bonanova Posted October 20, 2008 Report Share Posted October 20, 2008 A quarter is glued to the table. A dime is moved around the quarter once, returning to its original position. As it moves, the dime remains in contact with the quarter, without slipping. How many revolutions about its center does the dime make? For our purposes a dime has half the diameter of a quarter. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 quarter have circumference of Pi*d dime has diameter of Pi*(1/2d) dime will travel Pi*d distance and has a circumference Pi*(1/2d) so therefore has half the cirucumference compared to the quarter. dime will travel about its centre twice? lol is this the solution or am i just randomly guessing? Quote Link to comment Share on other sites More sharing options...
0 dms172 Posted October 20, 2008 Report Share Posted October 20, 2008 it will make about 4 revolution Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 (edited) 3 revolutions. After going halfway across the quarter, the dime has made a whole revolution, compared to the quarter. Since it is now on the other side, the total is 1.5 revolutions. Do that twice... Edit: Maybe a picture will clarify it better... Edited October 20, 2008 by Dimchord Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 Edit: Maybe a picture will clarify it better... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 (edited) Okay I see now. The dime is revolving around the center of the quarter so it will need to revolve an extra time. Edited October 20, 2008 by Diloul Quote Link to comment Share on other sites More sharing options...
0 Prime Posted October 20, 2008 Report Share Posted October 20, 2008 Great animation! But it looks like the dime is slipping to me... I think the trick here is... I think it is 5. The circumference of the quarter is 4 times that of a dime. So the dime makes 4 full rolling revolutions. And it also rotates in another direction -- about its face. As dime goes around the quarter, it faces different directions. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 Awesome. Quote Link to comment Share on other sites More sharing options...
0 Prime Posted October 20, 2008 Report Share Posted October 20, 2008 Clarification for my previous post. The assumption I'm making that the dime is standing vertically as it is rolled around the quarter. If OP meant that the dime is flat on the table, then the question is: revolutions from dime's perspective or from an outside observers'? Quote Link to comment Share on other sites More sharing options...
0 Prime Posted October 20, 2008 Report Share Posted October 20, 2008 oops... Where did I get 4 times circumference? The OP says it is one half... So it is 3 revolutions for an outside observer. For vertical placement of the dime 2 rolling and one turning about its face. And 2 times from the dime's perspective. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 agreed with that last one prime. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 I think reaymond got it, and inlcuded the math to prove it. The OP said "revolutions about its center" so the single circuit the dime makes around the quarter's face is irrelevant. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 It's 3 rotations. The key thing to notice in his animation is that the original bottom of the dime only touches the quarter at exactly the top and exactly the bottom of the quarter during the rotation. Despite that, the original top of the dime faces up a total of 3 times. Each point on the dime's perimeter touches the quarter twice, but notice the position of the dime at the bottom of the quarter. The contact point is the original bottom, but now the contact point is aimed up. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 It's 3 rotations. The key thing to notice in his animation is that the original bottom of the dime only touches the quarter at exactly the top and exactly the bottom of the quarter during the rotation. Despite that, the original top of the dime faces up a total of 3 times. Each point on the dime's perimeter touches the quarter twice, but notice the position of the dime at the bottom of the quarter. The contact point is the original bottom, but now the contact point is aimed up. I agree with your description of Dimchord's animation. I don't believe the animation is correct. the perimeter of the quarter is Pi*D. The perimeter of the dime is Pi*.5 *D. If you start with both coins face up, the bottom of the dime is touching the top of the quarter. One revolution of the dime covers half the perimeter of the quarter. This brings the top of the dime to touching the bottom of the quarter. One more revolution of the dime is the other half of the perimeter of the quarter, bringing them back to the original position with the bottom of the dime touching the top of the quarter. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 I agree with your description of Dimchord's animation. I don't believe the animation is correct. the perimeter of the quarter is Pi*D. The perimeter of the dime is Pi*.5 *D. If you start with both coins face up, the bottom of the dime is touching the top of the quarter. One revolution of the dime covers half the perimeter of the quarter. This brings the top of the dime to touching the bottom of the quarter. One more revolution of the dime is the other half of the perimeter of the quarter, bringing them back to the original position with the bottom of the dime touching the top of the quarter. Your mistake is here:"One revolution of the dime covers half the perimeter of the quarter. This brings the top of the dime to touching the bottom of the quarter." Imagine you are standing on the dime facing the quarter at the start. This means you are looking at the bottom of the dime, and the top of the quarter. After 1/2 the way around the quarter, you are again looking at the quarter. To you, it looks like you have made exactly one revolution. In fact, though, you have made 1.5 revolutions. Why? Because you are now looking the exact opposite direction of where you were looking at the start. The bottom of the dime (where you're looking) is now facing the bottom of the quarter. You were looking down at the quarter, now you are looking up. Look at the animation again. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 since dime has half the radius its perimeter is also half. so two revolutions come from here but also it is moving around the quarter and that will add one more revolution. So, my answer is 3. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 20, 2008 Report Share Posted October 20, 2008 I agree with your description of Dimchord's animation. I don't believe the animation is correct. the perimeter of the quarter is Pi*D. The perimeter of the dime is Pi*.5 *D. If you start with both coins face up, the bottom of the dime is touching the top of the quarter. One revolution of the dime covers half the perimeter of the quarter. This brings the top of the dime to touching the bottom of the quarter. One more revolution of the dime is the other half of the perimeter of the quarter, bringing them back to the original position with the bottom of the dime touching the top of the quarter. The animation is correct... I'm not sure how you could agree with my description but not the animation. Here's another way to think about it. If you imagine a string that's the circumference of the quarter sitting on a table, and you roll the dime over it, the dime does 2 complete rotations. Now, if take that string and wrap it in a circle, the fact that the path around the string is round, the dime does 3 complete rotations - 2 for the length of the string, and one for the fact that it's circular. And a third way. Imagine you have two dimes. Start the bottom of dime 1 at the top of dime 2, and spin dime 1 around dime 2. Each point on dime 1 only touches one place on dime 2, because they have the same circumference. When dime 1 gets to the bottom of dime 2, the top of dime 1 is what's touching the bottom of dime 2. In other words, the top of dime 1 is facing up again! Dime 1 has done a full rotation, but has only gone half way around dime 2. (The general formula for the number of times a coin with circumference X must rotate to spin around a coin with circumference Y is Y/X + 1.) And a fourth way. Imagine the quarter and the dime are both stationary, but spinning together like gears -- in opposite directions. If the quarter makes 1 full rotation, the dime makes 2, right? That's 3 total rotations to bring the two coins back to their starting position. This brain teaser is just like that, except the dime does all the moving, so it has to do all 3 rotations by itself. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2008 Report Share Posted October 21, 2008 I can't believe how did you manage to convert a simple problem to a whole mess. Trick is that: The dime travel around the quarter, but the road is that: Draw a circle around the quarter, which has a radius of quarter + dime. This is also the circle which passing through the quarters centre. The way that quarter travels is this. And If you divide its length to quarters perimeter, it is 3. Quote Link to comment Share on other sites More sharing options...
Question
bonanova
A quarter is glued to the table.
A dime is moved around the quarter once, returning to its original position.
As it moves, the dime remains in contact with the quarter, without slipping.
How many revolutions about its center does the dime make?
For our purposes a dime has half the diameter of a quarter.
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