Dimensional Analysis Digital Help - Scott A. Strong

### Dimensional Analysis (DA) — Summary  - DA is a powerful method used in science and engineering to simplify physical problems by analyzing the dimensions (units) of the variables involved.  - Key theorems such as Bridgman's theorem ensure dimensional homogeneity in equations, while the Buckingham $\pi$ theorem allows the systematic construction of independent dimensionless parameters that capture the essence of the physical system. --- ### DA - Uses  It helps to: - Check the consistency and plausibility of equations. - Reduce the number of variables in a problem by forming dimensionless groups.  - Reveal fundamental relationships between physical quantities. - Guide experimental design and data analysis by identifying key dimensionless parameters. --- #### DA - Bridgman Theorem  - States that any physically meaningful equation involving physical quantities must be dimensionally homogeneous; that is, all additive terms in the equation must have the same dimensions.  - This principle allows us to check the plausibility of equations and to derive relationships between variables by ensuring dimensional consistency.  --- #### DA - Buckingham theorem - States that if a physical problem involves $n$ variables and $k$ fundamental dimensions (such as mass, length, time), the variables can be grouped into $n - k$ independent dimensionless parameters, called $\pi$ terms. These $\pi$ terms capture the essential relationships among the variables and simplify the analysis of physical systems by reducing the number of variables. --- #### The Mystery Data Set - Set Up - The mystery data found on canvas, [shortLeaf.txt](https://www.dropbox.com/scl/fi/56zl5llye1otmeoba2wy8/shortLeaf.txt?rlkey=g472utsbrn4hdg4lzz1lgt684&dl=0), is the volume and diameter data associated with 70 [shortleaf pines](https://en.wikipedia.org/wiki/Pinus_echinata), reported on by C. Bruce and F. X. Schumacher in a 1935 book titled, "[Forest Mensuration](https://books.google.com/books/about/Forest_Mensuration.html?id=ISTxAAAAMAAJ)".  - With these data, you are asked to define a linear model relating the two variables.  --- #### The Mystery Data Set - Ideas and Results - You should see that while the linear model has a high coefficient of determination, i.e., $R^2=0.8926$, both the scatter plot and residuals suggest that a nonlinear fit would be more appropriate. At this point, we could transform, ([Box-Cox](https://en.wikipedia.org/wiki/Power_transform#Box%E2%80%93Cox_transformation), [log transform](https://medium.com/@kyawsawhtoon/log-transformation-purpose-and-interpretation-9444b4b049c9)), the data to understand an underlying nonlinear relationship, but we could also inform this process through the use of dimensional analysis.  --- - Plot of linear fit of volume-diameter data ![[VD_LM.png]] --- - Plot of its residuals ![[VD_LM_Res.png]] --- #### Continued Work This data is further analyzed in the  - [Continued work from Penn Stat course on applied regression analysis](https://online.stat.psu.edu/stat462/node/154/) --- #### The Complete Data Set - Having also found the complete data set, [shortleaf_pine_data.csv](https://www.dropbox.com/scl/fi/4wh2k8m3bz3ji3fd77h9j/shortleaf_pine_data.csv?rlkey=mz43hmkuunq3il9u0r76qcdlu&dl=0) , we have the linear model with standard errors of estimates in subscripts, which indicates that $h$ is not a significant predictor, and a plot of its residuals further suggests that a nonlinear fit would be reasonable to explore in this more general case. $\hat{v} = -43.0513_{(5.1535)} + 6.6625_{(0.5087)} × d + 0.0443_{(0.1166)} h$ --- #### Residuals of the linear model ![[LMResiduals.png]] --- #### Results of dimensional analysis  ![[DimAnalysisPlot.png]] --- #### Results of dimensional analysis  ![[DimAnalysisPlot_Residuals.png]]