Dimensional formula and dimensional equations
of a physical quantity are defined as the fundamental quantities that are
raised to powers to express the physical quantity. The basic dimensional
formula of mass is [M], the length is [L] and time is [T].
The other way of defining dimensional formula
is, the physical quantities are expressed in terms of their basic units. For
example, the dimensional formula of force is
given as:
F = [MLT^{2}]
The unit of force is Newton or kg.m.s^{2}.
For any physical quantity, the dimensional
formula is written when its relation with other physical quantities is known.
Following is a table of the dimensional formula of a few quantities.
Physical quantity

Relation with other
physical quantities

Dimensional formula

SI unit

Area

Length * Breadth

[L]*[L] = [M^{0}L^{2}T^{0}]

m^{2}

Volume

Length * Breadth * Height

[L]*[L]*[L] = [M^{0}L^{3}T^{0}]

m^{3}

Density

(Mass)/(Volume)

[M]/[L^{3}] = [ML^{3}T^{0}]

kg.m^{3}

Energy

Work

[ML^{2}T^{2}]

J

Impulse

Force * Time

[MLT^{2}]*[T] = [MLT^{1}]

N.s

What is dimensional equation?
The dimensional equation is defined as the
equation with dimensional formula. By equating the dimensional formula on the
righthand side and on the lefthand side, the dimensional equation is
obtained. This is proved with the help of the principle of homogeneity.
What is the principle of homogeneity?
The principle states that the dimensions on
the lefthand side of an equation must be equal to the dimensions on the
righthand side. Using the principle of homogeneity one can even convert the
units from one system to another. Example of the principle of homogeneity is as
follows.
Dimensional equation of v = u + at is:
[M^{0}LT^{1}] = [M^{0}LT^{1}]
+ [M^{0}LT^{1}] * [M^{0}L^{0}T] = [M^{0}LT^{1}].
What are the uses of dimensional equations?
Following is a list of uses of dimensional equations.
●
It is used to check the
correctness of the physical quantity.
●
It can be used to derive the
relation between different physical quantities.
●
The dimensions of constants can be
found using the dimensional equation.
What are the limitations of dimensional analysis?
Following are the limitations of dimensional
analysis.
●
There is no information about the
quantity being scalar or vector quantity.
●
The formula cannot be derived if
the quantity depends on more than three factors.
●
Dimensional analysis cannot be
carried for functions like logarithmic functions, exponential functions, and
trigonometric functions.
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