Limitations of Dimensional Analysis

Understanding dimensions is critical because it allows us to mathematically examine the basis of physical elements. Dimensions are fundamental in the sense that one can either subtract or add numbers having the exact same dimensions. Furthermore, if any two physical quantities have almost identical dimensions, they are considered to be equal. These fundamental concepts aid us in drawing a new relationship among physical quantities, which is analogous to units.

Dimensional analysis refers to the investigation of correlations between physical variables using their units and dimensions of measurement. To transform a unit from one type to another, dimensional analysis is employed. We must maintain the unit constant in science as well as mathematics applications in order to execute mathematical operations simply. There exist numerous limitations and applications of dimensional analysis.

Dimensional analysis is conducted to identify the dimensions of a physical parameter before writing its equation. The unit of the parameter is then obtained by substituting L, T, and M in the dimensional expression with the basic units of the necessary system. Nevertheless, we may attach a special label to the unit as well, such as Work = Displacement × Force.

So, [W] = [MLT-2] × [L] = [ML2T-2]

Dimensional Analysis

• Dimensional analysis refers to the activity of finding the measurement units as well as the dimensions of physical quantities in order to examine their relationships.

• Nevertheless, dimensional analysis is only feasible if all of the components along both sides of the problem have identical dimensions. The principle of homogeneity of dimensions is the name given to this condition. The concept is built on the idea that only two elements having the same dimension may be subtracted, added and compared.

• The uses of dimensional analysis are numerous. It is considered to be an excellent method for determining if computations are dimensionally accurate. 

• When we determine the quantities present, we may also utilise dimensional analysis to build credible equations. 

• Dimensional analysis is a technique for quantifying the size as well as the form of objects. 

• The dimensional analysis makes it feasible to conduct mathematical studies on the composition of substances today.

• Example – 

F = mv2/r

Upon using dimension analysis of these physical quantities, it can be written as:

[MLT-2] = [M] [LT-1]2 / [L]

i.e., [MLT-2] = [MLT-2]

Dimensional Analysis Applications

Real-world physics problems have been solved using dimensional analysis. We can state three uses of dimensional analysis as follows:

• Identifying the unit of a certain physical object in a particular unit system.

• Dimensioning physical constants and coefficients.

• Helps convert a physical amount from one type to another.

• Validating the dimensional validity of a specified physical relationship.

• Research tool for discovering new connections.

Dimensional Analysis Limitations

Both the limitations and applications of dimensional analysis exist. Even though dimensional analysis is known to be extremely useful, it has some drawbacks as well, including

• Because multiple physical quantities possess the exact same dimensions, the distinctiveness of physical quantities is not dependent on the dimensional equation. For instance, if a certain physical component’s dimensional formulation is [ML2T–2], the component might be energy, work, or torque.

• The dimensional techniques cannot infer a numeric constant K with no dimensions, like 1 or 2π.

• The dimensions approach could only be utilized if the equation possesses the product format. 

• For instance, s = ut+(1/2)at2 or y = asinωt

Because they contain sine and addition factors, the above-mentioned equations cannot be obtained through dimensional analysis.

• The dimensions technique cannot be used to calculate an equation wherein a certain physical quantity is dependent on many physical factors. We may, however, examine the accuracy of the supplied equation in terms of dimensions.

• Assume a physical quantity is determined by three factors, two of which are the same size. In that scenario, the formula linking the variables cannot be determined using dimensional analysis

For example, the expression for tuning fork frequency         

 f = (d/L2)v cannot be obtained using dimensional analysis but may be tested.

Conclusion

Dimensional analysis is considered to be an effective approach in physics. The essential premise of dimensional analysis is the very fact that every equation describing a universal principle must be autonomous of the unit system framework. This basic rule typically allows for the development of an equation’s formulation.

Regardless of the notion that highly intricate connections must typically be represented, dimensional analysis is seldom used in civil engineering. They are derived not just from the characterization of novel materials, but from newer, more sophisticated versions of existing materials as well. Hence, one can observe both limitations and applications of dimensional analysis.

A complicated model is often dependent on a wide range of variables as well as material properties. With an increasing number of factors and dependencies, it becomes hard to determine which qualities are reliant on which factors. The uses of dimensional analysis are multiple. It can aid in the selection of appropriate groupings of factors. More importantly, formulas of the values predicted by these groups may clearly highlight the model relationships and hence make them easier to grasp.