For Spun Silver -- Was Thomas Aquinas wrong?

[Dee4707]

I still say math is a religion and should be banned from public schools because they FORCE people to believe that the numbers do exactly what they say they do

Toni, you are just so much fun!! If that happened, then our lovable Mathman wouldn't have a job. He would be homeless. :agree: :agree:

Dee

 

[Tonichelle]

Well... seeing as how I'm a math atheist and all...

 

[Mathman]

Perhaps. But there is, and always be, a huge difference between the Pythagorean theorem and God. You can prove the Pythagorean theorem to someone who doesn't believe it.

Here is my favorite proof of the Pythagorean Theorem (scroll down to the picture in Proof #9).

http://www.cut-the-knot.org/pythagoras/index.shtml

The blue square is c².

The two red squares are a² and b².

On the left, the big square is c² plus four triangles. On the right, the same square is a² plus b² plus four triangles. Thus c² = a² + b².

Now here is my favorite proof of the existence of God (Saint Anselm's original version of the Ontological Argument):

By defintion, God is that Being than which no greater can be conceived.

We can conceive of a God that exists.

A God that exists is greater than a God that doesn't exist.

Ergo, God exists.

(But don't ask Emmanuel Kant what he thought of this logic, LOL.)

 

[PolymerBob]

Now here is my favorite proof of the existence of God (Saint Anselm's original version of the Ontological Argument):

By defintion, God is that Being than which no greater can be conceived.

We can conceive of a God that exists.

A God that exists is greater than a God that doesn't exist.

Ergo, God exists.

I think I see.

By defintion, the infinite axel (an axel with an infinite number of rotations) is that jump than which no greater can be conceived.

We can conceive of an infinite axel that exists.

An infinite axel that exists is greater than one that doesn't exist.

Ergo, the infinite axel exists.

 

[Mathman]

We can conceive of an infinite axel that exists.

Can we?

 

[PolymerBob]

We can conceive of a God that exists.

If we can conceive of an infinite being, we can conceive of an infinite axel.

 

[sillylionlove]

Here's what I concieve....math is infinitly a pain in my butt and will indefinitly make me confused!!

 

[Mathman]

If we can conceive of an infinite being, we can conceive of an infinite axel.

Actually, although we are joking around here on this thread, the question of "what we think we can conceive of" brings us back to the original topic.

Thomas Aquinas could not conceive of a triangle whose angles did not add up to 180 degrees.

People in Saint Anslem's time had no problem convincing themselves that they could conceive of a God who had all the attributes usually assigned to him (omniscience, omnipotence, omnipresence, etc.) -- and one more attribute as well, existence.

Nowadays I thnk we are not so confident about what is or is not conceivable. A figure skater twirling in the air forever? -- OK, I can buy that. But from a forward edge take-off? Come on!

I think I think, therefore I think I am...I think.

 

[PolymerBob]

Here's what I concieve....math is infinitly a pain in my butt and will indefinitly make me confused!!

Actually, I've put together a nice little presentation on the energy requirement of Evan's quintuple jump. It relates the kenetic energy of the vertical jump with the rotational energy of the spin. Then I use a mathamatical construction known as a "derivative" to find the minimum energy requirement.

Um ............. well .............. uh .......... maybe I better not.

 

[Dee4707]

Here's what I concieve....math is infinitly a pain in my butt and will indefinitly make me confused!!

The best answer in this whole thread.

Dee

 

[Tonichelle]

Here's what I concieve....math is infinitly a pain in my butt and will indefinitly make me confused!!

Amen and amen and amen!

 

[sillylionlove]

This is for polymerbob and mathman

Lets say I take Thomas Aquinas to the top of a 82 story building with the 13th floor missing. I then take polymerbob and mathman to the the top of a 73 story building. In thier hands they have all theries of math and all books written by Thomas Aquinas. Then everyone falls off the buildings. The question is................................................were they pushed off the building by people who hate math and at what speed did someone's foot hit thier backside?

 

[PolymerBob]

OK, so everyone seems to love math so much, I may as well give it a shot.

Let’s say Evan is gliding over the ice in preparation for his quintuple jump. He will have some horizontal velocity Vx. For the sake of discussion, let’s say this remains constant. Evan takes off, at which time two things happen simultaneously. He jumps up with a certain vertical velocity Vy, and starts rotating with a certain angular velocity, ω (omega), measured in radians per second. ( 2 π radians = 6.283 radians = 360 degrees )

The kinetic energy of Evan’s vertical jump is ½ m Vy^2, where m is his mass. The energy of his rotation is ½ I ω^2, where “I” is Evan’s “moment of inertia”. This quantity is the difficulty with which an object can be rotated. When a skater pulls in his arms and twists his legs close to each other, he is reducing his moment of inertia, or making it easier to spin.

So the total energy requirement is the sum of these two energies:

E = ½ m Vy^2 + ½ I ω^2

………………… to be continued ……………………

 

[Mathman]

I can help! Evan is listed at 6'11" and 160 pounds. (His height may not be relevant, but his wingspan should be about the same and might help us to estimate "I.")

 

[PolymerBob]

I can help! Evan is listed at 6'11" and 160 pounds.

Wow! 6'11", he could play for the Chicago Bulls.

OK, he's actually 6'1", or 1.854 meters tall. 160 lb is 72.73 kg. I will say his density is about that of water, giving him a volume of 0.07273 m^3.

I will estimate his shape as rectangular box roughly three times as wide as thick. This a very rough calculation of course, but I've little else to go on. This puts his dimensions at 1.854 m x 0.114 m x 0.344 m . His moment of inertia can then be calculated as

I = m / 12 * ( thickness^2 + width^2 ) =
= 72.73 kg / 12 * ( (0.114 m)^2 + (0.344 m)^2) = 0.796 kg m^2

................... to be continued ...............

 

[PolymerBob]

So the total energy requirement is the sum of these two energies:

E = ½ m Vy^2 + ½ I ω^2

So we see that the sum of the energies Evan needs to do his jump is the sum of the translational kenetic energy ( Et = ½ m Vy^2 ) and the rotational kenetic energy ( Er = ½ I ω^2 ). What we need to do is get these 2 energy terms as functions of the same variable. The variable we will use it time, t.

Sparing you all the boring derivations: when an object is thrown up into the air with a velocity Vy, the time it takes to strike the ground is :

t = 2 Vy / g ............ where g is the acceleration of gravity, 9.81 m / s^2.

This is the time Evan spends in the air. A little rearrangement :

Vy = 1/2 g t

Plug this back into the equation for translational energy Et = ½ m Vy^2 and we get :

Et = 1/8 m g^2 t^2

Now, what is the angular velocity, ω ? It is the number of radians Evan turns divided by the number of seconds he is in the air. If N is the number of turns :

ω = 2 π N / t

Plugging this into the equation for rotational energy : Er = ½ I ω^2 we get :

Er = 2 I π^2 N^2 / t^2 ........... so the total energy is Et + Er

E = Et + Er = 1/8 m g^2 t^2 + 2 I π^2 N^2 / t^2 ......... to be continued ....

 

[PolymerBob]

E = 1/8 m g^2 t^2 + 2 I π^2 N^2 / t^2

To make it easier to see what's going on, let's group a few constants.

A = 1/8 m g^2 ............... and ................... B = 2 I π^2 N^2

So .......... E = A t^2 + B / t^2

As we can see, total energy is the sum of 2 terms. One is proportional to time squared, the other is inversely proportional to time squared. This means the total energy requirement of Evan's jump will go to infinity as his time in the air gets very small, or gets very large. Does this make sense?

As t → 0, it means Evan is making very tiny jumps. If he only jumps a few millimeters or so high, he would spend only a tiny fraction of a second in the air. This requires him to spin at thousands of RPM to complete his rotations, so the required energy goes to infinity, or E → ∞ . As t → ∞, it means Evan is jumping thousands of feet in the air. He will rotate very slowly since he has all the time in the world to complete his rotations, but required energy once again goes to infinity.

OK, so if the energy requirement for Evan's jump goes to infinity as his time in the air goes to either zero, or infinity, this implies there is some intermediate time where the energy requirement will be a minimum.

Wait a second ............ sniff, sniff, .............. Do I smell a derivative?

 

[Lanie]

skimming this thread made my dumb brain explode.

 

[PolymerBob]

skimming this thread made my dumb brain explode.

That's OK, Lanie. Other than Mathman, who is a real expert at this stuff, I don't think too many people are following this. But the final results should be pretty understandable, even to those who don't get all the derivations.

So we have our equation for the total energy of Evan’s jump :

E = A t^2 + B / t^2 ……….. where A and B are groups of constants. We know that the required energy goes to infinity as the time in the air ( t ) goes to zero or infinity. So we must find the value of t than requires the minimum energy. To find this time, we use what is known as a “derivative”. This is a function that relates how another function changes over time.

So if E = A t^2 + B / t^2 …. then ….. dE/dt = 2 A t – 2 B / t^3 ….. Now we find the value of t that makes dE/dt = 0.

dE/dt = 0 = 2 A t – 2 B / t^3 …. so …. 2 A t = 2 B / t^3 ... A t = B / t^3…. A t^4 = B

t^4 = B / A ……. therefore ………. t = ( B / A ) ^ (1/4)

Plugging in terms for A and B …… t = ( 16 I π^2 N^2 / m g^2 ) ^ (1/4)

This is the time Evan must spend in the air to complete his 5 rotations with a minimum expenditure of energy. Any more or less time, and the energy requirement goes up.

Plugging in the actual values, time ( t ) = 0.8186 seconds. From this, all the other unknowns can be calculated.

Evan’s vertical speed at take off, Vy = ½ g t = 4.015 m/s
Maximum height of jump, h = Vy^2 / 2 g = 0.822 meters
Angular velocity, ω = 2 π N / t = 38.4 radians per second or 366 RPM.

But what we’re really interested is the energy. Recall the equation as a function of time :

E = Et + Er = 1/8 m g^2 t^2 + 2 I π^2 N^2 / t^2 ………… plugging actual values

E = 586.2 J + 586.2 J = 1172.4 J

What is interesting here is that when Evan does his quintuple jump at minimum energy, the energy of his vertical jump (Et) exactly equals the energy of his rotation (Er).

One more point, then I’ll shut up. How much more energy is required for a quintuple than a quadruple? Doing all the math for N = 4, we get an energy requirement of 938.0 J, which is an increase of 25.0 %, which exactly matches the increase in number of jumps. Finally, we get an intuitive result.

 

[Mathman]

I applaud!

But here is the truly amazing part.

Evan goes off his training diet and eats a jelly doughnut, 300 calories.

How many quintuple jumps will he have to do to work it off?

(1 "calorie" (actually, a kilocalorie in scientific language) = 4187 joules.)