Dimensional Variables

Almost every physical object in the world can be measured. This is done with measurements that humans have developed. Measurements are composed of two parts, a number (n) and a unit (u). Numbers are described in the unit as to what this number is and what it means. There can only be a single unit for each quantity. Those quantities are raised to certain powers based on the fundamental units involved. Such powers are referred to as dimensions. All of the fundamental units are combined into an expression at a certain power. That expression is referred to as the dimensional formula. 

Dimensional variables do not have fixed values, but they are variables with dimensions. For instance, velocity, acceleration, work, and power. In physical terms, dimensionless variables are those quantities with dimensions of the form [Ma Lb Tc]… [where M, L, and T are physical quantities relating to mass, length, and time. As long as a, b, and c are real numbers, they are variables.] Physical quantities whose numerical value depends on the unit of the fundamental quantity but not on the system under consideration are called dimensional constants.

Units and Dimensions

The two terms, dimensions and units, are frequently confused, even though a solution to every engineering problem requires units. A dimension is something that can be measured, whereas a unit of measurement is something that refers to a particular dimension. (For example, a measurement is a length, but a metre refers to a length relative to a standardized definition). It’s critical to note that one dimension may be described by many different units. A conversion factor is also applied to all units of the same dimension (e.g., 2.54 cm is the same as 1 in by definition). 

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Fundamental and Derived Units

Physical quantities are quantities that are capable of being measured, which means they are defined properly, have a proper unit system, and their value can be measured by an instrument. Fundamental quantities and derived quantities are two types of physical quantities. 

• Those quantities that are directly derived from measurement are called fundamental quantities. There is no relationship between them and other quantities, and neither are the units of measurement defined in relation to any other. Mechanically, length, mass, and time are all fundamental quantities.
• Defining derived quantities is the process of expressing quantities according to their fundamental counterparts. For instance, volume, velocity, pressure, etc.
• Associated physical quantities are called derived units since they can be derived from fundamental quantities.

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Dimensionless Variables

Dimensional variables are unitless values resulting from the multiplication and division of physical variables, parameters, and constants. The diameter to circumference ratio isπ . Thus, it does not have dimensions. Similarly, the Coefficient of Friction (μ) and Relative Density are dimensionless variables.

Dimensional Analysis

By using a set of units, dimensional analysis establishes the form of an equation or, more frequently, checks that the answer to a calculation is correct as a safeguard against many simple errors. The start text, SI, end text, SI measurement system use these units as their base units. Both the candela and mole are base units of luminous intensity and substance amount.

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Conclusion

The basis of physics is measurement, which is dependent on standards. To establish consistency and accuracy, as well as to facilitate communication and comparison, measures and standards are in place. The units used to represent quantities are known as standards.

A dimensional analysis involves studying the relationship between physical quantities in terms of dimensions and units of measurement. Dimensional analysis is helpful because it allows us to keep the units constant and, as a result, perform calculations smoothly. The dimensions of dimensional variables are more like variables than fixed values. The dimensions of acceleration, work, and power are common examples. Dimensionless variables are those variables that are measured in units other than units. They are produced by multiplying and dividing the physical variables, parameters, and constants.