Mathematical question

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Thread man

Thread man

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I've always regarded myself as being good at geometry (and trigonometry) and I suppose you have to be working with threads. However, and to my surprise, I got the wrong answer to this and I even regarded it as simple!

Use your head or pencil and paper but don't cut anything out. That'd be cheating ☺

How many revolutions of the small circle (R1) will it take to circle the circumference of the large circle (R3) to arrive back at the starting point?

Any answers?
 

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3. Circumference is Pi x D. As Pi is constant then it's simply Diameter 6 divided by Diameter 2. Same rules for cam gears on a four stroke which are 2:1 ratio on pitch diameter and number of teeth.
 
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TonyM said:
3. Circumference is Pi x D. As Pi is constant then it's simply Diameter 6 divided by Diameter 2. Same rules for cam gears on a four stroke which are 2:1 ratio on pitch diameter and number of teeth.
Click to expand...
I though so too but that's wrong.
 
stevehuckss396 said:
Why is that wrong? In the example r1 has a Dia of 2 and r3 has a Dia of 6 so 6/2 = 3. 3 revs and your back. Tony's answer sounds correct to me. What am I missing?
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You're missing the same as I missed. Take two identical coins and have them touch. Turn one of them around the circumference until back to the same point as you started. You'll find what should have taken one turn takes two. Regardless of the two circle diameters always add a 1.

To be honest I'm still trying to figure out why.

Dieselpilot and kvom got it right but didn't do as asked. If it were an exam I'd have failed them both for their "non answers" ;)
 
If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.

We as "engine guys" will probably never look at the problem and not imagine both circles rotating and maintaining there position.
 
stevehuckss396 said:
If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.

We as "engine guys" will probably never look at the problem and not imagine both circles rotating and maintaining there position.
Click to expand...
When I saw it on Youtube I didn't get the right answer and I am an engineer that's pretty good at geometry and trigonometry. I'm guessing I shouldn't post any more tricky questions (at the time I regarded is as interesting rather than insulting to"engine guys" ) if there are other "touchy feely" types like yourself.

Re "engine guys" then among the various jobs I had before I started my own company, I was Quality Engineer for a Danish company making parts for the F-16. The contract at the time stated that the Danish company would make half the number. Ended up that the Danish company made them all.
 
stevehuckss396 said:
If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.

My first reaction was "That's the answer" until, after reading again, it could be understood that the correct answer would be the "calculated" rotations multiplied by 2. It is in fact always the "calculated" result plus 1.
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stevehuckss396 said:
If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.

We as "engine guys" will probably never look at the problem and not imagine both circles rotating and maintaining there position.
Click to expand...
When I first read that I thought "That's the reason". I then thought that it could be read as the correct answer for the number of rotations necessary was twice that of the "calculated". It is in fact always the "calculated" (calculatted from the circumference distance) plus 1 regardless of the two circle diameters.

I don't remember at any time referring to the two circles as gears and neither does the Youtube video. It was to me a simple mathematical question that turned out to be harder than I thought LOL

I showed it to my wife and she said "It should be 3 but I feel it is more than 3". I've got a clever wife :oops: At least I think so.
 
I think what is tripping me up is that the smaller circle makes a full rotation when compared to horizontal but not along the surface of the circle. Make a small line where the two circles meet and then turn the small circle until the line touches the surface of the large circle and it will be 3 turns to make the trip. It's the fact that the small circle is displaced from the starting point that throws the brain off. Kinda like a rod in a radial engine has to be compensated so the pistons hit top dead center at the right time.
 
Interesting...Like steve h said, when circle A is one quarter of the way around circle B, it has rotated 360 degrees around it's own center but it's outer edge has not yet traveled 1 circumference distance around Circle B, that doesn't happen for another quarter turn in this case.
 
stevehuckss396 said:
If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.
Click to expand...

It's not really "that one needs to rotate twice to achieve the same as both rotating once".

The reason there's 1 additional rotation, is because the one that's moving is _rolling_around_a_circle_. If you linearize the circumference of the bottom circle, you get exactly the ratio of "diameters" that you would expect. When you roll it around a circle though, the path _itself_ contains one rotation, so you get the rolling rotations, plus the rotation of the path.
 
I haven't looked at anything and my answer is 4 - the smaller outer turns 3 revs against the inner but in so doing rotates an additional revolution.

Kind of like the plot kicker in "Around The World In 80 Days" - Fogg thinks he has lost the bet as he has seen 80 sunsets - but that was in 79 days (if my memory serves me correctly).

If you like geometric puzzles, here's one called Curry's Triangle :-
CURRYTRI-Model.jpg

Curry originally presented this to his lecturer as a cardboard cut-out - painted red one side and blue the other - presenting the red side first and then turning it over to the revised pattern which has a 2 unit "hole" in it - he told his lecturer that this proved a triangle had a different area when viewed from the front versus the back.

The triangle is 12 units tall and 10 units wide and therefore has an area of 60 square units - but the "blue" version is only 58 as two have gone missing.

As a curiosity arranging the bits into rectangle (R.H. image) occupies only 59 units.

Attached a printable version if you want to make the cardboard cut-outs and drive yourself nuts.

P.S. If you know the answer keep it to yourself for a while at least (it drove me nuts when I first came across it).

Regards, Ken
 

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dieselpilot said:
To lead the horse to water. What fun is a thread that's over in 3 posts?
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I'm assuming you know the rest of your quote? I didn't realise that replying to threads was supposed to be "just" for fun. Up until now I had the impression this forum was about asking for help and helping when possible. One of us is in the wrong forum and if it's me then I'll just quit.

Here's a joke with realism.


A man goes into a pet shop to buy a parrot.
There are 4, each on a perch and with prices on each perch. First one 1,000, second one 2,000, third one 5,000 and the fourth 25,000.
The man asks the pet shop owner what the parrot at 1,000 can do.
“Speaks one language fluently” is the reply.
“Impressive”, say the man. “What can the one at 2,000 do?”
“Speaks two languages fluently” is the reply by the pet shop owner.
“Gosh” says the man. “And the one at 5,000?”
“Not only can it speak three languages fluently but it can operate a computer”, replies the pet shop owner.
“I hardly dare ask” says the man “but what about the one at 25,000?”
The pet shop owner leans forward and whispers in the man’s ear, “I’ve never seen it do anything or heard it say anything but the other three call it “BOSS”.
 
Kvom: A good thread!
A friend answered thus...
"There are a number of answers :

1) Mathematical
Assming nothing slips you will only get back to the starting point after N rotations of circle R1 ( assuming the larger cicle stays still ) where
Circumfrence of Circle R3 =2 pi R3
Circumfrence of circle R1 = 2 pi R1
N = 2 pi R3 divided by 2 pi R1 = R3 divided by R1

2) Philosophical
When both circles have reached a mutually beneficial understanding

3) Fatalistic
Who cares ?

4) Quantum
They have already reached the starting point but we cannot percieve it."

So I sent him the explanation for "4"... (having got the wrong answer myself).
Thanks for the fun!
K2
 
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