Explain the principle of homogeneity of dimensions. What are its uses? Given example.

Explanation

The principle of homogeneity of dimensions states that the dimensions of all the terms in a physical expression should be the same. For example, in the physical expression `v^(2)=u^(2)+2as`, the dimensions of `v^(2), u^(2)` and 2as are the same and equal to `[L^(2)T^(-2)]`.
(i) To convert a physical quantity from one system of units of another.
This is based on the fact that the product of the numerical values (n) is a constant. i.e., `n[u]=" constant (or )"n_(1)[u_(1)]=n_(2)[u_(2)]`.
Using the method of dimensions 76 cm of mercury pressure into `"Nm"^(-2)`.
In CGS system 76 cm of mercury pressure `=76xx13.6xx"980 dyne cm"^(-2)`.
The dimensional formula of pressure P is `[ML^(-1)T^(-2)]`
`P_(1)[M_(1)^(a)L_(1)^(b)T_(1)^(c)]`
We have `P_(2)=P_(1)[(M_(1))/(M_(2))]^(a)[(L_(1))/(L_(2))]^(b)[(T_(1))/(T_(2))]^(c )`
`M_(1)=1g, M_(2)=1kg`
`L_(1)=1 cm, L_(2)=1m`
`T_(1)=1s, T_(2)=1s`
So `a=1, b=-1, and c=-2`
Then `P_(2)=76xx13.6xx980[(1g)/(1kg)]^(1)[(1cm)/(1cm)]^(-1)[(1s)/(1s)]^(-2)`
`=76xx13.6xx980[10^(-3)]xx10^(2)`
`P_(2)=1.01xx10^(5)Nm^(-2)`
(ii) To verify the dimensional correctness of a given physical equation.
Let us consider the equation of motion `v=u+at`
Apply dimensional formula on both sides
`[LT^(-1)]=[LT^(-1)]+[LT^(-2)][T]`
`[LT^(-1)]=[LT^(-1)]+[LT^(-1)]`
It is found that the dimensions of both sides are same. Hence the equation is dimensionally correct.
(iii) To establish relations among various physical quantities.
If the physical quantity Q depends upon the quantities `Q_(1), Q_(2) and Q_(3)` i.e., Q is proportional to `Q_(1), Q_(2) and Q_(3)`.
Then, `Q prop Q_(1)^(a)Q_(2)^(b)Q_(3)^(c )`
`Q=kQ_(1)^(a)Q_(2)^(b)Q_(3)^(c )`
Where k is a dimensionless constant. When the dimensional formula `Q, Q_(1),Q_(2) and Q_(3)` are substituted, then according to the principle of homogeneity, the powers of M, L, T are made equal on both sides of the equation. From this, we get the value of a, b, c. From these values the relations can be formed.