Dimensional Analysis and its Applications

Introduction:

Dimensional analysis and its applications 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. For instance, how can we determine the number of minutes in three hours? In general, we believe the following: 60 minutes equals one hour. 3 hr = 3 * 60 min = 180 min (the conversion factor is 60 minutes in this case) We employ appropriate conversion factors to ensure that the response is in the desired unit and that biassed results are avoided.

An illustration of dimensional analysis and its applications

The values must represent the supplementary quantities when employing a conversion factor. For instance, 60 minutes equals one hour, 1000 metres equals one kilometre, and 12 months equals one year.  Allow us to attempt to comprehend it in this manner. Consider that you have fifteen pens. If you double that number by one, you will still have fifteen pens. To determine how many packets of the pen equal 15 pens, you’ll need the conversion factor. Assume you have a boxed set of ink pens that comprises 15 pens in each container. Assume you have six packets. To determine the total number of pens, multiply the number of packets by the number of pens contained in each box. This proves to be true. 15 x 6 equals 90 pens Additional conversion factors that are employed in daily life include the following:

• The time period of a harmonic oscillator is a simple example

• A more complicated example is the energy contained within a vibrating conductor or wire

• A third example is a demand vs. capacity for a revolving disc

Dimensional Analysis in Practice

Dimensional analysis and its applications are critical components of measurement in physics. Dimensional analysis is mostly utilised for four reasons. They are as follows: To determine the accuracy of an equation or any other physical connection using the homogeneity principle. On both sides of the equation, there should be dimensions. If the L.H.S. and R.H.S. of an equation have equal dimensions, the dimensional relationship is accurate. If two dimensions are inaccurate, then the relations will be erroneous. The term “dimensional analysis” refers to converting the value of a physical quantity from one system of units to another.

• Dimensional analysis is used to represent the physical properties of objects
• Dimensional expressions are manipulable as algebraic quantities
• Formulas are fundamental and derived quantities using dimensional analysis

Dimensional Analysis Explained Limitations

• This approach cannot be used to determine dimensionless quantities. This approach cannot be used to find the constant of proportionality. They can be discovered either experimentally or theoretically
• This procedure does not work with trigonometric, logarithmic, or exponential functions
• This strategy will be difficult to apply to physical quantities that are reliant on more than three physical qualities
• In some instances, the constant of proportionality has dimensions as well. In such instances, we are unable to use this system.
• We cannot use this procedure to obtain an expression of one side of the equation comprises the addition operation of physical quantities

Quantities with dimensions, Quantities without dimensions, Principle of Homogeneity

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].

Conclusion

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.