Information on Dimensions of Physical Quantities

Measurement basics should be reviewed before getting into dimensions and dimensional equations. Seven fundamental (base) quantities may be used to express all physical values from the dimensions of the physical quantities list. These include mass, length, time, temperature, electric current, luminance, and the amount of material. These seven quantities make up the seven dimensions of physical quantities. 

The fundamental quantities must have enough strength to represent any derived quantity. The ability of a fundamental quantity to represent the unit-derived quantity is referred to as a dimension. The dimensions of physical quantities are determined by increasing the base quantities (basic quantities) to the appropriate power when expressed.

Symbols may be used instead of the names of the real numbers. The letters M, L, and T stand for mass, length, and time, respectively, in mechanical terms. They may also be denoted using the brackets [M], [L], and [T]. Other dimensions include K (temperature), I (electric current), cd (light intensity), and mol (mass) (for the amount of substance). The dimensions of a physical quantity and its unit are identical. The size of the unit is not indicated by the characters [M], [L], [T], and so on.

Dimensional formula and dimensional equation

The dimensional formula is a compound statement that explains how and which basic quantities are used to create a physical quantity. Physical quantities’ dimensional equations equate physical quantities to their dimensional formulas. For example, dimensional equations represent physical variables’ dimensions in terms of the fundamental quantities.

[M0L2T0], [M0L3T0], [M0L1T-2] are example of dimensional formulae and [A] = [M0L2T0], [V] = [M0L3T0], [a] = [M0L1T-2] are example of dimensional equation.

Different types of Dimensions

Physical quantities may be categorised into four groups based on the dimensional formula.

• Dimensional variables: Diffraction variables, physical quantities with dimensions but no fixed values, are dimensional variables. A few examples include speed, labour, and power.

• Dimensionless variables: Dimensionless (non-dimensional) variables are physical quantities that do not have dimensions but are variables because they may change over time. Example: strain, angle of inclination of a plane.

• Dimensional constants: Dimensional constants are physical quantities with a fixed value yet have dimensions of physical quantities. Planck’s constant (h), for instance, or the constant of gravitational gravity (G)

• Dimensionless constants: Numbers with no dimensions, such as 1,2,3, etc., are known as dimensionless (non-dimensional) constants. 

All dimensions of physical quantities can be derived from the fundamental 7 dimensions.

What is dimensional analysis?

The nature of physical quantities may be described by the dimensions of the fundamental quantities and the combinations of these dimensions. It is possible to utilise dimensional analysis to determine the relationships between different physical variables, verify equations’ dimensional correctness, etc. For an equation to be dimensionally consistent, it does not need to be an exact or accurate equation.

Analyses based on Dimensionality

Dimensional analysis has a wide range of useful applications. Some of these include:

• Translating physical quantities between different systems of measurement.

• To verify the accuracy of a given relationship.

• To establish a correlation between several physical properties.

• To verify the accuracy of a given relationship.

The dimensional formula of each term on each side of the relationship may be used to verify if the presented relation is valid. It’s safe to assume that the relationship is valid if the dimensions are equal. Homogeneity (e.g., the principle of equality) The equation should be checked for accuracy.

For example, check the correctness of the equation:

S = ut + 1/2at2

Dimensional formula of S = [ L ]

Dimensional formula of ut = [ L T-1] x [ T ] = [ L ]

Dimensional formula of 1/2at2 = [ L T-2] x [ T2 ] = [ L ] 

Here 1/2 is a constant and has no dimensions. The equation is accurate because all three terms have the same dimensions.

Limitations of Dimensional Analysis

Among dimensional analysis’s most egregious shortcomings are the following:

• This method makes no mention of dimensionless constants.

• S = ut + 1/2at2 is difficult to infer links using more than one word.

This method will fail for a connection with more than three unknown variables.

This strategy will not work if the connection comprises trigonometric, logarithmic, complex, or exponential functions. It’s tough to distinguish between so many different physical characteristics with the same dimensions.

Conclusion

The minimum number of coordinates needed to identify every point inside a mathematical space (or object) is how dimensions of physical quantities are defined informally in physics and mathematics. For example, the point at the end of the number line at 5 has one dimension since only one coordinate is needed to identify it (1D). Surfaces like planes, the surfaces of cylindrical bodies, and the surfaces of galaxies all have two dimensions since two coordinates are required to identify one point (2D). For example, latitude and longitude are both necessary to locate an object on the surface of a globe.