1.12 : Dimensional Analysis

All physical quantities can be expressed using either base quantities or derived quantities and each quantity is represented by a symbol, which defines its dimensions.

For instance, the speed of a car is defined as the distance divided by time. The term distance corresponds to the quantity length, denoted with L and time with T.

Hence, we can write the dimension of the quantity speed, as L divided by T or LT to the power of minus one.

For an equation to be dimensionally correct, it should obey two rules. Number one, the expressions on each side of the equality in an equation must have the same dimensions.

Number two, the standard mathematical functions in equations must be dimensionless

For example, we know the dimension of volume is L cubed. Now, consider a cylinder with radius r and height h.

We know that the volume of a cylinder is π r squared h. The term π is a constant, and it's a dimensionless quantity. The term r corresponds to the quantity length, and we can write its dimension as L squared, and the term h also corresponds to the quantity length, which gives the dimension of the volume of the cylinder as L cubed. Hence, the equation is dimensionally correct.

As long as we know the dimensions of the individual physical quantities that appear in an equation, we can check to see whether the equation is dimensionally consistent.

Another application of dimensional analysis is to remember an equation. For example, let's say that you don't remember whether speed equals time divided by distance or distance divided by time.

The dimensions of time, distance, and speed are T, L, and LT to the power of minus one respectively. Reducing both the equations to their fundamental units on each side of the equation, we get speed equals distance divided by time.