Question

If energy (E), force (F) and linear momentum (P) are fundamental quantities, then match the following and give the correct answer.

A	B
Physical quantity	Dimensional formula
a) Mass	d) \[{{\text{E}}^{0}}{{\text{F}}^{-1}}{{\text{p}}^{1}}\]
b) Length	e) \[{{\text{E}}^{-1}}{{\text{F}}^{0}}{{\text{p}}^{2}}\]
c) Time	f) \[{{\text{E}}^{1}}{{\text{F}}^{-1}}{{\text{p}}^{0}}\]

A. a-d, b-e, c-f
B. a-f, b-e, c-d
C. a-e, b-f, c-d,
D. a-e, b-d, c-f

Answer 1

Hint: The dimensional formulae are derived using the parameters mass, length and time. In this problem, we will derive the dimensional formula of the mass, length and the time using the dimensional formula of the energy, force and the linear momentum. So, we will be performing the reverse operation.

Complete step by step answer:
From given, we have the data,
The energy (E), force (F) and linear momentum (P) are fundamental quantities.
The dimensional formula of the energy is, \[\text{E}=[\text{M}{{\text{L}}^{2}}{{\text{T}}^{-2}}]\]
The dimensional formula of the force is, \[\text{F}=[\text{ML}{{\text{T}}^{-2}}]\]
The dimensional formula of the linear momentum is, \[\text{p}=[\text{ML}{{\text{T}}^{-1}}]\]
Consider the given options one by one and substitute the dimensional formulae of the energy, force and linear momentum.
The first option given is, \[{{\text{E}}^{0}}{{\text{F}}^{-1}}{{\text{p}}^{1}}\],
The energy is raised to 0, force is raised to -1 and linear momentum is raised to 1.
So, we have,
\[\begin{align}
& [{{\text{E}}^{0}}{{\text{F}}^{-1}}{{\text{p}}^{1}}]={{[\text{M}{{\text{L}}^{2}}{{\text{T}}^{-2}}]}^{0}}{{[\text{ML}{{\text{T}}^{-2}}]}^{-1}}{{[\text{ML}{{\text{T}}^{-1}}]}^{1}} \\
& \Rightarrow [{{\text{E}}^{0}}{{\text{F}}^{-1}}{{\text{p}}^{1}}]=[{{\text{M}}^{-1}}{{\text{L}}^{-1}}{{\text{T}}^{2}}][\text{ML}{{\text{T}}^{-1}}] \\
\end{align}\]
Therefore, upon further calculation, we get,
\[[{{\text{E}}^{0}}{{\text{F}}^{-1}}{{\text{p}}^{1}}]=[{{\text{M}}^{0}}{{\text{L}}^{0}}{{\text{T}}^{1}}]\]
So, the dimensional formula of the first option represents the dimensional formula of time.
Similarly, compute the other two given options.
The second option given is, \[{{\text{E}}^{-1}}{{\text{F}}^{0}}{{\text{p}}^{2}}\],
The energy is raised to -1, force is raised to 0 and linear momentum is raised to 2.
So, we have,
\[\begin{align}
[{{\text{E}}^{-1}}{{\text{F}}^{0}}{{\text{p}}^{2}}]={{[\text{M}{{\text{L}}^{2}}{{\text{T}}^{-2}}]}^{-1}}{{[\text{ML}{{\text{T}}^{-2}}]}^{0}}{{[\text{ML}{{\text{T}}^{-1}}]}^{2}} \\
&\Rightarrow [{{\text{E}}^{-1}}{{\text{F}}^{0}}{{\text{p}}^{2}}]=[{{\text{M}}^{=1}}{{\text{L}}^{-2}}{{\text{T}}^{2}}][{{\text{M}}^{2}}{{\text{L}}^{2}}{{\text{T}}^{-2}}] \\
\end{align}\]
Therefore, upon further calculation, we get,
\[[{{\text{E}}^{-1}}{{\text{F}}^{0}}{{\text{p}}^{2}}]=[{{\text{M}}^{1}}{{\text{L}}^{0}}{{\text{T}}^{0}}]\]
So, the dimensional formula of the first option represents the dimensional formula of mass.
Now compute the third option.
The third option given is, \[{{\text{E}}^{1}}{{\text{F}}^{-1}}{{\text{p}}^{0}}\],
The energy is raised to 1, force is raised to -1 and linear momentum is raised to 0.
So, we have,
\[\begin{align}
  & [{{\text{E}}^{1}}{{\text{F}}^{-1}}{{\text{p}}^{0}}]={{[\text{M}{{\text{L}}^{2}}{{\text{T}}^{-2}}]}^{1}}{{[\text{ML}{{\text{T}}^{-2}}]}^{-1}}{{[\text{ML}{{\text{T}}^{-1}}]}^{0}} \\
 & \Rightarrow [{{\text{E}}^{1}}{{\text{F}}^{-1}}{{\text{p}}^{0}}]=[{{\text{M}}^{1}}{{\text{L}}^{2}}{{\text{T}}^{-2}}][{{\text{M}}^{-1}}{{\text{L}}^{-1}}{{\text{T}}^{2}}] \\
\end{align}\]
Therefore, upon further calculation, we get,
\[[{{\text{E}}^{1}}{{\text{F}}^{-1}}{{\text{p}}^{0}}]=[{{\text{M}}^{0}}{{\text{L}}^{1}}{{\text{T}}^{0}}]\]
So, the dimensional formula of the first option represents the dimensional formula of length.

Therefore, the dimensional formula of the mass is, \[[\text{M}]=[{{\text{E}}^{-1}}{{\text{F}}^{0}}{{\text{p}}^{2}}]\]
Therefore, the dimensional formula of the length is,\[[\text{L}]=[\text{E}{}^{-1}{{\text{F}}^{0}}{{\text{p}}^{2}}]\]
Therefore, the dimensional formula of the time is, \[[\text{T}]=[{{\text{E}}^{0}}{{\text{F}}^{-1}}{{\text{p}}^{1}}]\]

If energy (E), force (F) and linear momentum (P) are fundamental quantities, then the correct match is a-e, b-f, c-d, thus, the option (C) is correct.

Note:
The things to be on your finger-tips for further information on solving these types of problems are: The units of the given parameters should be taken into consideration while solving the problem. The units are the source for computing the dimensional formulae.