Units and dimensions

Units and Dimensions is the study of connections between physical variables using their units and dimensions of measurement. We utilise dimensional analysis to translate a unit between several forms. We must maintain the same unit to facilitate mathematical operations in science and mathematics. 

Dimensional Analysis is often referred to as the factor label method or the unit factor method because we utilise a conversion factor to get the same units.

Unit

The unit is the reference standard used to measure physical quantities. The main characteristics of units are:

• The unit must be of a reasonable size.
• The unit must have a clear definition.
• The device should be easily replicable at any location.
• The unit must not alter with time.
• Physical circumstances such as temperature, pressure, and so on should not affect the unit.
• The unit must be easily compared empirically with physical quantities of equivalent magnitude.

There are two types of units – fundamental units and derived units.

• Fundamental units are the units that are defined for the fundamental quantities.
• The derived units are the units of all other physical quantities derived from the fundamental units.

System of units

In earlier times, scientists from various countries used distinct measurement systems. Three such systems, the CGS, the FPS (or British) system, and the MKS system, were widely used.

The following were the base units for length, mass, and time in these systems:

• In the CGS system, centimetre, gramme, were used
• In the FPS system, foot, pound were used; 
• In the MKS system, metre, kilograms were used.

The units included are:

These units are also known as SI measurement system base units. Base units are used to measure light intensity and material amount.

These units are also known as SI measurement system base units. Base units are used to measure light intensity and material amount.

Only a combination of the units above can measure a quantity. Density, for example, is measured in Kg/m3.

We would write this as [M]/[L3] or [M L-3] in dimensional symbols.

Dimension Analysis

Quantities may be classified into four groups based on their dimension. 

1. Dimensional variables

Dimensional variables are physical quantities that have dimensions and have changeable values. For instance, length, velocity, and acceleration are all examples. 

2. Dimensionless variables

Dimensionless variables are physical quantities that lack dimensions yet have changeable values. Specific gravity, strain, and refractive index are all examples. 

3. Dimensional Constant

Dimensional constants are physical quantities that have dimensions and have constant values. The gravitational constant, Planck’s constant, and others are examples. 

4. Dimensionless Constant

Dimensionless constants are quantities with constant values and no dimensions. Examples include e and numerals. 

Dimensional homogeneity principle 

The notion of dimension homogeneity requires that all words in a physical statement have the same dimensions. In the physical formula v2 = u2 + 2as, for example, the dimensions of v2, u2, and 2as  are identical and equal to [L2T-2].

Factor labelling using dimensional analysis

It is possible to convert units between different measurement systems. Factor label technique, unit factor method, or dimensional analysis are used to describe this approach.

Conversion factors are used to express the connection between different systems’ units, making it possible to move an item from one to another. Based on the fact that the ratio of each basic quantity in one unit with their counterpart in another unit is equal to one, the formula is used.

For instance,

1) How many hours and minutes are in a day?

Solution:

60 minutes is equal to one hour. Three hours equals 180 minutes.

We use conversion factors to ensure that the answer is correct and that conversion factors do not skew the results. The conversion factor is used to change the unit system of each basic quantity, such as mass, length, and time.

Conclusion :

In this chapter, you have read about Units and dimensions study material containing various types of formulas that even includes dimensional formulas, the system of units, CGS, MKS, etc system, Basic units, and derived units. Dimensional analysis is an incredible technique for determining the dimensions of physical quantities in order to verify their relationships. The homogeneity principle is founded on the concept that two quantities of the same dimension may only be added, subtracted, or compared. When the sort of quantities involved is known, the dimensional analysis may be utilised to build credible equations.