You are a promising theoretical physicist who does not believe that gravity is a distinct fundamental force but is instead related to the other forces by an all-encompassing relativistic, quantum-mechanical theory. In particular, you do not believe that the universal gravitational constant $\mathbf{G}$ is really one of nature's elite set of fundamental constants. You believe that $\mathbf{G}$can be derived from more-basic constants: the fundamental constant of quantum mechanics, Planck's constant$\mathbf{h}$ ; the fundamental speed limiting the propagation of any force, the speed of light$\mathbf{c}$ ; and one other- a fundamental length $\mathbf{l}$, important to the one unified force. Using simply dimensional analysis, find a formula for $\mathbf{G}$, and then an order-of magnitude value for the fundamental length.

Universal Gravitational constant $\mathrm{G}=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$.

Planck’s constant $\mathrm{h}=6.626\times {10}^{-34}\text{J}\cdot \text{s}$

Velocity of light $\mathrm{C}=3\times {10}^8\text{\hspace{0.17em}m}/\text{s}$.

The units for universal gravitational constant $\mathbf{G}$are:

$\begin{array}{c}\frac{\mathbf{\text{N}}\mathbf{\cdot }{\mathbf{\text{m}}}^{\mathbf{2}}}{\mathbf{}{\mathbf{\text{kg}}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{\text{kg}}\mathbf{\cdot }\mathbf{\text{m}}\mathbf{/}{\mathbf{\text{s}}}^{\mathbf{2}}\mathbf{\cdot }{\mathbf{\text{m}}}^{\mathbf{2}}}{\mathbf{}{\mathbf{\text{kg}}}^{\mathbf{2}}}\\ \mathbf{=}\frac{{\mathbf{\text{m}}}^{\mathbf{3}}}{\mathbf{}\mathbf{\text{kg}}\mathbf{\cdot }{\mathbf{\text{s}}}^{\mathbf{2}}}\end{array}$

So, by dimensional analysis, it is$\left[\mathrm{G}\right]={\left[\mathrm{L}\right]}^3{\left[\mathrm{M}\right]}^{-1}{\left[\mathrm{T}\right]}^{-2}$.

The units for Planck const is shown below:

$\begin{array}{l}\text{J}\cdot \text{s}=(\text{N}\cdot \text{m})\cdot \text{s}\\ \text{J}\cdot \text{s}=((\text{kg}\cdot \text{m}/{\text{s}}^2)\cdot \text{m})\cdot \text{s}\\ \text{J}\cdot \text{s}=\text{kg}\cdot {\text{m}}^2/\text{s}\end{array}$

So, by dimensional analysis, it is $\left[\mathrm{h}\right]={\left[\mathrm{L}\right]}^2{\left[\mathrm{M}\right]}^1{\left[\mathrm{T}\right]}^{-1}$.

The dimensional analysis of speed of light is $\left[\mathrm{c}\right]={\left[\mathrm{L}\right]}^1{\left[\mathrm{T}\right]}^{-1}$.

The dimensional analysis of the fundamental length is $\left[l\right]={\left[L\right]}^1$.

So, obtain:

$\begin{array}{l}\left[G\right]={\left[h\right]}^x{\left[C\right]}^y{\left[I\right]}^z\\ \left[G\right]={\left({\left[L\right]}^2{\left[M\right]}^1{\left[T\right]}^{-1}\right)}^x{\left({\left[L\right]}^1{\left[T\right]}^{-1}\right)}^y{\left[L\right]}^Z\\ \left[G\right]={\left[L\right]}^{2X+y+Z}{\left[M\right]}^x{\left[T\right]}^{-x-y}\\ \left[G\right]={\left[L\right]}^3{\left[M\right]}^{-1}{\left[T\right]}^{-2}\end{array}$

The universal gravitational constant $G$ in terms of$h$ is $G=h^{-1}{c}^3{l}^2$.

Rearrange above equation for $I$.

$l=\sqrt{\frac{Gh}{c^3}}$

Substitute $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$ for role="math" localid="1660036687213" $G,6.626\times {10}^{-34}\text{J}\cdot \text{s}$ for $h$, and $3\times {10}^8\text{\hspace{0.17em}m}/\text{s}$ for $c$ as:

$\begin{array}{l}\mathrm{l}=\sqrt{\frac{(6.67\times {10}^{-11}\text{\hspace{0.17em}N}\cdot {\text{m}}^2/{\text{kg}}^2)(6.626\times {10}^{-34}\text{J}\cdot \text{s})}{{(3\times {10}^8\text{\hspace{0.17em}m}/\text{s})}^3}}\\ \mathrm{l}=4\times {10}^{-35}\text{m}\end{array}$

Thus, the fundamental length should be around$4\times {10}^{-35}\text{m}$.