Dimensions, scaling, and similarity

Name the quantity and its unit

A unit is a reference used to express a quantity's value. The same supplied length can be written as 0.50 m = 50 cm, because 1 m = 100 cm. The numerical value changes with the unit; the length does not. Length and elapsed time are different quantity kinds, so a number without its quantity and unit is incomplete.

A dimension records a quantity's dependence on base quantities within a chosen system. It does not identify every quantity kind. In the SI, energy and torque have the same dimension but are different kinds; torque describes a force's turning effect. A dimension or a unit alone therefore does not supply a quantity's full meaning.

Build dimensions from simple operations

For the mechanical examples here, use capital L for the dimension of length, M for mass and T for time. These are dimension symbols, not the units metre, kilogram and second. In [q], the brackets mean the dimension of the quantity q. The SI uses other base dimensions too; this lesson needs only these three.

Multiplying two lengths gives an area dimension: L × L = L^2. Multiplying three gives L^3. A square means a quantity multiplied by itself; a cube means three factors. Dividing distance by elapsed time gives a speed dimension L/T. The table separates these dimensions from example units.

Check the dimensions of each term

In an equation between physical quantities, the dimensions on both sides must match. Terms being added or subtracted must also have matching dimensions. This is a necessary check; it does not replace checking the quantities' meanings.

For a rectangle with side lengths a and b, the expression ab has dimension L². The expression a + b has dimension L, so it cannot give that rectangle's area as written. Check any constants as well: a constant can carry units. A pure numerical factor such as 2 has dimension one and does not change the dimension of ab.

A consistent equation still needs justification

Passing a dimensional check does not determine a pure numerical coefficient or establish a physical law. Both A = ab and A = 2ab have matching area dimensions when 2 is a pure number. Dimensions alone cannot choose between them. The rectangle's geometry gives A = ab; a factor of 2 would count twice that area.

Keep the model and its conditions attached to a proposed relationship. The same dimensions can occur in different quantities, and an equation can pass the check while describing the wrong quantity or situation. Dimensional analysis can expose a mismatch; evidence and further reasoning are still needed for a physical prediction.

Scale the whole shape consistently

Geometric similarity means that corresponding lengths share one positive scale factor while the shape is retained. Define k = new length / original corresponding length, using the same unit. This length ratio is dimensionless: it has dimension one. Scaling only some lengths can change the shape. Include thicknesses and other relevant lengths when they belong to the shape being compared.

For corresponding cuboids, let the original lengths be a, b and c, and the new lengths be ka, kb and kc. A corresponding face changes from ab to (ka)(kb) = k^2 ab. Volume changes from abc to (ka)(kb)(kc) = k^3 abc. Thus k = 2 gives four times the face area and eight times the volume. These are geometry calculations under the stated similarity condition.

Add material assumptions before scaling mass

Volume scaling does not by itself give mass scaling. Mass density is mass per volume: rho = m/V. Here rho is a name for the density symbol, and m denotes mass rather than the metre unit. Rearranging gives m = rho V. For two filled blocks with the same stipulated uniform density, an eightfold volume gives an eightfold mass. If their densities differ, keep both density values in the comparison.

For an ideal cube of side a, each of its six faces has area a². Its total outer surface area is S = 6a^2, and its volume is V = a^3. Dividing gives S/V = 6/a. Doubling a therefore halves S/V, even though S itself increases fourfold. This ratio has dimension L^2/L^3 = 1/L; it is not dimensionless. A value stated per centimetre means divided by a centimetre.

The worked blocks below have supplied exact geometry and density in an invented teaching case. Their calculated digits do not establish the precision of real measurements. A hollow block, a different density or a thickness that stays fixed can change the comparison. State those choices before using a scaling rule.

Similar shapes do not establish similar behaviour

Geometric similarity does not by itself establish dynamic similarity. In mechanical model studies, dynamic similarity concerns corresponding force relationships. For fluid-flow comparisons, the relevant force ratios must match; matching lengths alone does not check them. This lesson introduces that distinction without deriving fluid-flow criteria.

Our cube calculations describe size, surface area, volume and mass under supplied assumptions. They supply no test of forces or motion. A prediction about behaviour needs an appropriate physical model, relevant properties and conditions. Identify what must correspond for that prediction before treating a scaled shape as a successful physical model.

Worked examples

Try it yourself

1. A supplied length is written as 0.50 m and as 50 cm. Has its dimension changed? Does either report describe an area?

   Show the answer and reasoning

   No. Both describe a length with dimension L, not an area.

   Changing the unit changes the numerical value while retaining the same length. An area would have dimension L² and an area unit, such as m².

2. For a positive distance travelled d and positive elapsed time t, compare d/t and d × t as candidate average-speed expressions. What are their dimensions?

   Show the answer and reasoning

   d/t has L/T, the speed dimension. d × t has LT and fails as speed.

   Divide a length dimension by a time dimension, or multiply them, as the expressions require. The dimensional test rules out the product. Using distance travelled divided by elapsed time also supplies the meaning of average speed.

3. For a rectangle with positive side lengths a and b, both A = ab and A = 5ab use a pure numerical coefficient. Can dimensions select the rectangle's area formula?

   Show the answer and reasoning

   No. Both have area dimension L², but the rectangle's geometry gives A = ab.

   A pure number has dimension one, so 5 does not change the dimensional product. Matching dimensions cannot determine that coefficient or establish the correct relationship.

4. Two ideal filled, uniform-density cubes have a new-to-original side ratio of 3. The second cube's stipulated density is half the first's. What are their volume and mass ratios?

   Show the answer and reasoning

   Volume ratio 27; mass ratio 13.5.

   Every corresponding cube length triples, so the volume ratio is 3 × 3 × 3 = 27. From m = rho V, the mass ratio is 0.5 × 27 = 13.5. Geometry alone does not supply the density ratio.

5. Compare a new-to-original side ratio with a cube's total-surface-area-to-volume ratio. Are both dimensionless?

   Show the answer and reasoning

   No. The side ratio has dimension one; surface area divided by volume has dimension 1/L.

   Matching length dimensions cancel in L/L = 1. For S/V, L^2/L^3 = 1/L. Keep the inverse-length unit and the specified surface and volume when comparing numerical values.

6. A model and another object have the same shape with every corresponding length in a fixed ratio. Has that alone established dynamic similarity?

   Show the answer and reasoning

   No. It establishes geometric similarity, while the relevant force relationships still need checking.

   A length comparison contains no force information. For the intended behaviour, specify the physical model, conditions and necessary correspondence; do not infer dynamic similarity from shape alone.

What to carry forward

• A unit gives a numerical reference; a dimension records dependence on base quantities.
• The same dimension does not necessarily identify the same quantity kind.
• A dimensional mismatch rules out an equation as written; a match does not fix coefficients or prove a law.
• A common length factor k gives k² face-area and k³ volume factors for corresponding cuboids.
• Mass comparisons also require density information, and surface area divided by volume is not dimensionless.
• Geometric similarity alone does not establish dynamic similarity or a prediction about behaviour.