Consider
the equation: s = vₒt + ½ at²
… (1)
Where
the letters have their usual meaning. Since it is meaningless to equate two
quantities of different nature, e.g., mass to a length or a temperature
to a force, it is evident that for the above equation to have any physical
meaning, the three terms combining it must have the same
dimensions. We know that the dimensional formula of s is
[L], that of t is [T], that of vₒ
is [LT⁻¹] and that of a is [LT⁻²¹].
Hence, the dimensional equation corresponding to
eqn. (1) which neglects all numerical coefficients may be
written as [L] = [LT⁻¹]
[T] + [LT⁻²¹] [T²] = [L] + [L]
i.e., the
dimensions of all the terms on the two sides of the equation are the same. This
property is true of all the equations representing different physical phenomena
and may be stated as:
“Physical equations must be
dimensionally homogeneous.” This is known as the principle of
homogeneity of dimensions and according to this principle.
When a physical equation consists of a number of
terms, each of these terms must be of the same dimensions in each of the
fundamental units.
One can state the principle in a
much simpler way as follows: The powers of mass, length and
time in each term on one side of a dimensional equation must always be equal to
their respective powers in each term on the other side of the dimensional
equation.
According to the principle of homogeneity, the
dimensions of each term on the LHS must be equal to the dimensions of the term
on the RHS.
Clearly [ML²T⁻²] a = [L²]
Or a = [L²]/[ML²T⁻²] = [M⁻¹T²]
And b = [L²]
Dimensional Analysis and its uses: Sometimes we might be required either to derive or to
check a specific relation. Even if we have forgotten the details of the
derivation, there is a useful and powerful procedure that can help us in this
matter. This procedure, which treats dimensions
as algebraic quantities is called dimensional
analysis.
Thus, dimensional
analysis is the process of
analysing a physical problem with the help of dimensional equations.
Following
are the three main uses of dimensional analysis (or of dimensional equations).
(a) Checking the dimensional correctness of a
physical equation: To
check the correctness of a physical equation, we use the principle of homogeneity of dimensions.
We calculate the dimensions of the quantities on both
sides of the equation in terms of M, L and T. If the dimensions obey the
principle of homogeneity, the equation is dimensionally correct, otherwise not.
(b) Deriving relationship between different physical
quantities: Using
the principle of homogeneity of dimensions, in many cases, we can find the
expression for a physical quantity, if we know the factors on which it depends.
(c) Conversion of one system of units into another: The magnitude of a quantity is the same, whatever
may be the system of its measurement. This fact helps us to change one system
of units into another.
Let [MᵃLᵇTᶜ] be the dimensional formula of that
quantity whose units are to be converted from one system into another.
Further, let M₁, L₁, T₁ represent the fundamental units of mass, length and time in one system of units and M₂, L₂, T₂ be the corresponding units in another system.
If u₁ and u₂ represent the units of the quantity in the two
systems, then
u₁ = [M₁ᵃL₁ᵇT₁ᶜ] and u₂ = [M₂ᵃL₂ᵇT₂ᶜ]
If n₁ (known) is the numerical value of the quantity
in one system and n₂ (unknown) is
the numerical value of the quantity in the other system, then
n₁u₁ = n₂u₂
Or n₁ [M₁ᵃL₁ᵇT₁ᶜ] = n₂ [M₂ᵃL₂ᵇT₂ᶜ]
Or n₂ = n₁ [M₁ᵃL₁ᵇT₁ᶜ]/[M₂ᵃL₂ᵇT₂ᶜ]
Or n₂ = n₁ [M₁/ M₂]ᵃ[L₁/ L₂]ᵇ[T₁/ T₂]ᶜ … (1)
Thus, if the fundamental units in both the systems,
dimension of the quantity and its numerical value in one system are known, we
can easily calculate n₂, i.e., the numerical value of the quantity in the other system.
It
should be clearly understood that eqn. (1) is
to be applied only after expressing the quantity in the absolute units.
No comments:
Post a Comment