More about Dimensional Formula in Physics

Dimensional analysis establishes the correlation or connection between one physical quantity and the other physical quantity. In mathematics, a dimension measures length, width, or height extended in a specific direction. According to the dimensions in physics definition, it is a measure of a spot or arc extended in one order. The degree to which the fundamental units are elevated to obtain one unit of any physical quantity is called its dimension. Let’s look at the dimensional formula in physics and see some examples at the conclusion. 

Seven Fundamental Quantities

There are seven fundamental underived physical quantities: length, mass, electric current, time, luminous intensity, amount of substance, and temperature. These seven fundamental quantities are also termed the seven dimensions of the physical world. Instead of using the names of the base quantities, we can utilize symbols. The dimensions of the three most common physical quantities i.e. mass, length, and time are predominantly used by [M], [L], and [T].

Other dimensions such as K for temperature because of Kelvin, I is used for electric current, Cd is used for luminous intensity and mole is used to denote the amount of substance.  A physical quantity’s dimensions and its unit’s dimensions are the same. The letters [M], [L], [T], and so on merely indicate the unit’s type, not its size.

What is the Dimensional Formula?

In terms of dimensions, a dimensional equation is an equation that connects fundamental and derived units. The three base dimensions in mechanics are length, mass, time, temperature, and electric current, with the fundamental units being the metre, kilogramme, second, ampere, kelvin, mole, and candela. In any dimensional equation, the dimensional formula of different quantities is employed to construct a link between them. The following is an example of a dimensional equation:

Dimensional formula(equation) for area of rectangle:

Area = length of rectangle × breadth of rectangle

= length(in meter) × length(in meter)

= [L] × [L]

= [L]2

⇒ Dimensional formula (equation) for area (A) = [L2 M0 T0]

What is Dimensional Analysis?

The nature of physical quantities is described by the dimensions of base quantities and their combinations. Dimensional analysis can be used to recheck the equations for dimensional consistency, deduce relationships between distinct physical amounts, etc.

 A dimensionally consistent equation must not be perfect or correct, whereas a dimensionally incorrect or inconsistent equation must be wrong.

Applications of Dimensional Analysis

Dimensional analysis has a wide range of applications.

To change a physical quantity’s value from one standard to another.

To ensure that a particular relationship is correct.

To come up with a relationship between different physical quantities.

Limitations of Dimensional Analysis

1. The dimensionless constants are not mentioned in this method.
2. It is impossible to derive relations with more than one term, such as s = ut + ½ at2.
3. If there are more than three unknown quantities in the relationship, this approach will fail.
4. If the relationship contains trigonometric functions, logarithmic, complex, or exponential parts, this method will not work.
5. Because there are so many physical quantities with exact measurements, it’s difficult to tell them apart just by looking at them.

Dimensional Formula Basic Structure

The dimensional formula of a derived formula for any physical quantity portrays the powers or exponential degree to which the fundamental units of mass, time and length must be increased or decreased to acquire a single unit of the derived physical quantity.

If Q is the unit of a derived quantity, MaLbTc is known as the dimensional formula, and the exponents a, b, and c are the dimensions.

Dimensional Formula and Dimensional Equations

The dimensional formula is a compound statement that explains how and which fundamental quantities are used to create a physical quantity.

A physical quantity’s dimensional equation is an equation that equates the physical quantity to its dimensional formula. In other words, the dimensional equations represent the dimensions of a physical quantity in terms of the fundamental quantities.

For example, dimensional formulae are expressions like [M0L2T0], [M0L3T0], [M0L1T-2], and dimensional equations are equations like [A] = [M0L2T0], [V] = [M0L3T0], [a] = [M0L1T-2], and so on.

Applications of Dimensional Formula

The dimensional formula is helpful in the following situations:

It is used to ensure that an equation is correct.

The dimensional formula aids in the calculation of relationships between physical quantities.

For any given quantity, convert from one system of units to another.

It is a mathematical expression that expresses a single quantity in terms of the fundamental units.

Let’s look at several solved cases to grasp the dimensional formula better.

Law of Homogeneity of Dimensions

The law of homogeneity of dimensions defines the equality of dimensional measurement comparing physical quantities on both sides of the equation. The size of all terms on both sides must be the same in any accurate equation describing the relationship between physical quantities. The dimensions equation for physical quantities separated by a ‘+’ or a ‘–’ should be the same.

When a physical quantity Q has dimensions p, q, and r in length (L), mass (M), and time (T), and n1 is its dimensionless numerical value in the system where the fundamental units are L1, M1, and T1, and n2 is its dimensionless numerical value in the system where the fundamental units are L2, M2, and T2, respectively, then,

n2 = n1 [L1/L2]a [M1/M2]b [T1/T2]c

Conclusion

The use of dimensional analysis to simplify an issue is becoming more widespread. It has become more efficient and practical in a wide range of research and development domains due to its ease of use in other derived physical quantities and multiple benefits. Dimensional analysis is used in many disciplines of science and exploration, including physics, chemistry, maths, fluid mechanics, space research, economics, finance, accountancy, biology, and others. The dimensional formula for all physical quantities is interlinked with seven fundamental quantities.