How Good Is Your Intuition, 2? (Another Mind-Teaser) - Page 2

jyl · jyl's Garage

I mean, make a dot on the table under the center of the small coin. Then start rolling the small coin around the large coin. The center of the small coin will move away from the dot, travel in a circle, and eventually return to the dot.

wdfifteen · wdfifteen's Garage

Quote:

Originally Posted by jyl

But - the edge of the large coin is not a line segment. It is a circular arc that makes a full 360 degree loop. Thus the small coin makes 1 full rotation just to follow this arc. Even if the small coin were slid around the large coin's edge (the same point on the small coin's edge is always touching the large coin), it would make 1 full rotation by the time it returned to its original location.

3 + 1 = 4. The small coin makes 4 full rotations around its own center.

Arggghhh!! You baxtard!

Maffett · 01-19-2011, 05:46 AM

I think the answer is two times.

Sorry for the poor quality sketch. The contact point between the coins "travels" a distance of 6pi. The smaller coin has a circumference of 2pi. Therefore the smaller coin "rolls" three times during the trip. But point B only circumnavigates point A twice. It would circumnavigate three times if the trip had been in a straight line. But one doesn't happen because the distance traveled was a circle.

rick-l · 01-19-2011, 06:16 AM

Here is a practical application of that problem.

Say you want to test your inertial measurement unit while it is standing still by seeing how fast the earth is going around. You want to figure the rotation speed out and don't have a WGS sitting around. The earth goes around the sun 1 time per year, and rotates relative to the sun once per day, 365.25 times a year. It keeps coming out wrong.

The earth makes about 366.25 rotations per year. The once around the sun is another rotation.

jyl · jyl's Garage

Quote:

Originally Posted by Maffett

I think the answer is two times.

Sorry for the poor quality sketch. The contact point between the coins "travels" a distance of 6pi. The smaller coin has a circumference of 2pi. Therefore the smaller coin "rolls" three times during the trip. But point B only circumnavigates point A twice. It would circumnavigate three times if the trip had been in a straight line. But one doesn't happen because the distance traveled was a circle.

Good diagram, but it shows 4 rotations. The point on the small coin marked "B" starts at 3 o'clock (relative to the center of the small coin), it passes through 3 o'clock at 1/4 of the way around the large circle, again at 1/2 way, again at 3/4 way, and finally when it completes the trip - that is 4 times.

Maybe think about it this way. Suppose the large coin had radius zero, it is just a point. Rotate the small coin around that point - it makes 1 rotation even though the circumference of the large coin is zero. So you have to add 1 rotation to the 3 rotations that the small coin has to do just to travel around the circumference of the large coin.

MRM · 01-19-2011, 06:52 AM

Quote:

Originally Posted by jyl

I don't even want to know how you did the earth-wire problem . . .

Me and Zarniwoop have a artificial universes in our offices. I synthesized a spare earth to do the experiment. Getting the wire the right length and fused without a visible joint was the hard part. The quarter/penny experiment was much easier.

masraum · 01-19-2011, 06:56 AM

Quote:

Originally Posted by Maffett

I think the answer is two times.

Sorry for the poor quality sketch. The contact point between the coins "travels" a distance of 6pi. The smaller coin has a circumference of 2pi. Therefore the smaller coin "rolls" three times during the trip. But point B only circumnavigates point A twice. It would circumnavigate three times if the trip had been in a straight line. But one doesn't happen because the distance traveled was a circle.

Nice try, but as has been said, your drawing is missing a couple of important points. I made two circles in a 3:1 diameter ratio and worked this out. Your rotation is off. If you start at the top of the big circle, then the spot where the two circles touch will be back at the bottom of the small circle when it is at the 3 o'clock position on the big circle. Each full rotation is at 12, 3, 6, and 9 o'clock positions.

Maffett · 01-19-2011, 07:00 AM

Reply to jyl: Thanks for the correction - Four times.

island911 · 01-19-2011, 07:39 AM

More visualization...

note the pelican would be on its head again before the first third.

. . .starting on its head, and going clockwise..

DARISC · 08:25 AM

No visualization needed.

Three circumferences of small coin = one circimference of large coin.

Rolling small coin for three circumferences (three revolutions) brings it to the end of the circumference of the the large coin.

Rolling it one more circumference (revolution) puts it back in its original position, relative to the large coin.

Four revolutions is the answer.

murphyjp · 01-19-2011, 08:10 AM

what coins?

island911 · 01-19-2011, 08:15 AM

updated. visualization...

island911 · 01-19-2011, 08:29 AM

Quote:

Originally Posted by DARISC

Last edited by DARISC; Today at 09:25 AM..
No visualization needed.

Three circumferences of small coin = one circimference of large coin.

Rolling small coin for three circumferences (three revolutions) brings it to the end of the circumference of the the large coin.

Rolling it one more circumference (revolution) puts it back in its original position, relative to the large coin.

Four revolutions is the answer.

Such strong (post edit) confidence you have. What happened to your 'oops' post?

rick-l · 01-19-2011, 08:45 AM

The earth does go around 366 1/4 times each year.

jyl · jyl's Garage

Suppose the small coin is rolled along a path whose length is N-times the circumference of the small coin, but that path is not the edge of the large coin - it is some other squiggly line. What's the rule that lets us figure out how many rotations the small coin makes?

DARISC · 01-19-2011, 08:57 AM

Quote:

Originally Posted by island911

Such strong (post edit) confidence you have. What happened to your 'oops' post?

Immediately after posting it I noticed a typo and at the same time I heard my doorbell. It was the termite inspector arriving for his 9:00 appt.

I had to show him where my concern was and rather than leave the faulty post up I changed it to "oops", let the termite guy in, showed him where the droppings were and he told me it wasn't termites, it was ants. Didn't take very long. he left, and I reposted.

When I read the OP I typed out my answer as concisely and quickly (thus the typo) as I could. That took, oh, two minutes?

How much time did you spend playing with your graphic?

DARISC · 01-19-2011, 09:01 AM

Quote:

Originally Posted by jyl

Suppose the small coin is rolled along a path whose length is N-times the circumference of the small coin, but that path is not the edge of the large coin - it is some other squiggly line. What's the rule that lets us figure out how many rotations the small coin makes?

Rotations = length of line / circumference of coin.

oldE · 01-19-2011, 10:00 AM

Quote:

Originally Posted by DARISC

Rotations = length of line / circumference of coin.

Then you have to consider whether the net direction change of the line is in the direction of rotation of the coin or opposite it.
If in the same direction, add one turn for every 360 degree net change in orientation of the line or subtract one turn for every 360 degree net change on the opposite direction.

So for a straight line: exactly what Darisc said.
For a line such as the one above, subtract about one rotation.

Les

DARISC · 01-19-2011, 10:36 AM

Who can argue with that?