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We are familiar with the two types of units, the fundamental units, and the derived units. The fundamental units are base units that are given by the International System of Units. These are kilogram, meter, second, kelvin, ampere, mole, and candela.
The derived units are those units that can be derived from the fundamental units. There are numerous derived units. These include units of speed, acceleration, force, energy, surface tension, pressure, power, and many more.
The dimensional analysis of a system builds a relationship of a physical quantity with their base units.
What are the Dimension Formulae of a Physical Quantity?
The formulae or the set of expressions that denote which fundamental quantities are present in a given physical quantity is known as a dimensional quantity of that physical quantity. It is always enclosed in square braces.
Dimensional formulae are helpful in deriving units from one system to another system. It forms the base of the measurement and is widely applicable in real life.
Consider a physical quantity X. Its dimensional formula is represented by [Ma Lb Tc]. M, L, and T are the base units that denote mass, length, and time with the respective powers a, b, and c.
What do you Mean by the Equation of Dimension of Physical Quantity?
The equation that we get after equating the physical quantity with its dimension formula.
For example,
- The equation of dimension of velocity is given by [M° L T-¹].
- The equation of dimension of force is given by [M L T-²].
- The equation of dimension of energy is given by [M L² T-²].
What are the Dimensions of a Physical Quantity?
The power to which the fundamental units are raised in the equation of dimension is known as the dimensions of that physical quantity.
Let a physical quantity have an equation of dimension as [M L T²]. Then the required dimensions of this physical quantity are given by 1, 1, 2.
Derivation of Formula for Surface Tension using Dimensional Analysis
Surface tension is given by T = Force x Length-¹ ——————(1)
As we know, Force = Mass X Acceleration
And Acceleration = Velocity X Time
Therefore, the dimensional formula for acceleration becomes [M0 L T-²].
Hence, the dimensional formula for force becomes [M] X [M° L T-²] = [M L T-²].
From equation number (1), we get:
Surface tension T = [M L T-²] X [L-¹] = [M L0 T-²].
Hence, the dimensional equation for surface tension is given by [M L° T-²].
Here, M stands for mass, L stands for length, and T stands for time.
Deriving the SI unit for velocity using Dimensional Analysis
The standard formula for velocity is given by v = Displacement x Time-¹ ——(1)
The dimensional formula for displacement is given by [M° L T°].
The dimensional formula for time is given by [M° L° T].
On substituting the above values in (1) we get,
Velocity v = [M° L T°] X [M° L° T]-¹
v = [M° L T-¹]
Hence, the dimensional equation for velocity is given by [M° L T-¹].
Here, M stands for mass, L stands for length, and T stands for time.
Advantages of the Dimensional Analysis
The following points listed below gives the advantages of the dimensional analysis:
- To determine the correctness of a given relation.
- To derive the relationship between various physical quantities.
- To find out the dimension of the unknown physical quantity.
- To convert one system of units into another system of units.
Conclusion
This article explains in detail about dimensional analysis and the equation of dimension. Numerous examples are given to clarify the doubts about the above topics. By going through this article, one can be able to understand the topics.
