Using the force constant in equations

[marcus]

...Suppose we go scuba diving on this planet. how deep do you go to get an increased pressure of one atmosphere? (it's the limit of a suction well-pump that we learned about in middleschool)

the relevant arithmetic fact 1/8 is 12.5 percent, you know: 8 and 0.125 are reciprocals

and the density of water, in natural units, is 1.225E-91
or to put it more humanly, 1.225 pound/pint.

In the round number ocean, pressure rises by one atmoshere for every 8 paces you go down.

why? on this planet one standard atmosphere is E-106
and gravity is E-50.
I want a depth D such that
D x E-50 x 1.225E-91 = E-106
D x E-50 = 8E-16
D = 8E34 = 8 paces

a eucalyptus across the street is 40 paces tall (I paced it off as that yesterday---to a point where its elevation was half a rightangle) and 40 = 5 x 8, so today I pictured being in clear water up to the top of the eucalyptus, which I guess was some kind of giant seaweed they have there. I was tankdiving at a depth of 40E34 and the pressure was 5 atmospheres. Amazing conditions! Water so clear I could see the top of the eucalyptus.

 

[marcus]

A couple of posts back I mentioned that the force I exert on the ground is 2E-40.
Just went for a walk up the hill behind campus. Still open undeveloped land? it is fairly steep and you can do a 200 pace change of altitude (500 feet if you like feet). along some steep firetrails. you go up 200E34 = 2E36.

I am curious about my "wattage" on this climb. In natural units there is a natural unit of power which is pretty huge and E-49 of that is like a 160 watt lightbulb. So E-49 power can be visualized, and there is a nice bit of serendipity that E45 time units is 4.5 minutes.

It just happens that I can do this climb of 2E36 in a time of 4E45 (four of those 4.5 minute periods). So I calculate the rate I do lifting work as
the force of my weight x height/time

2E-40 x 2E36 /4E45 = E-49

not surprising, knew I could probably crank 160 watts on a stationary bicycle.have to go, back later

 

[marcus]

inputs for the round number planet
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year
E-4 is the planet's orbit speed around its sun

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

well I've been using myself as a guineapig and it seems to me that this is a nicely proportioned set of natural units. one can calculate all sorts of human and earthly things with considerable ease, or so it seems to me, once one gets the hang of it.

I would like to know something. Suppose someone reading this thread wants to test-drive these units and see how they work for them. What sorts of things would you need to know to get started?

And what would it occur to someone to calculate?

So if anything occurs to you as you read, constructive suggestions of that sort would be welcome.
here are a few more convenient humanscale handles on the units, which I mentioned earlier and have been using
------
time E45 = 4.5 minutes
length E34 = pace (32 inches, 81 cm)
length E37 = halfmile
force E-40 = 480 Newton, like usual weight of 50 kg.
energy E-5 = roughly one food Calorie
power E-49 = 160 watt bulb
mass E8 = about one pound
voltage E-28 = one quarter volt
angular frequency E-39 = D on treble staff.

 

[marcus]

I tried the classic problem of the airplane flying over the N geomagnetic pole, which asks what is the voltage difference between the wingtips.

Maybe I made a mistake. i got that the voltage difference (for a 100 foot wingspan plane) was very small, like 0.6 of a conventional volt.

worked in natural units it went this way
Speed E-6
Wingspan 4E35 (this is 40 paces, about 100 ft)
Vertical component of geomagn. field 6E-58

multiplying these together gives 24E-29 for the potential difference betw wingtips of the aircraft.

this is only a couple of quartervolt (it comes to 2.4E-28, so 2.4 which is
like 0.6 conventional volts.) I don't know or can't remember if that is about right.

the natural unit of charge used is the electron charge
the form of the Lorentz force equation adopted is
F = q( E + beta X B)
this means that the natural units of electric and magnetic fields are the same unit---can be thought of as voltage/distance or force/charge

we should have an exercise about lightning.
he're link with some background
http://www.weatherwise.org/qr/qry.lightningpower.html
http://www.space.com/scienceastronomy/lightning_backgrounder.html

 

[marcus]

time E45 = 4.5 minutes
length E34 = pace (32 inches, 81 cm)
length E37 = halfmile
force E-40 = 480 Newton, like usual weight of 50 kg.
energy E-5 = roughly one food Calorie
power E-49 = 160 watt bulb
mass E8 = about one pound
voltage E-28 = one quarter volt
angular frequency E-39 = D on treble staff
magnetic field E-57 = gauss
magnetic field E-53 = tesla

another unexpected coincidence that makes natural units potentially easier to use is the fact that a tesla is very close to E-53 (force units per electron charge)
here is how close, in case you are curious.
1 Tesla = 0.9974 E-53 natural = 1.00E-53
1 gauss = 0.9974 E-57 natural = 1.00E-57
if one rounds to two decimal accuracy the relevant factor is just one!

I haven't been bothering to show precision in this thread since we rarely if ever need it, but it is always available
natural energy unit = 3.9018E8 joule
natural charge = electron charge = 1.602176E-19 coulomb
1 conventional volt = 4.1062E-28 natural voltage units
1 meter = 1.2342E34 natural length units.
Enough digits! the upshot is that a magnetic field that registers as 1 Tesla on a metric gauge will read 1.00E-53 on the natural scale.

And my handbook's value of 0.58 gauss for the Earth magnetic field at the north pole converts directly to 0.58E-57 natural.

 

[marcus]

we should have an exercise about lightning.
he're link with some background
http://www.weatherwise.org/qr/qry.lightningpower.html
http://www.space.com/scienceastronomy/lightning_backgrounder.html

Every minute we have 6000 flashes of lightning (worldwide).
this is because the Fat Men of Ornish are angry with us for not providing enough sour cream with the potato pancakes or enough other junk food in which they delight.

Each bolt of lightning releases some 1 to 25 natural units of energy, according to background info.

The captain of an Ornish battle cruiser wishes to hurl a lightning bolt at Trenton New Jersey to express his dissatisfaction with their offering of Tartar sauce with the fried scallops. Scallops require enough Tartar sauce.

He sets the voltage to 1.4E-20 and the current to 1.4E-19.
What duration of flash should he choose if he wishes exactly 20 units of energy to be delivered upon the helpless city?

answer. the pulse of current should last E40 time units
E40 x 1.4E-20 x 1.4E-19 = 20

-------------
How many lightning bolts occur in E45 time units?

recall that E45 = 4.5 conventional minutes. Multiply by 6000.
27,000 bolts
-------------
If the average energy per flash is 10 natural units, the Ornish ships must be expending energy on the Earth at the rate of 270,000 units per E45 interval.
The Ornish ships are powered by Cat Engine which converts a standard 10 pound cat (E9 mass units) entirely to energy in accordance with the usual emceesquare proportion. Over what length of time is one Catsworth of energy expended on lightning?

answer: a catsworth is E9, a billion units. divide E9 by 270,000 to learn the number of E45 intervals required to consume one cat.
3700 of these intervals. multiply by 4.5 if you want to know minutes.

 

[marcus]

inputs for the round number planet, and other benchmarks
easy numbers:
E-29 is global avg. surface temp
E-50 is a "gee"
E-106 is a "bar"----standard air pressure
E33 is the planet's mass
E50 is the year
E-4 is the planet's orbit speed around its sun

hard numbers:
5.7E-117 is the brightness of sunlight.
1.2E-91 is the density of water.
2.6E18 protons make one mass unit.
-------------------

convenient handles on the units:
time E45 = 4.5 minutes
length E34 = pace (32 inches, 81 cm)
length E37 = halfmile
force E-40 = 480 Newton, like usual weight of 50 kg.
energy E-5 = roughly one food Calorie
power E-49 = 160 watt bulb
mass E8 = about one pound
voltage E-28 = one quarter volt
angular frequency E-39 = D on treble staff
magnetic field E-57 = gauss (earth's field is about half a gauss)

----------
what about cyclotron frequency of the proton, in a field of some strength B?

say it's the geomagnetic field somewhere on Earth where it's between 1/2 and 1/3 gauss. plenty of places like that! (only gets strong like 0.6 gauss near the poles)

let's say it is (1/2.6) gauss

now a proton will spiral around in that field at an angular frequency (radians per unit time) called "cyclotron frequency". the stronger the field the higher the frequency and this is a nice weak field so the frequency should be low. maybe even audible!

cyclotron frequency = qB/m where q=1, and m= 1/(2.6E18) and B = (1/2.6)E-57

cyclotron freq. = (1/2.6)E-57 x 2.6E18 = E-39

that is the D on the treble staff (one line from the top) a soprano note.

definitely audible, maybe some radio noise frequencies come from spiraling particles.

 

[marcus]

measuring a 1 Tesla field with a stirrup gauge

classic gauge is like a stirrup, you lower it down into the magn. field B and run a current in the crossbar of the stirrup, and measure the force it gets pulled.

the crossbar of the stirrup can have several parallel conductors, it can be a sector of a coil in other words, but for simplicity just think of one conductor with a lot of amps

E-24 natural current unit is 0.6 amps, so let's say we put E-23 current thru the stirrup (6 amps). And suppose the crossbar is handbreath long, or E33

and the field is one Tesla, which is E-53 natural

then what is the pull?

E-23 x E33 x E-53 = E-43

that is half a Newton. So if you put E-23 current across a tesla field, a conductor which is E33 long will experience half Newton force.

 

[marcus]

another unexpected thing. it turns out that at standard conditions of temperature and pressure (E-29 and E-106) the mass of air is
about 1 pound per cubic pace. I calculate it's 1.115 E8 mass units.

the number of molecules in (E34)³ volume

PV/T = n
E-106 x E102/E-29 = E25

the mass of E25 molecules, each 29/(2.6E18).
E25 x 29/(2.6E18) = 1.115E8, to humanize it, call it 1.1 "pound"

--------------
It was a crisp Fall day, the temperature was E-29.
The hills of rural Vermont had turned bright colors.
A dog and a goat wished to take a ride in a balloon, so they went
to the goat's barn and got out his hot-air balloon.

the mass of gear and passengers is 400 pounds----that is 4E10 natural.

the goat asked the dog, who was a physicist,
how much they would have to heat the air to get lift-off.

What is the volume of the balloon? said the dog.
8000 cubic paces, replied the goat, naturally that is 8E105.

Well, said the dog, who enjoyed off-the-cuff order-of-magnitude calculation, the mass of air at ambient conditions is about 8000 pounds.
We have to heat it by about 5 percent, to lighten it by 5 percent, which is the weight of us and our equipage.

Excellent said the goat, as he opened the propane valve and pressed the igniter. We will raise the temperature in the bag from 1.00E-29 to 1.05E-29. Being a Vermonter, I call that 25 Fahrenheit degrees----shouldn't take too long!

 

[marcus]

Lo Fat was a pirate in the South China Sea who practiced piracy in an environmentally sustainable manner. He and his men ate organic vegan food and their ship was powered by oars.

Lo Fat had a crew of blond well-muscled young Republican captives to row the vessel. He motivated them by giving lectures on dismantling social security and the graduated income tax. The pirate vessel cut swiftly through the waves, searching for merchant prey.

After many years of successful piracy, Lo Fat noticed that the air temperature was more often than not 1.10E-29---which is human body temperature and eventually makes a man long for nice chilly air.

So he decided to equip the captain's cabin on his ship with air conditioning.
You may recall picturing a power of E-49 natural as a 160 watt lightbulb, in which case you know what it means for the pirate deciding on a 5E-49 airconditioner model for his cabin.

Now because of his committment to sustainable piracy, and thorough-going rejection of fossil fuels, the AC unit had to be solar powered.

And because of 10 percent efficiency, the PV panel had to get 5E-48 of sunlight!

Now the problem is how big a panel does the comfort-seeking pirate require?

 

[marcus]

remember that the solar constant (sunlight power per unit area) is
5.7E-117
in the South China Sea let us suppose that 5E-117 is available during the hours of peak demand. comparing this to 5E-48 we see that the area must be E69. My goodness that is ten square paces (a square pace area is (E34)²)

 

[marcus]

... at standard conditions of temperature and pressure (E-29 and E-106) the mass of air is
about 1 pound per cubic pace. I calculate it's 1.115 E8 mass units.

the number of molecules in (E34)³ volume

PV/T = n
E-106 x E102/E-29 = E25

the mass of E25 molecules, each 29/(2.6E18).
E25 x 29/(2.6E18) = 1.115E8, to humanize it, call it 1.1 "pound"

--------------
It was a crisp Fall day, the temperature was E-29.
The hills of rural Vermont had turned bright colors.
A dog and a goat wished to take a ride in a balloon, so they went
to the goat's barn and got out his hot-air balloon.

the mass of gear and passengers is 400 pounds----that is 4E10 natural.

the goat asked the dog, who was a physicist,
how much they would have to heat the air to get lift-off.

What is the volume of the balloon? said the dog.
8000 cubic paces, replied the goat, naturally that is 8E105.

Well, said the dog, who enjoyed off-the-cuff order-of-magnitude calculation, the mass of air at ambient conditions is about 8000 pounds.
We have to heat it by about 5 percent, to lighten it by 5 percent, which is the weight of us and our equipage.

Excellent said the goat, as he opened the propane valve and pressed the igniter. We will raise the temperature in the bag from 1.00E-29 to 1.05E-29. Being a Vermonter, I call that 25 Fahrenheit degrees----shouldn't take too long!

an obvious followup concerns how much propane they are going to burn in the initial heating, to raise the temperature in the 8000 cubic pace bag by 0.05E-29---or 5E-31
we just calculated that the number of molecules in a cubic pace is E25
so we are talking about some 8000E25---or 8E28---molecules.

In natural units, heat capacities are generally easy because k=1, so for most metals it is around 3k per atom, for water 9k per molecule,...I will drop the k since it is one... for monatomic gasses at constant pressure 5/2
for biatomic gasses 7/2, per molecule always.

so the energy to heat the air in that balloon is about
(7/2) x 8E28 x 5E-31 = 140E-3 = 0.14 energy unit.

we just multiplied the delta-tee by the number of molecules by the heat capacity per molecule. In cases like this there is nothing to look up.

How much propane does the goat require, to get lift-off?

 

[marcus]

In answer to this inquiry, the dog recited a poem:

"whether it's to burn or eat
the Oh-Two count will tell the heat
on the average each Oh-Two
releases 17 eekyoo."

Now a propane is C3H8, and it uses 5 Oh-Twooz when it burns,
so it releases 85 eekyoo which is 85E-28 natural energy unit.
Moreover a propane is 36+8 = 44 baryons, and therefore a pound (mass E8) is 2.6E18 x E8 baryons. Accordingly the dog calculated that a pound is (2.6E26)/44 propanes and must therefore supply 85E-28 x 2.6E26 x (1/44) = 0.05 energy units.

To get off the ground, said the dog, we will need 0.14 unit of heat.
Behold! said he, we will burn 3 pounds of propane for lift-off!

That is fine, said the goat, who had experience in these matters: after that we will not need so much, because we will mostly just be keeping the air in the bag warm. The propane tank will be ample for our trip.

 

[marcus]

The captain of a million-pound scout vessel of the Ornish fleet is searching for planets rich in junk food for his men to plunder. As the ship comes out of warp, he discovers that it is bearing directly down on Atlantic City New Jersey at a speed of E-4.
With no time to spare he must order a photon pulse to cancel the ships momentum.

How many standard 10 pound cats will be consumed?

----
answer: a pound is E8 so the ship mass is E14 and the speed (30 km/second) is E-4, so the momentum to be canceled is E10 natural momentum units.
The light pulse with this momentum, directed at Atlantic City so as to avoid collision, delivers E10 natural energy units.

This will of course vaporize the famed vacation spot and some of the surrounded ocean. Of interest to the captain, however, is how many cats need to be converted to supply the energy.

the mass of a 10-pound cat is E9 and therefore, when converted in accordance to the emcee-square rule, the cat will yield E9 units of energy. Therefore 10 cats are needed from the ships fuel reserves to accomplish this maneuver.

 

[marcus]

It might be of interest to judge the effects of the scout ship's maneuver on the environs of Atlantic City. The boardwalk and casino Mecca is located on a narrow sandbar with water on both sides. For simplicity let us assume that the pulse of light misses the ciity and is entirely absorbed by the adjacent ocean.

the handbook figure for the latent heat of vaporization of water (2260 joules per gram) translates into one natural unit of energy vaporizing 399 pounds of water, let us say 400 pounds for round numbers. If we allow for some of the energy to go into preliminary heating then we can estimate that one unit is sufficient to vaporize 300 plus pounds.

We may estimate that ten billion units from the Ornish ship, on being absorbed, then vaporizes 3E12 pounds of water. It is clear why junk food pirates are generally looked on with disfavor.

 

[nightcleaner]

Hi Marcus.

I am very much enjoying your tour de force. But I wonder if you would address a question about natural units which has been bothering me.

The Planck mass, as I recall, is the mass that would be required, if compacted somehow into the size of a proton, for the creation of a mini-black hole. Wikipedia suggests this mass is about the mass of a small flea.

Wouldn't it be more in line with the other natural units if the basis of mass were made to be the amount of mass required, if compacted to form a black hole, into the volume of the Planck space? Then, one Emass would be the mass of one Evolume at the birth of a nascent black hole.

Just wondering what you would think.

Thanks for all this,
as well as for tickling the peach blossums,

nc

 

[marcus]

... mass were made to be the amount of mass required, if compacted to form a black hole, into the volume of the Planck space? Then, one Emass would be the mass of one Evolume at the birth of a nascent black hole.

Hi nc, good to hear from you. what you what to be the case IS almost the way things are with conventional Planck units except for a factor of 2. It is normal for physicists not to worry too much about "factors of order one" (that is to say like 2 or 1/2pi and suchlike smallish numerical factors) especially in these extreme situations.

So one can say that if the Planck mass were compressed down to a ball with radius equal to the Planck length then its own gravity (which increases the closer you get to centerpoint) would be so strong that it would take charge and collapse the mass to a black hole.

Actually, putting in the factor of two, one can say that as soon as Planck mass is compressed to a ball with radius TWICE Planck length then a black hole will result.

It might be only a fanatical perfectionist would want to change the Planck length to get rid of that factor of 2. If everybody could be satisfied with the extent that we already have harmony between Planck units and black holes, it would avoid unnecessary difficulty of trying to eliminate the last factors of pi and 2pi and 2 and so on.

In the variant of Planck units I am working with, the Schwarzschild radius of the mass unit is 1/(4pi) of the length unit. this factor of one over 4pi does not bother me at all. I am very happy with the other neat things that happen with this version of Planck units. Like the clean form of the Einstein equation

 

[marcus]

The goat was flying his balloon over central Vermont, with his friend the dog. the two admired the Fall colors.

At our house, said the dog pensively, when someone goes to put air in the tires it is called "weighing the family car".

This is because, the dog continued, passing the binoculars over to the goat, we know the footprint of a properly inflated tire is 3E66.

(here the dog held out his paws to show a square about 3 human palms in area) and the combined footprint on the pavement of all four tires is 12E66.
Moreover it is our practice to inflate the tires to 2.8E-106 natural pressure, according to the gauge.

Therefore, declared the dog, the weight of the car is discovered by multiplying these two numbers
2.8E-106 x 12E66 = 34E-40

the dog did not explain, but this force is the weight of a 3400 pound mass in E-50 gravity. That mass is 34E10 natural and one multiplies it by "gee" or E-50 to find the weight.

After that it was deemed proper to unpack the sandwiches: liverwurst for the dog and cucumber for the goat.

 

[marcus]

At our house, said the goat when each of the friends had finished his sandwich, it is our custom to drink a gin-and-tonic on hot afternoons, and the preparation of a gin-and-tonic is called "measuring the height of the clouds".

this is because we believe that a proper drink of that sort should be just cold enough to make the glass sweat. We add just enough ice to make that happen. The person making the drinks can always measure Delta-T, the difference between ambient and making the glass sweat.

So you know how cold the air must get for condensation to start, observed the dog, and of course you also know the rate temperature falls off with altitude.

Yes, agreed the goat, ballooning has at least taught me that. In these parts on days like this lapse rate is about 3E-68. So we just divide Delta-T by the lapse rate and it tells you how high you'd have to go for the air to be as cold as a gin-and-tonic. That is where the clouds form.

 

[marcus]

by a Planck accident, lapse rate 3E-68 = 30E-69 = 30E-32/E37 is thirty halfdegrees per half mile, and in humanly familiar terms that is simply thirty degrees per mile. The goat remembers this so that he always knows what sweaters and down jackets to take with him on a balloon flight. If the lapse rate is 3E-68, he knows it will be 30 degrees colder one mile up.

On a day when the gin-and-tonic temperature differential Delta-T is 30 degrees the clouds are a mile high.

To be pedantic about it, if Delta-T is 6E-31
and the lapse rate is 3E-68 we just divide
Delta-T/lapse rate = 6E-31/3E-68 = 2E37 = two halfmiles = height of clouds

there are some unstated assumptions, like convection is occurring and cumulus clouds are forming locally (not just drifting in) but these conditions are not uncommon in Vermont on warm summer days.

 

[marcus]

the gypsy's keepsake

Artem Starodubtsev had a brief passionate love affair with a gypsy girl while her tribe was visiting the solar system.
On the day she left, she gave Artem a memento to remember their time together:
it was a black hole with the same mass as the earth.
How wide was the black hole?

Imagine Artem holding it in his palm and compare the width of the black hole (in it's protective jacket) with the breadth of his palm. The hole's diameter is:
A. 1/20 of his palm
B. 1/5 of his palm
C. 1/2 the width of his palm
D. exactly the width of his palm

 

[marcus]

the Au Pair Girl business

Besides raiding planets for their junk food, the Men of Ornish run an au pair girl business.

The have arranged an exchange between Earth and a planet inhabited by air-breathing giant squid.

An Ornish troop transport has been diverted from raiding in order to transfer 1000 young Republican women from Iowa to the Squid planet, where they will care for cephalopod children and attend high school.

On its return the ship will bring 1000 girl squid to look after Earth children.

The ship is currently in geosynchronous orbit around earth, and the young Republicans have been beamed on board.

How fast is the ship going relative to Earth?

 

[marcus]

the cube of geosynchronous orbit speed

there are several ways to calculate the speed in synchronous orbit from the planet's mass (1.38E33) and rotational period (3.19E47)
to be brief, one way is the cube of the speed is the planet's mass divided by 4 times the period, assuming things are expressed in natural units

v³ = mass/(4 period) = 1.38E33/(1.28E48) = 1.08E-15

the cube root of that is about E-5, more precisely 1.02E-5

so the Ornish craft was going one 100 thousandth of the speed of light.

that is the same speed that communication satellites in synchronous orbit go as well, so as to remain over one spot on the equator.

The captain prepares the ship to clear Earth's gravitational field and enter warp. The soon-to-become au pair girls are happily discussing impending bankruptcy of national healthcare and consequent opportuntities for private enterprise.

 

[marcus]

the spherical mirror in Artem's hand

the gypsy has placed in Artem's hand a ball with a perfect mirror coating. this is the tight-fitting protective jacket surrounding a black hole with the mass of the earth

they have chosen a place on the outskirts of the solar system for their last meeting, but the protective jacket isolates the holes gravity and inertia from the outside world so that in any case planet orbit would not be perturbed by her gift. (love is like this, so momentous that it should turn planets from their tracks and yet does not)

expressed in natural terms, the diameter of a black hole is simply equal to the mass divided by 2pi.

the Earth's mass is 1.38E33 and dividing that by 2pi gives 0.2E33
or about one fifth the width of Artem's hand.

for a moment, they watch their two small reflections mirrored on the ball

 

[marcus]

how the giant squid heat their hot tubs

It is widely known that the proper temperature for the hot tub is 1.11E-29.

expressed like this in natural units, it is an eternal number, and will no doubt be remembered (and respected by hot tubbers) long after the metric system and other arbitrary human constructs are forgotten

what is not so well known is how the giant airbreathing squid heat their tubwater to this very temperature

they have a bed of encapslated black holes each of which is at hawking temperature 1.11E-29
the tight-fitting reflective jacket surrounding each small hole protects the outside world from its gravity and inertia, but let's the nice warmth of the hawkingradiation escape into the water. this brings the water to exactly the right temperature

what is the mass and diameter of a black hole whose temperature is thus?

 

[marcus]

although the formula for the hawking temperature of a mass M hole is rather complicated and hard to remember when written in conventional metric format, it is quite simple in natural terms.

the temperature is simply the reciprocal of the mass

so if one wants the temperature of the hole to be 1.11E-29 then one makes the mass of the hole be the reciprocal of 1.11E-29, which is
0.9E29 natural mass units.

to humanize this, since E8 mass units is a pound, this is 0.9E21 pounds.
which is why the squid encapsulate these small black holes in a protective jacket, isolating the effects of their gravity and inertia.

Remember that the diameter of a hole is 1/2pi times its mass, when we are using this system, as appeared in the story of Artem and the gypsy.
To find the diamter of each tiny micro-capsule, we divide 9E28 by 2pi and get some 1.4E28 natural length units.

(this is dust particle size, about one micron)

 

[marcus]

you are orbiting a small planet at a steady speed of 67 miles per hour, and after you've circled the planet 4 times the clock says you have been in orbit 7 and an half hours.

what is the mass of the planet?

 

[marcus]

4 x orbit period = 4 P = E47
speed = 67 mph = E-7

mass M = 4P v³ = E47 (E-7)³ = E(47-21) = E26


to put the planet mass E26 in more familiar terms call it E18 pounds, a quintillion pound planet.
Or since the Earth mass is around E33, perhaps think of it as E-7 (one ten millionth) of the mass of earth.
to get a handle on the time E47, think E45 = 4.5 minutes, so 100E45 is 100x4.5 = 450 minutes = 7 and 1/2 hours. this is just footnotes.
the main thing is that with any circular orbit you have

mass = 4 x period x speed³

 

[marcus]

Imagine a planet of so little mass that you can orbit close to the surface at only 6.7 miles per hour

Perhaps the exceptionally benign atmosphere offers no air resistance but is good to breathe and of a comfortable temperature. In that case you can orbit without using any kind of spacecraft ---in your street clothes.

Let us suppose you can smell plumblossom and magnolia as you orbit (just grazing the hilltops) seeing everything on the planet at the speed of a run.

You are invited to calculate the mass of this planet, using one additional piece of information: In an orbit with a steady speed of 6.7 mph it takes
one and 7/8 hours to go full circle around the planet.
Calculate the mass any units you please. I've stated it in common units so it should make no difference.

 

[marcus]

about this thread, and natural units

I am trying out these natural units----like ordinary Planck but with |F|=1 instead of the more usual |G|=1

They do seem to work better than conventional Planck and this is consistent with what I notice in Quantum Gravity research papers. Often I see a kappa ("gravitational constant") which is 8piG, which can be set to equal one to further simplify the equations.

the moment one sets
|F|= |c|=|hbar|=|k|=|e|=1
one has a fairly universal set of units and it is interesting to see what some familiar quantities come out to be.

Here are some rough sizes of familiar things expressed in the units

rough sizes:

pound E8
year E50
handbreadth E33
pace (32 inch) E34
halfmile E37
lightyear E50
food Calorie E-5
lab calorie E-8
quartervolt E-28
tesla E-53
green photon energy 10E-28
average Earth surface temp E-29
2/3 mph E-9
67 mph E-7
cold air speed of sound E-6
D on treble clef E-39
one "gee" acceleration E-50
weight of 50 kg sack of cement E-40
power of 160 watt bulb E-49

some constants (approx.):

reciprocal proton mass 2.6E18
electron mass 2.1E-22
Hubble time 1.6E60
Lambda 0.85 E-120
rho-Lambda 0.85 E-120
rho-crit (critical density) 1.16 E-120
more exact Earth year 1.1676 E50
more exact lightyear 1.1676 E50
avg Earth orbit speed E-4
earth mass 1.38 E33
earth radius 7.86 E40
sun mass 4.6 E38
solar surface temp 2.0E-28
sun core temp 5E-25
CMB temperature 9.6E-32
earth surface air pressure 1.4E-106
earth surface gravity 0.88E-50
fuel energy released by one O₂ 17E-28
density of water 1.225 E8/E99

timescale:

1/222 of a minute E42
4.5 minutes E45
As a handle on the natural timescale, imagine counting out loud rapidly at the rate of 222 counts a minute, each count is E42 natural time units. A thousand counts is 4 and 1/2 minutes. It just happens that one year is roughly E8 counts, or E50 natural.