Using Dimensional Analysis to derive an equation (Walter Lewin video)

[Chenkel]

TL;DR Summary

Mr Lewin's dimensional analysis

Hello everyone. I was watching the Walter Lewin lectures and I noticed in the talk he used something called dimension analysis to study the time it takes for an object to drop based on differing heights.



I'm 22:07 minutes into the video.

With some guess work on what's proportional to what, he wrote
$$t = k{h}^\alpha{m}^\beta{g}^\gamma$$Where k is a constant, h is the height, m is the mass, and g is the gravity.

He argued that the dimensional equivalents are $$[T]=[L]^\alpha[M]^\beta\frac {[L]^\gamma} {[T]^{2\gamma}}$$
He said that $\alpha + \gamma = 0$ and $\beta = 0$ and $1 = -2\gamma$ so $\gamma = \frac {-1} {2}$ and $\alpha = \frac {1} {2}$

Plugging this values into the first equation we have $$t=k\frac{\sqrt{h}}{\sqrt{g}}$$
I'm wondering about the theory of this approach, how do we know that the equation $t=k{h}^\alpha{m}^\beta{g}^\gamma$ is right? It seems the approach used by Mr Lewin requires a little bit of guess work, but I'm wondering what the logic behind the guess work is, how do we know for example that there isn't some addition or subtraction of certain multiplications happening on the right side, how can we assume this simple formula before solving for the exponents.

Any feed back on this would be helpful, thank you!

 

[.Scott]

I often use dimensional analysis to check my calculations.
What Mr Lewin is suggesting is a bit weak. How he came up with Mass, Gravity, and Height as candidate terms for time-to-fall is certainly very ill-defined.

 

[robphy]

Other possibly interesting sources:

• https://www.compadre.org/nexusph/course/Example_Creating_equations_using_dimensional_analysis
• Section 10 on Mechanical Similarity in Landau's Mechanics
• What decisions do experts make when doing back-of-the-envelope calculations?
  Gabriel Murillo-Gonzalez and Eric W. Burkholder
  Phys. Rev. Phys. Educ. Res. 18, 010125 – Published 5 April 2022
  https://journals.aps.org/prper/abstract/10.1103/PhysRevPhysEducRes.18.010125

 

[Chenkel]

Other possibly interesting sources:

• https://www.compadre.org/nexusph/course/Example_Creating_equations_using_dimensional_analysis
• Section 10 on Mechanical Similarity in Landau's Mechanics
• What decisions do experts make when doing back-of-the-envelope calculations?
  Gabriel Murillo-Gonzalez and Eric W. Burkholder
  Phys. Rev. Phys. Educ. Res. 18, 010125 – Published 5 April 2022
  https://journals.aps.org/prper/abstract/10.1103/PhysRevPhysEducRes.18.010125

I followed the compadre example of a ball being throw upward, I think there was a mistake in the example they gave.

They wrote $$[m^a(v^y)g^c] = (M)^a(\frac L T)^b(\frac L {T^2})^c = M^aL^{b-c}T^{-b-2c}$$

I believe they meant $$[m^a(v^y)g^c] = (M)^a(\frac L T)^b(\frac L {T^2})^c = M^aL^{b+c}T^{-b-2c}$$

I suppose they are trying to say $h = km^a(v^y)g^c$ and solve for the exponents that will create a dimensionality of L

So are they guessing that the following equation is true? $h = km^a(v^y)g^c$ it seems possible but I don't know how reasonable this assumption is; it feels easy for me to imagine getting wrong answers with dimensional analysis because something could have correct dimensions but incorrect formula, is this a reasonable concern?

 

[f95toli]

Summary: Mr Lewin's dimensional analysis
I'm wondering about the theory of this approach, how do we know that the equation $t=k{h}^\alpha{m}^\beta{g}^\gamma$ is right? It seems the approach used by Mr Lewin requires a little bit of guess work, but I'm wondering what the logic behind the guess work is, how do we know for example that there isn't some addition or subtraction of certain multiplications happening on the right side, how can we assume this simple formula before solving for the exponents.

You can't know that for sure. Dimensional analysis is a very powerful technique; but it is primarily used when designing experiments (and checking the final answer).
That is, using dimensional analysis you can write down a relationship between different variables; based on this you can then figure out which measurements you need to do in order to determine the unknowns.
It could very well be that you need to subtract/add terms but if that is the case it should become obvious as you do the measurements.
Dimensional analysis can quite dramatically reduce the the number of measurements you need to do; but it can't replace experiments.

 

[hutchphd]

So are they guessing that the following equation is true? ... it seems possible but I don't know how reasonable this assumption is

Two points:
-It is the "simplest", and hence by Occam's Razor the most compelling, candidate.
-No theory is ever proven "true". They are all guesses, although "seeing " some overarching structure in a theory seems to bolster Occam, they are all subject to scrutiny via experiment. We call it science.