Mechanics Made Easy: Dimensional Analysis

Physics Tutorial

Mechanics Made Easy: Dimensional Analysis

Intro

In physics, a situation or formula can be evaluated by its dimensions (i.e. Length[L], Time[T], Mass[M], etc). Dimensional analysis, as it is called, checks to see if the dimensions of a formula are balanced on each side. This method of evaluation can be useful because it ensures both sides of an equation have matching units. This is done by looking at the units and determining which dimensions they represent. For example, an immobile paperclip that weighs 1 gram on Earth has a dimension of mass(M) because the gram is a unit of mass.

More complexly, The force of gravity on earth on that paperclip would be roughly .00981 Neutons. Well what dimensions go into that? As a reminder, force equals acceleration times mass(M), and acceleration equals change in position(Length[L]) divided by time(T) travelling squared(^2). So from this, we know the dimensions of that free-falling paperclip are (Mass*Length/Time^2).

Sample Problem

Michelle knows that velocity is proportional to an object’s distance traveled, and inversely proportional to the time it spent travelling. She figures from the following formula that velocity must operate proportionally with the mass dimension and inversely with the time dimention:

v = Δx/t

Check her formula to see if dimensional analysis confirms her theory.

Solution

We know Δx represents change in position, which is a dimension of length(L) and t represents time. From the formula we see that velocity is proportionally related to change in position, and inversely related to time:

v = Δx/t = L/T

Therefor, her theory is supported by dimensional analysis.



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