Question

The density of material in the CGS system of units is $4g/c{{m}^{3}}$. In a system of units in which unit of length is 10cm and unit of mass is 100g, the value of density of material will be
A. 0.04
B. 0.4
C. 40
D. 400

Answer 1

We are given the density of a material in the CGS system and also the unit of length and mass in some other system of units. To find the value of density in the new system we need to find the value of 1 unit of length and mass in the new system and equate it with density in the CGS system. Thus we will get the required solution.

Complete step by step answer:
In the question we are given the density of a material in the CGS system as $4g/c{{m}^{3}}$.
Therefore we can write, $\rho =4g/c{{m}^{3}}$, where '$\rho$’ is density.
In the question we are also given the unit of length as 10 cm and the unit of mass as 100g.
Therefore we can write,
$L=10cm$
From this we can find 1 cm as,
$\Rightarrow 1 cm=\dfrac{L}{10}$
$m=100g$
From this we will get 1 g as,
$\Rightarrow 1g=\dfrac{m}{100}$
We have the density as, $\rho =4g/c{{m}^{3}}$. We know that this means,
$\Rightarrow \rho =4\times \dfrac{1g}{1c{{m}^{3}}}$
From earlier calculations we have,
$1cm=\dfrac{L}{10}$, $\Rightarrow 1c{{m}^{3}}={{\left( \dfrac{L}{10} \right)}^{3}}$ and $1g=\dfrac{m}{100}$. By substituting these values in the equation for density, we will get
$\Rightarrow \rho =4\times \dfrac{\left( \dfrac{m}{100} \right)}{{{\left( \dfrac{L}{10} \right)}^{3}}}$
By solving this we will get,
$\Rightarrow \rho =\dfrac{\left( \dfrac{4}{100}\times m \right)}{\left( \dfrac{{{L}^{3}}}{1000} \right)}$
$\Rightarrow \rho =\left( \dfrac{4}{100}\times m \right)\times \left( \dfrac{1000}{{{L}^{3}}} \right)$
$\Rightarrow \rho =\left( 4\times m \right)\times \left( \dfrac{10}{{{L}^{3}}} \right)$
$\Rightarrow \rho =4\times 10\times \left( \dfrac{m}{{{L}^{3}}} \right)$
$\therefore \rho =40\times \left( \dfrac{m}{{{L}^{3}}} \right)$
We know that ‘m’ is with respect to ‘g’ and ‘L’ is with respect to ‘cm’. Therefore we will get the above equation as,
$\therefore \rho =40g/c{{m}^{3}}$
Therefore we get the value of density of the material as 40 units.

So, the correct answer is “Option C”.

Note:
A system of units is a collection of set units where certain units are selected as fundamental and the others will be derived from these fundamental units. The CGS system of units is the centimeter – gram – second system where the units of length, mass and time are centimeter, gram and seconds respectively. Some other systems of units are SI system, MKS system, FPS etc.