Question

Drive the relation for conversion of one unit system to another using the dimensions.

Answer 1

Conversion of units of physical quantities from one system to another system: The method of dimensional analysis can be used to obtain the value of the physical quantity in some other system when its value in one system is given.

As discussed earlier, the measurement of a physical quantity is given by:

Q = nu

If the unit of a physical quantity in a system is $u_1$, and the numerical value is $n_i$, then:

$Q=n_1{u}_1$ ................(1)

Similar in the other system if the unit is $u_2$ and magnitude is $n_2$ then:

$Q=n_2{u}_2$. .......(2)

From Eqs. (1) and (2)

$n_1{u}_1=n_2{u}_2$ .................(3)

If a,b,c are the dimensions of a physical quantity in mass, length and time, then:

$n_1\left[M_1^a{L}_1^b{T}_1^c\right]=n_2$

Here $M_1,L_1,T_1$ and $M_2,L_2,T_2$ are the units of mass, length and time in the two systems, then

$n_2=n_1\left[\frac{M_1}{M_2}{]}^a\right[\frac{L_1}{L_2}{]}^b[\frac{T_1}{T_2}{]}^c$

The equation can be used to find out the value of a physical quantity in the second or the new system, when its value in first system is known.