Question

If C the velocity of light, h Planck’s constant and G gravitational constant are taken as fundamental quantities, then the dimensional formula of mass is :
$
  {\text{A}}{\text{. }}{h^{\dfrac{{ - 1}}{2}}}{G^{\dfrac{{ - 1}}{2}}}{C^0} \\
  {\text{B}}{\text{. }}{h^{\dfrac{1}{2}}}{C^{\dfrac{1}{2}}}{G^{\dfrac{{ - 1}}{2}}} \\
  {\text{C}}{\text{. }}{h^{\dfrac{{ - 1}}{2}}}{C^{\dfrac{1}{2}}}{G^{\dfrac{{ - 1}}{2}}} \\
  {\text{D}}{\text{. }}{h^{\dfrac{{ - 1}}{2}}}{C^{\dfrac{{ - 1}}{2}}}{G^{\dfrac{{ - 1}}{2}}} \\
$

Answer 1

Hint: Every physical quantity can be expressed in terms of dimensions of fundamental quantities like mass, length, time, etc. If we know the dimensions of the three given quantities then we can express mass in terms of these quantities easily.

Detailed step by step solution:
Dimensional formula: A dimensional formula of a physical quantity is an expression which describes the dependence of that quantity on the fundamental quantities.
We are given three quantities: c = velocity of light, G=universal gravitational constant and h = Planck’s constant. Their dimensions are given as
$
  c = \left[ {{M^0}{L^1}{T^{ - 1}}} \right]{\text{ }}...{\text{(i)}} \\
  h = \left[ {{M^1}{L^2}{T^{ - 1}}} \right]{\text{ }}...{\text{(ii)}} \\
  G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]{\text{ }}...{\text{(iii)}} \\
$
From the first equation (i), we get the following expression
$
  \left[ {L{T^{ - 1}}} \right] = c \\
   \Rightarrow \left[ L \right] = c\left[ T \right]{\text{ }}...{\text{(iv)}} \\
$
Now, by using the equation (iv) in equation (ii), we get
$
  \left[ {M{L^2}{T^{ - 1}}} \right] = h \\
   \Rightarrow {c^2}\left[ {M{T^{2 - 1}}} \right] = h \\
   \Rightarrow \left[ {MT} \right] = \dfrac{h}{{{c^2}}} \\
   \Rightarrow \left[ M \right] = \dfrac{h}{{{c^2}}}\left[ {{T^{ - 1}}} \right]{\text{ }}...{\text{(v)}} \\
$
Now by using the equation (iv) and (v) in equation (iii), we get
$
  \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right] = G \\
  \dfrac{{{c^2}}}{h}\left[ T \right]{c^3}\left[ {{T^{3 - 2}}} \right] = G \\
  \left[ {{T^2}} \right] = \dfrac{{Gh}}{{{c^5}}} \\
  \left[ T \right] = \dfrac{{{G^{\dfrac{1}{2}}}{h^{\dfrac{1}{2}}}}}{{{c^{\dfrac{5}{2}}}}}{\text{ }}...{\text{(vi)}} \\
$
Using equation (vi) in equation (v), we get
$
  \left[ M \right] = \dfrac{h}{{{c^2}}}\left[ {{T^{ - 1}}} \right] = \dfrac{h}{{{c^2}}}\dfrac{{{c^{\dfrac{5}{2}}}}}{{{G^{\dfrac{1}{2}}}{h^{\dfrac{1}{2}}}}} \\
   = {G^{\dfrac{{ - 1}}{2}}}{h^{1 - \dfrac{1}{2}}}{c^{ - 2 + \dfrac{5}{2}}} \\
   = {G^{\dfrac{{ - 1}}{2}}}{h^{\dfrac{1}{2}}}{c^{\dfrac{1}{2}}} \\
$
Hence, the correct answer is option B.

Additional information:
All physical quantities can be expressed in terms of certain fundamental quantities. Following table contains the fundamental quantities and their units and dimensional notation respectively.

No.	Quantities	unit	Dimensional formula
1.	Length	metre (m)	$\left[ {{M^0}{L^1}{T^0}} \right]$
2.	Mass	kilogram (g)	$\left[ {{M^1}{L^0}{T^0}} \right]$
3.	Time	second (s)	$\left[ {{M^0}{L^0}{T^1}} \right]$
4.	Electric current	ampere (A)	$\left[ {{M^0}{L^0}{T^0}{A^1}} \right]$
5.	Temperature	kelvin [K]	$\left[ {{M^0}{L^0}{T^0}{K^1}} \right]$
6.	Amount of substance	mole [mol]	$\left[ {{M^0}{L^0}{T^0}mo{l^1}} \right]$
7.	Luminous intensity	candela [cd]	$\left[ {{M^0}{L^0}{T^0}C{d^1}} \right]$

Note:
1. Mass, length and time are most commonly encountered fundamental quantities so they must be specified in all dimensional formulas. The square bracket notation is used only for dimensional formulas.
2. A physical quantity of desired dimensions can be constructed from given quantities by using their dimensional formulas. The dimensional formulae of given quantities can be used to obtain relation between interdependent quantities.