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A dimensional formula is a mathematical formula that represents a physical quantity in terms of fundamental quantities such as mass, length, and time. With the help of dimensional formulae, the fundamental quantities can be raised to different powers to build a new physical quantity. The powers of fundamental physical quantities are referred to as dimensions.
Relative Density
The relative density of a material is the ratio of its density to that of a reference substance. When the reference material is water, we may refer to it as “specific gravity.”
The phrase, relative density, refers to the ratio of two densities; however, it may often be measured by selecting a quantity (sample) of the material and employing proper physical balance. So, the relative density of a substance can be given as:
Relative Density = Density of a Substance / Density of Relative Substance
Dimensional Formula
The Dimensional Formula of any bodily amount is defined as the expression that represents how and which of the bottom portions are protected in that amount. It is denoted by enclosing the symbols for base portions with suitable strength in rectangular brackets – [ ].
An example is the Dimension Formula of Mass, which is given as [M].
Examples of Dimensional Formula
Let’s take the formula of the area of the rectangle
Area of the rectangle = length x breadth
= L x L ( where breadth is also showing the length of the side)
= [L1] X [L1]
= [L2]
Here, we can see the length to the power of 2, but we cannot find the dimension of mass and time.
Hence, the dimension of the area of a rectangle is written as [M0 L2 T0]
Let’s take the formula of speed
Speed = Distance / Time
The distance can be written in length [L]
Time can be written as [T]
Thus, the dimensional formula of speed would be [ M0 L1 T-1]
Hence, we can conclude that the speed is dependent on only length and time, not mass.
Dimensional Equation
The dimensional formula depicts the dependency of physical quantity with fundamental physical quantity, along with the powers.
The physical quantity is equated with the dimensional formula to get the dimensional equation.
For example, Velocity = [ M0 L1 T-1]
Uses of Dimensional Formula
The following are the uses of the dimensional formula:
It can prove to be a helpful tool to check the consistency and coherency of your dimensional equation
The dimensional formula will be used to establish the correlation between the physical quantities of physical phenomena
These formulas can be used as a device to change the units from one system to another
Application of Dimensional Analysis
In real-life physics, dimensional analysis is a crucial part of the measurement. We use dimensional analysis for the following reasons:
To ensure that the dimensional equation is consistent
To determine the relationship between physical quantities in physical phenomena
To switch from one system to another’s units
To develop a fluid phenomena equation
To reduce the number of variables necessary in an equation
Limitations of Dimensional Formulas
It is not concerned with the dimensional constant
The formula that contains functions like trigonometric, exponential, logarithmic and the like cannot be derived
In the context of a physical quantity being scalar or vector, it does not provide any information about whether the quantity is physical or not
Derivation of Dimensional Formula of Relative Density
To find the dimensional formula of relative density, write its formula in terms of fundamental quantities:
Relative Density = Density of a Substance / Density of Relative Substance
The unit of density is kg / m3
So, Relative Density = kg/m3 / kg/m3
Now, write the units of density in terms of [M], [L], and [T]
Relative Density = [M1L-3] / [M1L-3]
Since the numerator and denominator are the same, they can be cancelled out.
Therefore, Relative Density = 1
As the right-hand side is a constant number 1, it can be represented in terms of fundamental quantities as:
Relative Density = [M0L0T0]
So the dimensional formula of relative density is [M0L0T0].
Conclusion
The ratio of the density of a material to the density of a relative substance is known as relative density. Any quantity’s dimensional formula expresses it in terms of the fundamental quantities. In order to find the dimensional formula of relative density, replace the units of density with fundamental units. Since both the numerator and denominator have density, it cancels out, due to which it is a dimensionless quantity. So the dimensional formula of relative density is [M0L0T0].
