Units and Measurements: Applications

1. To check the correctness of a physical relation

Example: Show that the expression of the timer period T of a simple pendulum of l given by
T = 2π√l/g is dimensionally correct

2. To convert one system of units to another

This is based on the fact that the product of the numerical value (n) and its corresponding unit (u) is a constant.  Thus n[u] = constant or n₁[u₁] = n₂[u₂]

Let us consider a physical quantity whose dimensional formula is [M^aL^b T^c].  Let n₁ & n₂ be its numerical value in two systems having different units u₁ & u₂

n₁u₁ = n₂u₂

(In 1st system of units)   (In 2nd system of units)

u₁ = [M₁^aL₁^b T₁^c] and u₂ = [M₂^aL₂^b T₂^c]

Thus, n₁ [M₁^aL₁^b T₁^c] = n₂  [M₂^aL₂^b T₂^c]

∴ n_{2 =} n₁ [M₁/M₂]^a[L₁/L₂]^b[T₁/T₂]^c

Thus, if we know n₁ we can calculate n₂. 

3. To find the dimensions of constant in a given physical relation

For this, we again use the principle of homogeneity of dimensions i.e. only like quantities can added or subtracted

4. To derive the relationship between different physical quantities

If we know the factors on which a given physical quantity depends, we can find the relation between the physical quanity and those factors on which it depends by using the principle of homogeneity of dimensions.

4. To derive the forgotten formula

Suppose a person is sure of physical quantities involved in a relation but is doubtful about the powers or a person is reading a book in which an equation has been written whose power are not legible.  In such situations, we can use the principle of homogeneity of dimensions to obtain the correct relation.