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PostPosted: Mon Jul 25, 2016 2:16 pm 
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HI All,

I am trying to explain DD a question in CSSE 2008 - Q26.

The diagram shows a circle with circumference of 1cm being rolled around an equilateral
triangle with sides of length 1cm.
How many complete turns does the circle make as it rolls around the triangle (without
slipping) to its original starting position?

The Image has a Triangle with sides as 1cm and a circle, as in question, is of circumference as 1 cm.

The answer I could think of by default is '3 complete turns'.

Can someone explain any reason why it is not '3' ?

The answer sheet shows something else (which I will reveal after getting further answers from experts).

Thanks in advance,
Praveen


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PostPosted: Mon Jul 25, 2016 2:23 pm 
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We've had this question before - draw a triangle and roll a coin around the edge - what has to happen at the vertices (corrners)?


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PostPosted: Mon Jul 25, 2016 2:35 pm 
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I think I missed it or made a bad search in forum, which I did and couldnt find any.

' what has to happen at the vertices (corrners)?'
It rolls over, I suppose.

Since the ball touches the same tip of point once it completes on on side, it just rolls 3 times in total.


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PostPosted: Mon Jul 25, 2016 3:00 pm 
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Sorry to duplicate the post:

Got some answers from previous posts:
viewtopic.php?f=2&t=5383&p=53829

But I am not at all convinced. ( I am sorry but I am bit adamant on things that doesnt convince me :roll: )
Rotating round a corner, which is infinitesimally small, results in 1/3 extra length ?? But the point of contact on the tip of the triangle is a negligible value.

A circumference, by its definition, is length around. Whether I wrap it around like a circle, or a square or a single straight line, it doesn't matter but will remain same.

Now rolling a ball on top of another surface, with its contacts touching with infinitesimally small, has no difference in its length.


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PostPosted: Mon Jul 25, 2016 3:11 pm 
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I think I am getting the point now. The important word is 'complete turns'.
There are 2 parts in the problem.
(i) Length of sides vs circumference of circle - 3 sides so 3 whole turns
(ii) Now it is about the word 'Equilateral triangle'
Exterior angle of corner is '360-60' = 300 degrees.
3 corners = 3 * 300 = 900, which is 2 and 1/2 turn

But now I am getting total of 5 turns and 1/2 turn. As it asks for 'complete turns', now I can see 5 complete turns.

Even now, my answer is not 4 turns :cry:


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PostPosted: Mon Jul 25, 2016 3:33 pm 
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Try it with a coin marked with a dot so you can see what happens at the corner - the sum of exterior angles is 360 - the angle it turns around at every corner is 120 degrees. [The angle from going straight on to going down the next side]


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PostPosted: Tue Jul 26, 2016 9:58 am 
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This is a tricky question indeed. I do not have 2008 Maths Q paper, but good learning for me.


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PostPosted: Tue Jul 26, 2016 5:13 pm 
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such a great question! Albeit a bit of a trick.


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PostPosted: Wed Jul 27, 2016 5:32 pm 
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I wonder if this helps:
Imagine a string 3cm long round the edge of the triangle. You stick one end to the edge of the coin and start rolling it round, winding the string round it as it goes. When it reaches the top of the triangle it will not start rolling the next bit of string until it has been pivoted round 1/3 of a turn so it is in line with the next bit of string. All three corners add up to one whole turn.


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