## The Gravitation Constant
Newton had no estimate for its value, rather he worked in terms of ratios of forces but it was measured in a laboratory by the very odd Henry Cavendish about a century later. It is a fundamental constant of nature. It. Just. Is. There’s no deriving it. Were it different by a little, our whole universe would be very different.
>The gravitational force is very weak and characterized by a single constant of nature, $G$.
>**Gravitation Constant: big G**
$$G=6.67 \times 10^{-11}\text{ N m}^2\text{kg}^{-2}$$
> It's also that collection of constants in Equation 7.8:
$$G=\left(M_E \frac{4\pi^2}{k_E} \right) \nonumber $$
> **Wait.** What happened to $k_E$?
> **Glad you asked.** Notice that our use of $k_E$ gets swallowed up into $G$. But there's also $M_E$ in there. Were we talking about, say the Earth-Sun system, then there'd be a different $k_S$, but then the $M_S$ would be upstairs and their ratio would still be the same as for the Earth-Moon. So the little $k$'s were placeholders for a bigger idea.
In Equation (1) the force of attraction *on* 1 due to 2 is $F_{1,2}$ while the force of attraction *on* 2 due to 1 is $F_{2,1}$. From Newton’s Third law? They’re equal in Equation Equation \@ref(eq:universal): $F=F_{1,2}=F_{2,1}$.
```{admonition} Please answer Question 1 for points:
:class: danger
Space station weightlessness?
```
### Little *g* Again {#cos2gagain}
Now we can understand Galileo’s result from a modern point of view. With the Universal law of Gravitation and Newton’s Second law, the acceleration due to a gravitating body can be isolated from Newton’s rule by finding the "$a$” and the “$m$.” To see what I mean, look at this figure:
An apple sitting on top of the world.
Now you might be confused about the sudden rebranding of the acceleration of the Moon to the Earth when the Moon was brought to the Earth in Newton's imagination. What happened to little $g$??
```{admonition} Please study Example 3:
:class: warning
What about $g$ and $G$?
```
So in fact $g$ is just an approximate acceleration that just happens to be the value of the Universal Gravitational acceleration for objects near the earth.
It's where our weight comes from. The Earth attracts us with a force that’s $F=mg$, which is a effectively a constant—on the surface of the Earth. When you step on a scale, it pushes back and is calibrated to read back how much spring-force is required to balance your weight.
Look at this drawing. (Yes, I know that's not an apple tree. And yes, I know that trees don't grow gigantically on the North Pole. It's an analogy.)
An apple, Earth, and an airplane (which might have an apple on board).
```{admonition} Please study Example 4:
:class: warning
An apple in a tree is further from the center of the Earth than one on the ground:
For all practical purposes? Even in a tree, the acceleration has not changed by much:
$$
a(\text{in a tree}) = g.
$$
```
How far up do we have to go in order to see $g$ start to get smaller? Let's take a trip:
```{admonition} Please study Example 5:
:class: warning
What about as high as any of us will probably go?
```
```{admonition} Please answer Question 6 for points:
:class: danger
Astronauts' weight on the way to the Moon:
```
```{admonition} Please answer Question 7 for points:
:class: danger
Satellite in orbit?
```
```{admonition} Please answer Question 8 for points:
:class: danger
Sun's attraction to me:
```
### Teachable Moment: Was Galileo Wrong?
> **Wait.** We’ve been saying that Galileo showed that the acceleration due to gravity is a constant. Now you’re saying that it depends on how far away one is? Which is it?
> **Glad you asked.** Yes. Galileo’s $g$ is really not a constant, but it varies very little...even for large distances above the Earth. So for all practical purposes, we can consider it to be a constant.
Here we have a situation that’s going to repeat itself over and over in the history of physics. Galileo said that the acceleration due to gravity was constant. Then along came Newton who showed that this wasn’t right in the strictest sense: that the acceleration due to gravity varies as you move away from the gravitating object.
Was Galileo wrong?
If there ever was a Law of Nature, Gravitation was it. But then it was shown to be the case only in a restricted domain...in this case, when you’re close to the surface of the Earth. There’s really no circumstance that you or any of us (except for a handful of astronauts) will ever experience in which Galileo's conclusion was practically incorrect.
This is the essence of a model: a model makes predictions about nature, but with the caveat that the precision of that prediction depends on one's ability to measure.
The scenario runs like this: First, Theory A explains a feature of the world and establishes a fact of nature and a mathematical Model that uses it. Then along comes Theory B that shows that the facts and the models of Theory A are not universally applicable. Typically, the facts of Theory A and the models in Theory A are included in the facts and the models of Theory B *within a domain of experience that’s smaller than the domain that Theory A describes*. When this happens we’d say two things: first, Theory B is more inclusive than Theory A. It explains more about the universe and over a wider domain. And second, Theory A is still the case when applied to the restricted domain of experience that’s a subset of the domain of Theory B.
In this case, Newton’s theory (B) explains gravitation everywhere. Galileo’s theory explains gravitation only in the region near the surface of the Earth (A). We still happily—and reliably—use a constant $g$ in the design of any structure or vehicle, for example.
In the Enlightenment, and for centuries after, if there *ever* was a Law of Nature, Newton's Gravitation was it. It's basically where the capital "L" idea for physics originated.
But, Newton’s gravitation Law...will become a “Theory” in a few hundred years at the hand of Albert Einstein.