Welcome to our Physics lesson on Dimensional Analysis, this is the fourth lesson of our suite of physics lessons covering the topic of Length, Mass and Time, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Dimensional analysis is a method of analysis in which physical quantities are expressed in terms of their fundamental dimensions that is often used when there is not enough information to set up precise equations.
For example, let's suppose you don't remember the equation of velocity v in a uniform motion (the correct equation is v = Δx / t where Δx is the displacement and t is the moving time). You are somehow puzzled whether the correct formula is the abovementioned one, or it is v = Δx × t or v = t /Δx. In this case, you can use the dimensional analysis to remove the doubts if you know the unit of velocity. Thus, giving that the unit of velocity is metre per second [m / s] where metre is the unit of displacement Δx and second is the unit of time t, it is obvious the equation of velocity is v = Δx / t.
However, Dimensional Analysis has an important drawback. Through this method, we are not able to identify dimensionless constants in a formula. For example, if we know the unit of energy is joule [J], we are not able to find the correct formula of the kinetic energy (KE) as
(Remember that 1 N = 1 kg × m/s2). Giving that [kg] is the unit of mass m and [m/s] is the unit of velocity v, we may (wrongly) conclude that
However, the correct formula of kinetic energy (see the article: Kinetic Energy), is
Hence, we missed the constant dimensionless constant 1/2 when tried to find the equation based on the dimensional analysis. Therefore, to be correct in our response, we must write
(the symbol ̴ means "is proportional to"). Thus, the correct way to write the equation of KE found by using the dimensional analysis method is
where C is a dimensionless constant. In the specific case, C = 1/2.
A student believes the speed of sound v in a gas depends on three factors: pressure p density of the medium ρ and the volume V of the medium. He also has determined the approximate equation
where C is a dimensionless constant.
First, let's write all units in terms of metre, kilogram and second. We have
Giving that speed and velocity have the same unit [m/s], we obtain after substituting the symbols of the respective units
The same method is used for the other quantities as well. Thus,
In symbols, we have
where [m3] is obviously the unit of volume. Hence, combining all these units, we obtain
Hence, we obtain three different equations:
From the third equation, we can find the value of x (x = 1/2). Also, since in the first equation we have x + y = 0=>y = - x = - 1/2.
Therefore, substituting the above two values in the second equation, we obtain
Since the volume's exponent is zero (z = 0), the speed of sound does not depend on the volume. This means the volume does not appear in the formula of the sound speed in a gas.
The correct formula is
Giving that m1/2 = √x, we obtain
where C is a dimensionless constant.
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|1.2||Length, Mass and Time|
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