1.4 Dimensional analysis

The dimension    of any physical quantity expresses its dependence on the base quantities as a product of symbols (or powers of symbols) representing the base quantities. [link] lists the base quantities and the symbols used for their dimension. For example, a measurement of length is said to have dimension L or L ¹ , a measurement of mass has dimension M or M ¹ , and a measurement of time has dimension T or T ¹ . Like units, dimensions obey the rules of algebra. Thus, area is the product of two lengths and so has dimension L ² , or length squared. Similarly, volume is the product of three lengths and has dimension L ³ , or length cubed. Speed has dimension length over time, L/T or LT ^–1 . Volumetric mass density has dimension M/L ³ or ML ^–3 , or mass over length cubed. In general, the dimension of any physical quantity can be written as ${\text{L}}^a{\text{M}}^b{\text{T}}^c{\text{I}}^d{\text{Θ}}^e{\text{N}}^f{\text{J}}^g$ for some powers $a,b,c,d,e,f,$ and g . We can write the dimensions of a length in this form with $a=1$ and the remaining six powers all set equal to zero: ${\text{L}}^1={\text{L}}^1{\text{M}}^0{\text{T}}^0{\text{I}}^0{\text{Θ}}^0{\text{N}}^0{\text{J}}^0.$ Any quantity with a dimension that can be written so that all seven powers are zero (that is, its dimension is ${\text{L}}^0{\text{M}}^0{\text{T}}^0{\text{I}}^0{\text{Θ}}^0{\text{N}}^0{\text{J}}^0$ ) is called dimensionless    (or sometimes “of dimension 1,” because anything raised to the zero power is one). Physicists often call dimensionless quantities pure numbers .

Physicists often use square brackets around the symbol for a physical quantity to represent the dimensions of that quantity. For example, if $r$ is the radius of a cylinder and $h$ is its height, then we write $[r]=\text{L}$ and $[h]=\text{L}$ to indicate the dimensions of the radius and height are both those of length, or L. Similarly, if we use the symbol $A$ for the surface area of a cylinder and $V$ for its volume, then [ A ] = L ² and [ V ] = L ³ . If we use the symbol $m$ for the mass of the cylinder and $\rho$ for the density of the material from which the cylinder is made, then $[m]=\text{M}$ and $[\rho ]={\text{ML}}^{\mathrm{-3}}.$

The importance of the concept of dimension arises from the fact that any mathematical equation relating physical quantities must be dimensionally consistent    , which means the equation must obey the following rules:

• Every term in an expression must have the same dimensions; it does not make sense to add or subtract quantities of differing dimension (think of the old saying: “You can’t add apples and oranges”). In particular, the expressions on each side of the equality in an equation must have the same dimensions.
• The arguments of any of the standard mathematical functions such as trigonometric functions (such as sine and cosine), logarithms, or exponential functions that appear in the equation must be dimensionless. These functions require pure numbers as inputs and give pure numbers as outputs.