----THE HANGING OF J.R.TOLKIEN----
The characters of J.R. Tolkien were furious with their author for never putting any sex scenes in his books, so to get revenge they decided to hang him.

 "He has us do all that silly stuff and screw around with rings and that," said an elderly Hobbit who was wiser than the rest, "hanging is too good for such rascals."

 The Hobbit search team found the fugitive author on a small planet where the air had no resistance and you could go into orbit skimming over the surface at 67 miles per hour. When they arrested him, he was modifying an old Chevy pickup truck to go into orbit.

 J. R. Tolkien stood bravely on the gallows with the noose around his neck.
 Please don't hang me, he said, I will write a new Hobbit book with plenty of wonderful sex scenes.

 The Hobbits considered his offer.

 And you get twenty percent of the movie rights, conceded the famous author.

 All right, said the wisest Hobbit, if you can tell us how much this planet would weigh in its own gravity, we will let you go. We will reconsider your case after the film release.

*****
the above is the problem. you can solve it if you feel like it.
then comes a little discussion. and afterwards I will work the exercise in case you did not, so it doesnt remain unanswered.

How would you weigh a planet in its own gravity anyway? Dig up a chunk which you think is a millionth of the planet, weigh that with giant scales sitting on the surface of the planet, and multiply by a million.

 or if a millionth is still too much, use a smaller fraction and multiply by more. Like weigh a billionth part of it, and then multiply by a billion.

 How much would the moon weigh in its own gravity? How much would Mars weigh, in its?

  How can Tolkien escape the Hobbit's clutches?
*****
solution: 67 mph is E-7
you just have to raise that to the fourth power (to get E-28) and multiply by 8pi
In natural force units, the weight of the planet in its own surface gravity is 8pi E-28

if you want a humanscale interpretation of that force, recall from the introduction page that the weight of
a 50kg sack of cement at the earth's surface is  E-40
so this weight force would be 8pi E12 times that one-----about 25 trillion times the sack of cement.

---------GYPSIES STEAL THE MOON---------

Some gypsies were roaming this part of the galaxy and on their way thru the solar system they stole the moon and replaced it by a black hole of the same mass

 Dont worry, said the gypsies (when the people complained) you will still have tides and everything will be the same because the black hole we put in has the same mass----1.7E31 natural units---as the moon.

 The people appointed John Baez the noted explorer to negotiate.

 Give us back our moon, Baez told the gypsies, this black hole you gave us will eventually evaporate. It is not a fair exchange, you have cheated us.

 The gypsies thought about it some, then they said "All right, we will give you one chance. If you can tell us how long the black hole will take to evaporate we will swap the moon in for the black hole and take our black hole away with us!"

 John baez cubed 1.7E31 to get 4.9E93

 then he multiplied that by (80/π) to get 125E93

 He knew gypsies like to use natural units so he said "1.25 x 10⁹⁵ time units."

 Very good, said one of the gypsies, but you are from Earth and measure time in minutes. Tell us the time in minutes so we can be sure you really understand.

 John Baez knew that E45 natural is 4.5 minutes, so he divided by E45, multiplied by 4.5 and said
 "5.6 x 10⁵⁰ minutes"

 All right said the gypsies and they swapped the familiar moon back into its place

------------WHICH PLANET HAS THE PARTY?---------------

An explorer approaches a giant planet with three large moons, each of a different temperature. According to native informants, there is a party in progress on one of the moons. The explorer flies by the nightside of each moon  to gauge the infrared heat radiation from each, and finds the thermal brightness of each,  as follows:

1.4E-117
1.9E-117
and 2.5E-117

The question is, what is the temperature of each moons and where is the party?

You multiply each number by 6 and take the fourth root (press square root twice)
well officially it is not 6,  what you are supposed to multiply each number by is 60/π²
but π-square is almost the same as ten, and 60/10 is six.

Let's do it the official way and multiply each moon's brightness by 60/π²

8.511 E-117
11.55 E-117
15.20 E-117

then we press the squareroot button twice and get the temps

0.960 E-29
1.037 E-29
1.110 E-29

The first is below freezing, the second is room temperature, and the third is the perfect temperature for a hot tub (104 on the Fahrenheit scale) but the wrong temperature for a party!

So the moral is that if it glow 1.9 E-117 in the infrared, it's the right moon.

----------FAST FOOD EXERCISE------
According to Defense Analyst Daniel Pinkwater, the earth is in danger of invasion by the Fat Men.
These are a space-faring race of creatures who plunder other planets for their fast food.

Their planet, planet Ornish, is deficient in basic resources like french fries, mayonaise, potato pancakes, take-out fried chicken, and sour cream, which has forced them into a life of nomadic piracy.

The Fat Men's spacewar uniforms consist of loud plaid sports-jackets, loafers, and eyeglasses with thick black plastic frames. They travel by the millions in a fleet of  troopships shaped like enormous bagels.

Professor Pinkwater fears that, before long, hordes of Fat Men will descend on earth and ravage our fast food outlets.

The defense analyst has calculated that one raid by the Fat Men could deplete the earth of a million pounds of its mayonaise. How much energy does this represent?
 ********
this is a very easy problem because if you just go into the kitchen and look on a jar of Best Foods Real Mayonaise it will say that a 13 gram serving has 90 food Calories.
One of our pounds is 434 grams so it contains almost exactly 3000 Calories. (A "pound" is just a nickname for E8 natural mass units in this context.)

Also to a reasonably close approximation, the natural unit of energy which is built into the universe (derived by setting the speed of light and other basic constants equal to one) is 100 thousand food Calories.

 So a million pounds of mayonaise, with its 3 billion Calories, represents 30 thousand natural energy units.

------THE CAT ENGINE OF THE SHIPS OF ORNISH----

The ships of Ornish are driven by Cat Motors
 which consume cats as fuel by converting each cat entirely into energy.

 The captain of a ten-million-pound troopship wishes to achieve a speed of E-3 (one thousandth of the speed of light, about 300 km/second) in order to depart a plundered system. Once in the clear he will enter warp and rendezvous with the rest of the pirate band. The initial change of velocity is accomplished by the ship's efficient photon drive.

 How many standard 10 pound cats must be converted?

 *********
 well clearly the mass of each cat is one billion mass units (10 pounds is 10 E8 units) so since c = 1 each cat yields one billion natural energy units.
 The ship mass is 10 million pounds (E15), so the momentum change is E15 x E-3 = E12 momentum units. This requires discharging a photon pulse with E12 energy units, which consumes 1000 cats.


-----------LOW ORBIT AND PLANET DENSITY-------------

An explorer visits a system in which there are three planets. He wants to get close in to take pictures with his digital camera, so he enters low orbit around all three in turn, finding each planet more delightful than the one before it. In his enthusiasm, while skimming around the third, he cannot resist calling you on the cellphone.
Hello, says the explorer, at each planet it took a different interval of time to go one radian in low orbit. It took
7E45
4E45
and 3E45
on planets A, B, and C respectively.
What do you make of this?

the formula for the density is D = 6/T²
you just divide 6 by the square of the radian time, so the three densities are
6/(49E90) = 1.224E-91
6/(16E90) = 3.75E-91
and 6/(9E90) = 6.66E-91

Planet A, you tell the explorer, has virtually the same density as water.
planet B, on the other hand, is slightly over 3 times the density of water---not terribly different in density from the earth's moon and suchline orbs.
planet C, however, is 5.4 times the density of water, quite close to earth itself!
Indeed, says the explorer, that is within experimental error. I believe I am just passing over Sausalito.

On the main  page introduction E45 natural time units was listed as lasting 4.5 minutes, so the radiantime for low orbit in the earth-like case, namely 3E45, becomes 3 x 4.5 = 13.5 minutes (multiply 13.5 minutes by 2pi to get the period) for simplicity we're neglecting atmosphere and considering idealized low orbits.

-------THE GREENISH STAR------

You have traveled to a distant planet to study with Yoda.
 The brightness of sunlight there is the same 5.7E-117 as at earth so much the same wildlife has evolved to grow there. In fact everything looks pretty similar to Marin County in Northern California.

 Yoda teaches you to see better, not only in normal colors but also in different wavelengths, like infrared and ultraviolet, and to be in harmony with the Force and stuff like that. The usual guru business.

 One night you have a dream of being on a strange planet where the sun appeared very small in the sky-----the angular diameter was only 1/350 radian, and the radius was of course half that: 1/700. After morning meditation, you tell Yoda about it and he asks what color the sunlight was, in the dream.
 "The average photon is green," you tell him, because the average sunlight photon energy was 10E-28 in your dream. Yoda's teaching has enabled you to judge the average energy and wavelength, just by looking at the light, which is pretty amazing and almost worth the trip on its own account.

 Then Yoda asks about the brightness of the sunlight on the planet in your dream, was it brighter or dimmer than the 5.7E-117 here?

 You remember perfectly how things looked in the dream but you want to calculate as a check, so you divide the color 10E-28 by 2.701 to get the temperature of the star's surface 3.7E-28 square twice multiply by the usual π²/60, divide by 700²
 It comes out 6.3E-117. So it is a little brighter there than what we have here, and on Yoda's planet, with our 5.7E-117 sunlight

--------------WHAT THE MOON THINKS IT WEIGHS---------------

The well-known category theorist Baez was once in low circular orbit around the moon in a small, but nicely appointed, spacecraft.
Wishing to get a good view of the terrain, Baez had put the craft in the lowest practical orbit, just skimming the tops of lunar ridges.
Happening to glance at the speedometer, he notices that his speed is 5.6 E-6  (5.6 millionths of the speed of light,  roughly comparable to 5.6 times the speed
of sound in the earth's atmosphere). What would the weight of the moon be in its own surface gravity?

Squaring 5.6E-6 twice gives E-21, multiplying by 8pi gives 25E-21.

------------FUEL AND FOOD------------------

as humanscale handles on the natural unit of energy, it is about E5 food Calories, E8 lab calories, and E28 "eekyoo". An eekyoo or electron-quartervolt (eQ)
is a surrogate small unit serving similar purposes as the conventional eevee (eV) except that it is about one quarter the size.  the energy of a green photon is typically around 10 eQ and the fuel value of a range of common fuels and foodstuffs is such that about 17 eQ is released for every oxygen molecule used in combustion or metabolism. To help remember this quantity of energy 17E-28, or 17 eQ, there is a small rhyme:

Whether it's to burn or eat,
the Oh-Two count will tell the heat.
On the average each Oh-Two
releases seventeen eekyoo.

--------SPEED OF SOUND ON TITAN----

Titan atmosphere is mainly nitrogen and the temperature is 0.33E-29 natural.
this compares with a typical temp on earth's surface 1.00E-29 (actually fairly close to reported global average)

What is the speed of sound? In natural units one divides the temperature by the mass of a nitrogen molecule, which is 28/(2.6E18)

speed²= 7/5  x T/(mass of N2) = 7/5 x (0.33E-29) x (2.6E18)/28 = 4.3 x E-13

the speed is therefore the squareroot  0.65 E-6.  A millionth of the speed of light is not uncommon for sound, above the cloud layer in the earth's atmosphere.
but 1.1E-6  is more usual at the earth's surface, so Titan sound is probably noticeably slower than what we're used to.

Here's a link to an ESA page of facts about Titan
http://www.esa.int/SPECIALS/Cassini-Huygens/SEMMF2HHZTD_0.html

-------GRAVITY ON TITAN-----

That ESA link says Titan radius is 0.4 earth's, and Titan mass is 1/45 of earth. this means surface gravity is around 13.8 percent of earth
What is Titan gravity in natural units terms?

As a crutch I remember an order of magnitude "gee" is E-50 natural acceleration units-----official earth surface gravity is 0.88E-50.
So  titan is 0.12E-50

------CONVECTION ON TITAN----

wind is the result of convection which happens when the temperature gradient (cooling off with altitude) exceeds the threshold gradient called "lapse-rate"

in natural units terms the lapse-rate gradient (rate of change in temp with altitude) is just 2/7  x wt of molecule

the weight of a nitrogen molecule = 0.12E-50 x 28/(2.6E18) = 1.29E-68
and 2/7 of that is 0.37E-68 = 3.7E-69
if you want a humanscale handle on that, E-32 on the temp scale is half a Fahrenheit degree and E37 on the distance scale is half a mile
so this is saying 3.7 fahrenhalfsteps per halfmile.  it is a very slight gradient, because of the weak gravity, by earth standards. so convection happens a lot and there is a lot of wind action.

what is the lapse rate in dry air, on earth?  In this case the molecule mass is average 29 (instead of 28) but that is about the same, what is different is the gravity, which is 0.88E-50.

------UNDERGRADUATE LIFE AT YALE-----

In an attempt to clear his head, Mr. B, the well known Yalie fratboy, drinks a solution containing one tenth of a pint of ethanol.   (in natural units a pint is E99------a cube one handbreadth E33 on a side). How many molecules of oxygen will be required to fully metabolize the ethanol? How much energy are we talking about (how many Calories)?

Roughly speaking a pint is a pound-----E99 volume of ethanol has a mass of E8. So the ethanol mass absorbed by the fratboy is E7.
an individual ethanol molecule mass is 46/(2.6E18).  therefore the fratboy has imbibed (2.6/46)E25 molecules.  Each of these requires 3 oxygen molecules for proper metabolism. therefore 3 x(2.6/46)E25  oxygens will be required, which is 1.7E24.

One may also wish to know the energy involved, which is 17E-28 x 1.7E24 = 29E-4 natural. Since a food Calorie is about E-5, that is some 290 food Calories.

------HOT TUB EXERCISE-----
Common dimensions for a hot tub are about meter radius and meter deep----3.14 cubic meters and about 3140 kilograms of water.  As a crutch, the "pound" like mass of 434 grams is E8 natural. The hot tub water mass is 7.235E11.  How many molecules of water is this?  How much energy is required to heat this amount of water from typical outdoor ambient temp 1.00E-29 to the perfect hot tub temp 1.11E-29?

the mass of a water molecule is 18/(2.6E18)----dividing the hot tub water mass by this gives 1.0E29.  To a good approximation there are E29 molecules in the tub.

the heat capacity of a water molecule in the ordinary liquid phase is about 9k, where k is boltzmann. But boltzmann is 1 in natural units. therefore the heat capacity is 9  per molecule and 9E29 for the whole tub. On the other hand the temperature interval is from 1.00E-29 to 1.11E-29-----which is 0.11E-29. One natural unit of energy is required to heat the tub.

-----SUNLIGHT IS HOW MANY OCTAVES HIGHER THAN A SOPRANO----

a nice inflammatory note in the soprano range is the D on the treble clef
and the frequency of that note is E-39 on the natural frequency scale.
(freqs. always given in angular format here, so that means E-39 radians per natural time unit).

The temperature of the sun's surface is 2E-28
which means that a fairly typical energy for a photon of sunlight is 2E-28
and such a photon has frequency 2E-28
(on the natural scales it is always the same number)

 HOW MANY OCTAVES are there between the soprano and sunlight?
 that is, about how many doublings between the two typical frequencies
 E-39 and 2E-28?
 ---answer---
looks like a factor of 2E11, so the number of doublings is one more than
log-base-two of E11.