Uses of Dimensional Equations:
(i) To test the correctness of a physical equation.
Test the correctness of the equation: $v^2 = u^2 + 2as$
u = initial velocity has dimensions [$L\,T^{-1}$]
v = final velocity has dimensions [$L\,T^{-1}$]
a = acceleration has dimensions [$L\,T^{-2}$]
s = Distance travelled has dimensions [L]
Writing in the Dimension form;
$[L\,T^{-1}]^2 = [L\,T^{-1}]^2 + 2[L\,T^{-2}]\,[L]$
or, $[L^2\,T^{-2}] = [L^2\,T^{-2}] + 2[L^2\,T^{-2}]$
Here, 2 is a dimensionless constant. Therefore, dimensions of L.H.S. = dimension of R.H.S. Hence above equation is dimensionally correct.
(ii) To derive the relation between various physical quantities.
(iii) To convert value of physical quantity from one system of unit to another systems.
(iv) To find the dimension of constants in a given relation.
Limitation of Dimensional Analysis:
- It does not give information whether the physical quantity is scalar or vector.
- It does not give information about the dimensionless constant.
- If a quantity depends on more than three factors having dimension, the formula can not be derived.
- This method can not establish relation between more than three physical quantities because one can form only three simultaneous equations by equating the dimensions of [M], [L] and [T].
- It is not applicable to the trigonometric and exponential function.
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