Question

If density of kerosene in C.G.S is 0.8 gm/cm\[^3\] then density in S.I is
A 800kg/m$^3$
B 1000kg/m$^3$
C 600kg/m$^3$
D 1200kg/m$^3$

Answer 1

Hint: We use dimension to convert the unit of density from one system (here CGS) to another (SI).Then we use simple conversions like 1kg$ = 1000 gm$ and 1m$ = 100cm$.Now we can unitary method to convert the given quantity in SI unit.

Step by step answer: Density is defined as mass per unit volume.$density = \dfrac{{mass}}{{volume}}$.
Therefore the dimension of density is $ = \dfrac{{\left[ {mass} \right]}}{{\left[ {volume} \right]}} = \dfrac{{\left[ M \right]}}{{\left[ {{L^3}} \right]}} = \left[ {M{L^{ - 3}}} \right]$
Unit of given density is gm/cm$^3$ $ = (gm){(cm)^{ - 3}}$
Unit in which density is to be converted is kg/m$^3 = (kg){(m)^{ - 3}}$
Dividing both the units we get $\dfrac{{gm/c{m^3}}}{{kg/{m^{^3}}}} = \left( {\dfrac{{gm}}{{kg}}} \right){\left( {\dfrac{{cm}}{m}} \right)^{ - 3}}$
We know that 1kg$ = 1000gm$and 1m$ = 100cm$.
Substituting these values, we get
$\dfrac{{gm/c{m^3}}}{{kg/{m^3}}} = \left( {\dfrac{1}{{1000}}} \right){\left( {\dfrac{1}{{100}}} \right)^{ - 3}}$
$\Rightarrow \dfrac{1}{{1000}} \times {\left( {100} \right)^3} = 1000$
Therefore, 1gm/cm$^3 = 1000kg/{m^3}$
Using unitary method,
0.8gm/cm$^3 = 1000 \times 0.8kg/{m^3}$
$\therefore$ 0.8 gm/cm\[^3\] $ = 800kg/{m^3}$
Therefore the density in the SI unit is 800kg/m$^3$.

Hence the correct option is A.

Additional information: The study of the relationship between physical quantities with the help of dimensions and units of measurement is termed as dimensional analysis. Dimensional analysis is essential because it keeps the units the same, helping us perform mathematical calculations smoothly. The main idea in Dimensional Analysis is to create a conversion ratio (unit factor) which has the units you want in the numerator and the units you already have in the denominator. It may be necessary to multiply by more than one conversion ratio in more difficult problems.

Note: The study of the relationship between physical quantities with the help of their dimensions and units of measurement is called dimensional analysis. We use dimensional analysis to convert a unit from one form to another.