Question

The density of brass is $8.89\,g\,c{m^{ - 3}}$ . What is the density in kilograms per cubic meter?

Answer 1

Hint:We simply have to convert the units from $g\,c{m^{ - 3}}$ to $kg\,{m^{ - 3}}$ for which we will need the conversion factors to convert g to kg and cm to m. We will express the density in the fractional form along with the units and then multiply with the conversion factors to get the answer. This is a part of dimensional analysis.

Complete step by step answer:
Given that the density of brass is $8.89\,g\,c{m^{ - 3}}$ , it means that in $1\,c{m^3}$ of volume occupied by the brass object, $8.89\,g$ is occupied by the brass metal and rest by air.The density along with the units can be written as $\dfrac{{8.89g}}{{c{m^3}}}$. The conversion factor for g to kg is $\dfrac{{1\,kg}}{{1000g}}$. Multiplying the density with the conversion factor, we have
$\dfrac{{8.89g}}{{c{m^3}}} \times \dfrac{{1\,kg}}{{1000g}}$

Now the conversion factor for cm to m is $\dfrac{{1m}}{{100\,cm}}$
So, the conversion factor for $c{m^3}$ to ${m^3}$ is ${(\dfrac{{1{m^3}}}{{100\,c{m^3}}})^3}$
Multiplying the density with the conversion factor, we have
$\dfrac{{8.89g}}{{c{m^3}}} \times \dfrac{{1\,kg}}{{1000g}} \times {(\dfrac{{100\,c{m^3}}}{{1\,{m^3}}})^3}$
Here the conversion factor will be reciprocated first and then multiplied since the quantity to be converted is in the denominator here.

Now cancelling the common units in the numerator and the denominator, we have,
$\dfrac{{8.89}}{1} \times \dfrac{{1\,kg}}{{1000}} \times \dfrac{{100 \times 100 \times 100\,}}{{1\,{m^3}}}$
Separating the units and the magnitude we have,
$\dfrac{{8.89 \times 100 \times 100 \times 100}}{{1000}} \times \dfrac{{\,kg}}{{{m^3}}}$
Further solving this we get,
$8.89 \times 1000 \times \dfrac{{\,kg}}{{{m^3}}} = 8890\,kg\,{m^{ - 3}}$

Hence,the density in kilograms per cubic meter is $ 8890\,kg\,{m^{ - 3}}$.

Note:In this question, we took the cube of the conversion factor for cm to m because to achieve volume, the unit of length has to be raised to a power of three. So, this can also be understood as multiplying each side with the conversion factor or taking the cube altogether. In any case, the dimensions of the physical quantity should not change while converting from one unit to another.