Demonstrating Dimensional Analysis

Based on their units and dimensions, dimensional analysis studies the relationship between physical quantities. It is used to change the shape of a unit. To make it easier to solve mathematical issues, it is important to maintain the units consistent.

Dimensional analysis is important, but what is its significance to you? Well! For example, the link between length, mass, time, and electric current and the units of measurement like miles vs. kilometers or pounds vs. kilograms is described in engineering and science through the use of dimensional analysis.

Fundamental and derived physical quantities are both studied in physics. The seven fundamental units of measurement include mass, length, amount of substance, time, luminous intensity, and current. The derived quantities, on the other hand, are formed when two or more fundamental units are combined.

Is Dimensional Analysis Possible?

Conversion of Weights and Measures

It is also known as the Unit Factor Method or Factor Label Method because the units are evaluated using a conversion factor.

Suppose there are a total of 4 km of distance, and we want to know how many meters that is.

On a regular basis, we compute-

A kilometer is equal to a thousand meters.

4 km = 1000 × 4 = 4000 meters

(In this case, 1000 meters is the conversion factor.)

Applications of Dimensional Analysis

Quantitative dimensional analysis is an important part of the measurement, and it has numerous applications in the physical sciences. Dimensional analysis is employed primarily for five reasons, which include the following:

For the purpose of determining the accuracy of an equation or any other physical relationship based on the principle of homogeneity, on both sides of the equation, there should be dimensions to consider. In an equation with identical dimensions on both sides of the LHS and RHS, the dimensional relation will be correct. If the dimensions on two sides are erroneous, it is likely that the relationships will be inaccurate as well.

Dimensional analysis is used to translate the value of a physical quantity from one system of units to another system of units. Dimensional analysis can be applied to any physical quantity.

 It is used to represent the nature of physical quantity in its various forms.

Dimensional expressions can be changed in the same way as algebraic quantities can.

Formulas are derived through the application of dimensional analysis.

Limitations of Dimensional Analysis

The following are some of the major limitations of dimensional analysis:

·  It is not possible to determine the dimensional constant using dimensional analysis techniques.

·  Dimensional analysis is unable to extract trigonometric, exponential, or logarithmic functions from their input data.

· However, it does not provide information on the scalar or vector identity of a given physical quantity.

The following is an example of a Dimensional Formula: the derivation of Kinetic Energy

It is the mathematical equation that represents the powers to which the fundamental units (mass M, length L, and time T) must be raised in order to obtain one unit of a derived quantity for any physical entity. The dimensional formula of every physical entity is represented by the following:

Let us now look at an example of the dimensional formula to better comprehend it. Now that we know that kinetic energy is one of the essential components of Physics, we can see why its formula is important in so many other disciplines of study. In order to do so, let’s look at the dimensional formula of kinetic energy.

The kinetic energy has a dimensional formula that looks like this:

[ML2T-2]

Where,

M is the object’s mass in kilograms.

L is the object’s overall length.

T is the amount of time it took.

Derivation

Using the formula [Mass x Velocity2] = 12, we may calculate kinetic energy (K.E.). In other words: (I)

Mass has the dimensional formula =[M1L0T0]. —— (ii)

We are well aware of this,

Distance x Time-1 = Velocity

the product of length and time one-one (dimensional formula)

There is a mathematical formula for velocity. [M0L1T-1] —– (iii)

As a result of substituting the following equations into the original one, we get

[Mass x Velocity2] x 12 = Kinetic energy (K.E.)

Alternatively, K.E = [M1L0T0] [M0L1T-1] [M (0 +1) L (1 + 1) T (-1 + -1)] is the formula for 2

The dimensional formula for the kinetic city is [M0L2T-2] after solving.

We may deduce from this that a set of units is used in dimensional analysis to assist us to determine the equation’s form and ensure that the result is free of even the smallest inaccuracies.

Conclusion

To finish up, we had learned about dimensional analysis and its application which means an analytical technique whereby physical quantities are expressed in terms of their fundamental dimensions, which is frequently employed when there is insufficient data to put up accurate equations.