10.5. Now Think Big!

So during the plague, down on the farm in 1666 he used some incorrect numbers and was still working out the mathematics—inventing it as he went along. Even though he never published his results, he worked on the Moon idea off-and-on for years. But eventually, he started to think about the actual force that the Earth would exert on the Moon and the apple.

Now it’s model-building time with the crucial idea that the centripetal force that the Moon feels in its orbit is a “regular” acceleration. So with his Second law for the force that the Moon feels is:

  Pens out!

\[F=m_Ma_C\]

and using his derived centripetal acceleration from Equation 7.5, $\(a_C(\text{M}) = \frac{4\pi^2}{k_E}\frac{1}{D_M^2}\)$

we would find that the force of attraction by the Earth on the Moon is:

\[\begin{split} \begin{aligned} F_{\text{Moon due to Earth}} &= m_Ma_c \\ F\text{ tentatively } &=m_M\frac{4\pi^2}{k_E}\frac{1}{D_M^2} \end{aligned} \end{split}\]

But from his Third law, the Moon must exert an identical (in magnitude, but oppositely-directed) force on the Earth,

\[\begin{split} \begin{align*} |F_{\text{Moon due to Earth}}| &= |F_{\text{Earth due to Moon}}| \\ & \propto M_E\frac{4\pi^2}{k_E}\frac{1}{D_M^2} \end{align*} \end{split}\]

and the only way that can happen is if both forces are proportional to both masses:

\[ F_{\text{Moon due to Earth}} = F_{\text{Earth due to Moon}} = M_E m_M\frac{4\pi^2}{k_E}\frac{1}{D_M^2} \nonumber \]

Now let’s re-think the force on the Moon:

\[\begin{split} \begin{align} F_{\text{Moon due to Earth}} &= m_Ma_C \nonumber \\ F_{\text{Moon due to Earth}} &= m_M \left[\left(M_E \frac{4\pi^2}{k_E} \right) \frac{1}{D_M^2}\right] \end{align}\end{split}\]

in which everything in the square brackets is a constant…just a number.

The normal way of writing this is to take all of the constants in the smooth brackets in Equation @ref(eq:earthmoonUG) and give them a name: \(G\). And therein begins a long history of a constant of nature—still a fundamental constant: $\( F_\text{M-E}=F_\text{E-M}=G\frac{M_M m_E}{D_M^2} \)$