Question

:The density of a material in SI units is $128\,kg/{m^3}$ . In certain units in which the unit of length is 25 cm and the unit of mass is 50 g, the numerical value of density of the material is
A) 410
B) 640
C) 16
D) 40

Answer 1

Hint Use the formula of density to determine the relationship between density, mass, and volume. Take the ratio of density in the two unit systems to determine the new numerical value of density
$\Rightarrow Density = \dfrac{{Mass}}{{Volume}}$

Complete step by step answer
We’ve been given the density of a material in SI units as $128\,kg/{m^3}$. Now to find the value of density in the new scale, we first have to convert the SI units (kilogram) and (meter) into their respective units in the new system (gram) and (centimeters). So,
$\Rightarrow {M_1} = 1kg = 1000g$
$\Rightarrow {L_1} = 1m = 100cm$
And in the new unit system, we have
$\Rightarrow {M_2} = 50g$
$\Rightarrow {L_2} = 25\,cm$
We know that the dimensional formula of density $\rho = {M^1}{L^{ - 3}}$
So taking the ratio of the two densities, we get
$\Rightarrow \dfrac{{{\rho _2}}}{{{\rho _1}}} = \left( {\dfrac{{{M_2}}}{{{M_1}}}} \right){\left( {\dfrac{{{L_2}}}{{{L_1}}}} \right)^{ - 3}}$
Placing the values of ${\rho _1} = 128$, ${M_2} = 50g$, ${L_2} = 25\,cm$ , ${M_1} = 1000g$ , ${L_1} = 100cm$, we can calculate
$\Rightarrow {\rho _2} = 128 \times \left( {\dfrac{{1000}}{{50}}} \right){\left( {\dfrac{{100}}{{25}}} \right)^{ - 3}}$
$\Rightarrow {\rho _2} = 128 \times \dfrac{{20}}{{64}} $
$\therefore {\rho _2} = 40 $
Hence the density in the new system is 40 units which correspond to option (D).

Additional Information
The SI unit system is the standard unit of measurement consisting of 7 quantities which is a set of quantities from which any other quantity can be derived and used internationally as a common unit system. The units and their physical quantities in the SI unit system are the second for time, the meter for measurement of length, the kilogram for mass, the Ampere for electric current, the Kelvin for temperature, the mole for amount of substance, and the candela for luminous intensity.

Note
The dimensional formula of density can be calculated since we’ve been given that we’re using the SI unit system which uses length and mass as base quantities. We must be careful in converting the quantities in the SI unit to the units in the new system before taking the ratio of the densities.