capital letters matter: m and M are not interchangeable prefix symbols.

Build useful units from simpler ones

The International System of Units, abbreviated SI, uses seven base units: second (s), metre (m), kilogram (kg), ampere (A), kelvin (K), mole (mol) and candela (cd). You do not need to master every quantity now. Length, mass and time are enough for this lesson’s calculations; later chapters introduce current, temperature, amount of substance and luminous intensity when needed.

Area combines two lengths, so its SI unit is square metre, m². Volume combines three lengths, giving m³. Speed compares distance with elapsed time, giving m/s. A mass expressed in kilograms is distinct from weight, which is a force expressed in newtons. A newton can also be written kg·m/s²; the forces chapter explains the relationship.

Use dimensions as a first check

A dimension describes the kind of quantity, independently of the unit chosen. Metres and millimetres both describe length. In dimensional notation length is L and time is T, so speed has dimension L/T. Here the upright dimensional symbol L is a category, while the earlier shelf symbol L names one quantity; words around the symbols prevent confusion.

You may add lengths after expressing them in compatible units. You cannot add 2 m to 3 s and call the result a length. Multiplication and division can create a new kind of quantity: length multiplied by length gives area. In the diagram, six one-square-metre cells fill the rectangle. Counting only its edge lengths would answer a different question.

A consistent unit does not prove a correct model

If distance d is calculated as speed v multiplied by time t, its unit is (m/s) × s = m. Writing d = v/t would instead produce m/s² and fail a length check. But d = 2vt also has units of length. Units alone cannot tell whether the extra factor 2 belongs in a particular situation. Check the physical assumptions, numerical factors and direction of the relationship too.

Find a rectangular panel’s area

An invented panel is 3 m long and 2 m wide. For a rectangle, area equals length × width: A = (3 m) × (2 m) = 6 m². The unit is squared because both measured factors carry a length unit.

Adding the two lengths gives 5 m, which is a length, not an area. If we need the distance around the outside, the perimeter is 2 × (3 m + 2 m) = 10 m. The question determines which calculation is meaningful.

Try the reasoning

1. Does 1.2 m describe a shorter shelf than 1200 mm?

   Show the answer and reasoning

   No. They express the same length using different units. Comparing numerical values alone would be misleading.

2. Someone calculates area as 3 m + 2 m. How can units expose the mistake?

   Show the answer and reasoning

   The sum has unit m, whereas area needs a length squared, such as m². Adding the sides does not count the surface.

3. Does a formula with the correct units have to be physically right?

   Show the answer and reasoning

   No. Dimensional consistency is necessary, but a wrong numerical factor or an unsuitable physical model can still have correct units.

What to carry forward

Carry the quantity name and unit through the reasoning. Dimensional consistency can reject some mistakes, but cannot validate a model by itself.