Question

State the principle of homogeneity of dimensions. Test the dimensional homogeneity of the following equation:
$$$$$h = {h_0} + {v_0}t + \dfrac{1}{2}g{t^2}$

Answer 1

Hint- To solve this question we use the basic concept of dimension analysis. according to this; we can add and subtract only those quantities that have the same dimensions. For example we add three quantities ${h_0}$ , ${v_0}t$ and $\dfrac{1}{2}g{t^2}$ That means the dimensions of these all three are the same in this case.

Complete step-by-step answer:

Principle of Homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another. To better understand the principle, let us consider the following example:
Testing the dimensional homogeneity for
$h = {h_0} + {v_0}t + \dfrac{1}{2}g{t^2}$
We have, As per dimensional Notation,
h = ${\text{L}}$
v = ${\text{L}}$ ${{\text{T}}^{{\text{ - 1}}}}$
t = ${\text{T}}$
g = ${\text{L}}$ ${{\text{T}}^{{\text{ - 2}}}}$
The principle of homogeneity states that the dimensions of each the terms of a dimensional equation on both sides are the same.
Using this principle, the given equation will have the same dimension on both sides.
On left side: h=[L],
On right hand side,
RHS= ${\text{L}}$+ ${\text{[L}}{{\text{T}}^{{\text{ - 1}}}}{\text{AT] + }}\dfrac{{\text{1}}}{{\text{2}}}{\text{[L}}{{\text{T}}^{{\text{ - 2}}}}{\text{A}}{{\text{T}}^{\text{2}}}{\text{]}}$
= ${\text{L}}$+ ${\text{L}}$+$\dfrac{1}{2}$[${\text{L}}$]
= ${\text{L}}$
Since All are of Dimension L , the result of sum is also equal to L.

Therefore,
LHS= RHS Dimensionally.
So, the Relation is correct.

Its application-
The dimensional equations have got following uses:
To check the correctness of a physical equation.
To derive the relation between different physical quantities involved in a physical phenomenon
To change units from one system to another

Note- Dimensional homogeneity is the concept where the dimensions of variables on both sides of an equation are the same. An equation could be dimensionally homogeneous but invalid if the equation is also not fully-balanced.