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.
1.12:
Dimensional Analysis
The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that mathematical equations must meet the following two rules. The first rule is that, in an equation, the expressions on each side of the equal sign must have the same dimensions. This is fairly intuitive since we can only add or subtract quantities of the same type (dimension). The second rule states that, in an equation, the arguments of any of the standard mathematical functions like trigonometric functions, logarithms, or exponential functions must be dimensionless.
If either of these two rules is violated, the equation is dimensionally inconsistent, hence it cannot be a representation of the correct statement of any physical law. Dimensional analysis can check for mistakes or typos in algebra, help remember the various laws of physics, and even suggest the form that new laws of physics might take.
Let us understand the effect of the operations of calculus on dimensions. The derivative of a function is the slope of the line tangent to its graph, and slopes are ratios. Thus, for physical quantities, say v and t, the dimension of the derivative of v with respect to t is the ratio of the dimension of v over that of t. Similarly, since integrals are just sums of products, the dimension of the integral of v with respect to t is simply the dimension of v times the dimension of t.
This text is adapted from Openstax, University Physics Volume 1, Section 1.4: Dimensional Analysis.