Obtain the relation between the units of some physical quantity in two different systems of units. Obtain the relation between the $MKS$ and $CGS$ unit of work.
(N/A) The unit of work in the $MKS$ system is Joule $(J)$ and in the $CGS$ system is erg. The relation between Joule and erg is derived as follows:
Dimensional formula for work is $W = [M^1 L^2 T^{-2}]$.
Let $n_1$ and $n_2$ be the numerical values and $u_1$ and $u_2$ be the units in the two systems.
$n_1 u_1 = n_2 u_2$
$n_2 = n_1 [M_1/M_2]^1 [L_1/L_2]^2 [T_1/T_2]^{-2}$
In $MKS$ system: $M_1 = 1 \text{ kg} = 10^3 \text{ g}$,$L_1 = 1 \text{ m} = 10^2 \text{ cm}$,$T_1 = 1 \text{ s}$.
In $CGS$ system: $M_2 = 1 \text{ g}$,$L_2 = 1 \text{ cm}$,$T_2 = 1 \text{ s}$.
$n_2 = 1 \times [10^3 \text{ g} / 1 \text{ g}]^1 \times [10^2 \text{ cm} / 1 \text{ cm}]^2 \times [1 \text{ s} / 1 \text{ s}]^{-2}$
$n_2 = 10^3 \times (10^2)^2 \times 1 = 10^3 \times 10^4 = 10^7$.
Therefore,$1 \text{ Joule} = 10^7 \text{ erg}$.
Dimensional formula for work is $W = [M^1 L^2 T^{-2}]$.
Let $n_1$ and $n_2$ be the numerical values and $u_1$ and $u_2$ be the units in the two systems.
$n_1 u_1 = n_2 u_2$
$n_2 = n_1 [M_1/M_2]^1 [L_1/L_2]^2 [T_1/T_2]^{-2}$
In $MKS$ system: $M_1 = 1 \text{ kg} = 10^3 \text{ g}$,$L_1 = 1 \text{ m} = 10^2 \text{ cm}$,$T_1 = 1 \text{ s}$.
In $CGS$ system: $M_2 = 1 \text{ g}$,$L_2 = 1 \text{ cm}$,$T_2 = 1 \text{ s}$.
$n_2 = 1 \times [10^3 \text{ g} / 1 \text{ g}]^1 \times [10^2 \text{ cm} / 1 \text{ cm}]^2 \times [1 \text{ s} / 1 \text{ s}]^{-2}$
$n_2 = 10^3 \times (10^2)^2 \times 1 = 10^3 \times 10^4 = 10^7$.
Therefore,$1 \text{ Joule} = 10^7 \text{ erg}$.
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