Dimensional Analysis
Any physical quantity can be expressed as a product of a combination of the basic physical dimensions.
Learning Objectives
Calculate the conversion from one kind of dimension to another
Key Takeaways
Key Points
- Dimensional analysis is the practice of checking relations amount physical quantities by identifying their dimensions.
- It is common to be faced with a problem that uses different dimensions to express the same basic quantity. The following equation can be used to find the conversion factor between the two derived dimensions: [latex]{\text{n}_2=\frac{\text{u}_2}{\text{u}_1}*\text{n}_1}[/latex].
- Dimensional analysis can also be used as a simple check to computations, theories and hypotheses.
Key Terms
- dimension: A measure of spatial extent in a particular direction, such as height, width or breadth, or depth.
Dimensions
The dimension of a physical quantity indicates how it relates to one of the seven basic quantities. These fundamental quantities are:
- [M] Mass
- [L] Length
- [T] Time
- [A] Current
- [K] Temperature
- [mol] Amount of a Substance
- [cd] Luminous Intensity
As you can see, the symbol is enclosed in a pair of square brackets. This is often used to represent the dimension of individual basic quantity. An example of the use of basic dimensions is speed, which has a dimension of 1 in length and -1 in time; [latex]\displaystyle \frac{[\text{L}]}{[\text{T}]} = [\text{LT}^{-1}][/latex]. Any physical quantity can be expressed as a product of a combination of the basic physical dimensions.
Dimensional Analysis
Dimensional analysis is the practice of checking relations between physical quantities by identifying their dimensions. The dimension of any physical quantity is the combination of the basic physical dimensions that compose it. Dimensional analysis is based on the fact that physical law must be independent of the units used to measure the physical variables. It can be used to check the plausibility of derived equations, computations and hypotheses.
Derived Dimensions
The dimensions of derived quantities may include few or all dimensions in individual basic quantities. In order to understand the technique to write dimensions of a derived quantity, we consider the case of force. Force is defined as:
[latex]\small{\text{F}=\text{m}\cdot\text{a}\\\text{F}=[\text{M}][\text{a}]}[/latex]
The dimension of acceleration, represented as [a], is itself a derived quantity being the ratio of velocity and time. In turn, velocity is also a derived quantity, being ratio of length and time.
[latex]\small{\text{F}=[\text{M}][\text{a}]=[\text{M}][\text{vT}^{-1}]\\\text{F}=[\text{M}][\text{LT}^{-1}\text{T}^{-1}]=[\text{MLT}^{-2}]}[/latex]
Dimensional Conversion
In practice, one might need to convert from one kind of dimension to another. For common conversions, you might already know how to convert off the top of your head. But for less common ones, it is helpful to know how to find the conversion factor:
[latex]\small{\text{Q}=\text{n}_1\text{u}_1=\text{n}_2\text{u}_2}[/latex]
where n represents the amount per u dimensions. You can then use ratios to figure out the conversion:
[latex]\displaystyle \small{\text{n}_2=\frac{\text{u}_2}{\text{u}_1}\cdot\text{n}_1}[/latex]
