Dimensional Analysis deals with checking the consistency of the physical dimensions of quantities in expressions and equations. It can be used to find dimensions of quantities as in the following example. Consider Newton’s law of gravitation which quantifies the force F between two massive objects of mass M and m which are a distance r apart:
We can use dimensional analysis to find out the physical dimensions of the gravitational constant G. To do this, we first assign the proper dimensions to the other quantities in that equation and then simplify the resulting expression (Note: Length [ L ], Mass [ M ], Time [ T ] are the three basic dimensions in Newtonian mechanics
From the equation, we get
So in SI units, where length is measured in meters (m), mass in kilograms (kg), and time in seconds (s)
What this means is that when G has those dimensions, the right-hand side of the gravitational law equation will have dimensions of Force. This is consistent since we have a Force on the left-hand side of that equation. Dimensional analysis ensures that both sides of the equation are consistent in terms of its physical dimensions
Establishing relationships with dimensional analysis
Very often one can gain insight about a particular process just by playing around with the physical dimensions involved in that process. For example, consider the simple pendulum problem where a mass m in a gravitational field g is suspended by a cable of negligible mass of length l. We would like to find how the period of oscillation (the time it takes the pendulum to make one full swing) is related to the other variables in this problem. We can make use of dimensional analysis to find the relationship.
First, assign the corresponding dimensions to the main variables
We can try and combine the given variables l, g, and m in such a way that the combined form gives out dimensions of Time. Since the period t has dimensions of Time we know for sure that the correct form of the expression must also yield dimensions of time. Mathematically, combining means making products and divisions of the given variables (we cannot add different dimensions together).
The following equation must be satisfied:
We need to solve for the powers a, b, c such that the dimensions simplify to just time. We can see right away that since there is no dimension of mass on the righthand side, c = 0. Likewise, we can see that
The first equation ensures that here is no length, and the second equation ensures that there is exactly one time. Solving these equation gives
We now substitute these values to get
Indeed, you can check that the final relationship has dimensions of time, the period relates as follows:
It turns out that this is the correct functional form for the period of the simple pendulum. To get the actual form involves solving what is called a second order linear differential equation that arises from applying Newton’s Second Law to this problem, giving
As you can see applying dimensional analysis allows us to get the functional form and we do not need to know anything about the physics of the problem whatsoever. We could get the numerical value of the constant by doing experiments, measuring the period for different lengths and then plotting the data to find the value of the constant. This is why dimensional analysis can be very useful, it can allow us to get some insight into some process while being completely ignorant about the process itself.
Hint for AP Candidates
On the physics multiple choice section, very often some of the choices contain expressions that are dimensionally inconsistent. For example, a question may ask you to find an expression for a distance and there will be choices which will have expressions containing dimensions other than length; thus, you can immediately rule them out.