Question

The ratio of the values in SI units to the values in CGS units of density is:
(A). ${{10}^{3}}:1$
(B). ${{10}^{2}}:1$
(C). ${{10}^{-2}}:1$
(D). ${{10}^{-3}}:1$

Answer 1

In order to measure physical quantities, we use different unit systems. The SI system is the most important and widely used system while the CGS system is a smaller unit system than SI. When we convert between different systems, we multiply when going from a smaller to a larger system and divide when we go from a larger to a smaller system.

Formulae used:
$1kg=1000gm$
$1{{m}^{3}}=1000l$

Complete step by step solution:
Density of a body is defined as the mass content per unit volume. Its SI unit is $kg\,{{m}^{-3}}$.
SI system of units is the most widely used metric system in the world. It has been recognized as the standard system of units as it is followed in most countries.
The CGS system or units is also called the centimeter-gram-second system of units.
The CGS system is smaller than the SI system so, when we convert a value from CGS to SI, we divide.
For mass,
CGS unit- $gm$
SI unit- $kg$
$gm$and $kg$are related as-
$1kg=1000gm$ - (1)
For volume,
CGS unit- $l$
SI unit- ${{m}^{3}}$
$l$and ${{m}^{3}}$are related as-
$1{{m}^{3}}=1000l$ - (3)
Let ${{d}_{1}}=kg\,{{m}^{-3}}$and ${{d}_{2}}=gm\,c{{m}^{-3}}$
The ratio of $\dfrac{{{d}_{1}}}{{{d}_{2}}}$is-
From eq (2) and eq (3), we get,
$\begin{aligned} & \dfrac{{{d}_{1}}}{{{d}_{2}}}=\dfrac{kg\,{{m}^{-3}}}{gm\,c{{m}^{-3}}} \\\ & \Rightarrow \dfrac{{{d}_{1}}}{{{d}_{2}}}=\dfrac{1000gm\,\times {{(100cm)}^{-3}}}{gm\,c{{m}^{-3}}} \\\ & \dfrac{{{d}_{1}}}{{{d}_{2}}}={{10}^{-3}} \\\ & \therefore \dfrac{{{d}_{1}}}{{{d}_{2}}}=\dfrac{1}{{{10}^{3}}} \\\ \end{aligned}$

Therefore, the ratio between SI unit and CGS unit of density is $1:{{10}^{3}}$. Hence, the correct option is (D).

Note:
For different systems of units, there are some prefixes which are used to represent parts of a unit. For example- kilo represents ${{10}^{3}}$ parts, hecto represents ${{10}^{2}}$ parts, centi represents ${{10}^{-2}}$ etc. There are seven fundamental units in SI system from which the others units are derived, they are; mass (kilogram, $kg$), time (seconds,$s$), temperature (kelvin,$K$), distance (metres,$m$), current (ampere,$A$), amount of a substance (mole, $mol$), luminous intensity (candela,$Cd$).