Question

Name the units of some physical quantities are given in List-I, and their dimensional formulae are given in List-II. Match the right pair of the lists.

List-I	List-II
(a) Pa-s	(e) $\left[ {{{\rm{L}}^2}{{\rm{T}}^{ - 2}}{{\rm{K}}^{ - 1}}} \right]$
(b) NmK-1	(f) $\left[ {{\rm{ML}}{{\rm{T}}^{ - 3}}{{\rm{K}}^{ - 1}}} \right]$
(c) Jkg-1k-1	(g) $\left[ {{\rm{M}}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}} \right]$
(d) Wm-1k-1	(h) $\left[ {{\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}{{\rm{K}}^{ - 1}}} \right]$

A) a-h, b-g, c-e, d-f
B) a-g, b-f, c-e, d-h
C) a-g, b-e, c-h, d-f
D) a-g, b-h, c-e, d-f

Answer 1

Hint: To solve this problem first we derived all dimensional formulas according to given units and found out which dimensional formula matches to the given units. Dimensional formulae defined as an expression for the unit of a physical quantity in terms of the fundamental quantities. The fundamental quantities are represented as mass (M), length (L), and time (T). A dimensional formula is expressed in terms of powers of $\left[ {{{\rm{M}}^a}{{\rm{L}}^b}{{\rm{T}}^c}} \right]$ which are called dimensions.

Complete step by step answer:
(a) Pa-s here Pa is the unit of pressure and second for time.
$ \Rightarrow \;\left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 2}}} \right]\;\left[ {{{\rm{T}}^1}} \right]\; = \;\left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 2 + 1}}} \right]\; = \;\left[ {{{\rm{M}}^1}{{\rm{L}}^{ - 1}}{{\rm{T}}^{ - 1}}} \right]$
(b) NmK^-1 here N is the unit of force, m for length and K is the kelvin unit of temperature.
$ \Rightarrow \;\left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 2}}} \right]\,\left[ {{{\rm{L}}^1}} \right]\;\left[ {{{\rm{K}}^{ - 1}}} \right]\; = \;\left[ {{{\rm{M}}^1}{{\rm{L}}^{1 + 1}}{{\rm{T}}^{ - 2}}{{\rm{K}}^{ - 1}}} \right]\; = \;\left[ {{{\rm{M}}^1}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}{{\rm{K}}^{ - 1}}} \right]$
(c) Jkg^-1K^-1 here J is the unit of energy, kg is the unit of mass, and K is the unit of temperature.
$ \Rightarrow \;\left[ {{{\rm{M}}^1}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}} \right]\,\left[ {{{\rm{M}}^{ - 1}}} \right]\;\left[ {{{\rm{K}}^{ - 1}}} \right]\; = \;\left[ {{{\rm{M}}^0}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}{{\rm{K}}^{ - 1}}} \right]$
(d) Wm^-1K^-1 Here W is the unit of power, m is length, and K is the unit of temperature.
$ \Rightarrow \;\left[ {{{\rm{M}}^1}{{\rm{L}}^2}{{\rm{T}}^{ - 3}}} \right]\;\left[ {{{\rm{L}}^{ - 1}}} \right]\;\left[ {{{\rm{K}}^{ - 1}}} \right]\, = \;\left[ {{{\rm{M}}^1}{{\rm{L}}^{2 - 1}}{{\rm{T}}^{ - 3}}{{\rm{K}}^{ - 1}}} \right]\;\; = \;\left[ {{{\rm{M}}^1}{{\rm{L}}^1}{{\rm{T}}^{ - 3}}{{\rm{K}}^{ - 1}}} \right]$

When we observe all these dimensional formulae, therefore the correct option is (D).

Additional information: The fundamental units of fundamental quantities form the basis from which various other quantities are derived. Mass, length, and time are most commonly used fundamental quantities so they must be specified in all dimensional formulas. All the dimensional formulae must be written in square brackets.

Note: There are some limitations of dimensional analysis are:
 i) This method cannot be applicable to a trigonometric, exponential, and logarithmic function.
 ii) If the physical quantities are dependent on more than three physical quantities than it is not easy to find the dimensional formula.