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Suppose there in a new system of units energy (E) , density (d) and power (P) are taken as fundamental units, then the dimensional formula of universal gravitation constant G will be A. $\left[ {{E}^{-1}}{{d}^{-2}}{{P}^{2}} \right]$ B. $\left[ {{E}^{-2}}{{d}^{-1}}{{P}^{2}} \right]$C. $\left[ {{E}^{-2}}{{d}^{-1}}{{P}^{-1}} \right]$D. $\left[ {{E}^{2}}{{d}^{-2}}{{P}^{-2}} \right]$

Last updated date: 23rd Jul 2024
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Hint: In order to solve this question we have to know about the dimension of these fundamental units. We know the dimension of the energy ,density, power , gravitational constant are respectively , $[M{{L}^{2}}{{T}^{-2}}]$, $[M{{L}^{-3}}]$, $[M{{L}^{2}}{{T}^{-3}}]$ and $[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]$.

According to this question energy (E) , density (d) and power (P) are taken as fundamental units .
We know dimension of energy (E) =$[M{{L}^{2}}{{T}^{-2}}]$
density (d) =$[M{{L}^{-3}}]$
power (P) =$[M{{L}^{2}}{{T}^{-3}}]$
gravitational constant =$[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]$

Let us assume $[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]$ is equal to ${{E}^{a}}{{d}^{b}}{{P}^{c}}$ where the power of the dimensions are assuming respectively a,b,c.
We know that the dimension of both sides of the equation must be the same. Now, applying the concept of dimensions analysis –
\begin{align} & \Rightarrow [{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]={{E}^{a}}{{d}^{b}}{{P}^{c}} \\ & \Rightarrow [{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]={{\left[ [M{{L}^{2}}{{T}^{-2}}] \right]}^{a}}{{\left[ [M{{L}^{-3}}] \right]}^{b}}{{\left[ [M{{L}^{2}}{{T}^{-3}}] \right]}^{c}} \\ & \Rightarrow [{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]=\left[ {{M}^{a+b+c}}{{L}^{2a-3b+2c}}{{T}^{-2c-3c}} \right] \\ & a+b+c\Rightarrow -1 \\ & 2a-3b+2c\Rightarrow 3 \\ & -2a-3c\Rightarrow -2 \\ & a=-2 \\ & b=-1 \\ & c=2 \\ \end{align}
By comparing the power of both side of equation, we get –
\begin{align} & a+b+c\Rightarrow -1 \\ & 2a-3b+2c\Rightarrow 3 \\ & -2a-3c\Rightarrow -2 \\ \end{align}
Therefore , the power of gravitational constant are-
\begin{align} & a=-2 \\ & b=-1 \\ & c=2 \\ \end{align}
So , we can write the dimensional formula of gravitational constant is option (C) $\left[ {{E}^{-2}}{{d}^{-1}}{{P}^{2}} \right]$

Note: Dimensional investigation is regularly used to check the believability of interred conditions and calculations. It likewise used to frame sensible speculations about complex actual circumstances that can be tried by exploring or by more created hypotheses of the wonder , and arrange kinds of physical quantity and units dependent on their relations.