The density of material in the CGS system of units is $4 ext{ g/cm}^3$. In a system of units in which unit of

Question

The density of material in the CGS system of units is $4	ext{ g/cm}^3$. In a system of units in which unit of length is $10	ext{ cm}$ and unit of mass is $100	ext{ g}$, the value of density of material in this system will be:

Accepted Answer

The magnitude of a physical quantity remains constant regardless of the system of units used. Thus, $n_1 u_1 = n_2 u_2$.
The dimensional formula for density is $[M L^{-3}]$.
We have $n_1 = 4$, $M_1 = 1	ext{ g}$, $L_1 = 1	ext{ cm}$.
In the new system, $M_2 = 100	ext{ g}$, $L_2 = 10	ext{ cm}$.
Using the relation $n_2 = n_1 \left[\frac{M_1}{M_2}ight]^a \left[\frac{L_1}{L_2}ight]^b$, where $a = 1$ and $b = -3$:
$n_2 = 4 	imes \left[\frac{1	ext{ g}}{100	ext{ g}}ight]^1 	imes \left[\frac{1	ext{ cm}}{10	ext{ cm}}ight]^{-3}$
$n_2 = 4 	imes \left(\frac{1}{100}ight) 	imes (10)^3 = 4 	imes \frac{1000}{100} = 40$.
Therefore, the value of density in the new system is $40$.

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