Question

The unit of relative density is ${M^x} {L^y} {T^z}$. Find the x+y+z.
(A). -2
(B). -1
(C). 0
(D). 2

Answer 1

The dimension formula for the physical quantity is expressed in terms of the power of the fundamental quantities. There are seven fundamental quantities: length, mass, time, temperature, current, amount of substance and the luminosity. The units of these quantities are fundamental units.

Complete step by step answer:
The density of the material is the ratio of the mass and the volume of the substance. The density of different materials is different. The SI unit of density is kg.m$^{-3}$.
The ratio of density of the substance with the ratio of the water is the relative density of the substance. The relative density of the substance is given as:
$\Rightarrow {{\rho }_{r}}=\dfrac{{{\rho }_{s}}}{{{\rho }_{w}}}$

Where, ${{\rho }_{s}}$ is the density of the substance and ${{\rho }_{w}}$ is the density of the water.

Thus, the relative density of the substance is the fraction of the density. The ratio of the same quantity is the unitless quantity. So, the relative density is the unitless and dimensionless quantity.
The dimension of the relative density is given as:
$\Rightarrow {{M}^{0}}{{L}^{0}}{{T}^{0}}$ … (I)

Given:
Dimension of the relative density is given as:
$\Rightarrow {{M}^{x}}{{L}^{y}}{{T}^{z}}$ … (II)
Comparing equation (I) and (II):
x=0
y=0
z=0
Thus,
x+y+z=0

Note:
The ratio of the same quantity is always unitless and dimensionless.
The unit of relative density is given as:
$\Rightarrow \dfrac{\text{kg}\cdot {{\text{m}}^{\text{-3}}}}{\text{g}\cdot \text{c}{{\text{m}}^{\text{-3}}}}$
Thus, the numerator and denominator cancel each other.