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Math Help - cog and wheel

  1. #1
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    cog and wheel

    Dear sir ,
    I would appreciate if anyone can show me the solution to the below question.
    thanks
    Kingman

    A cog-wheel of 12 teeth rotates on its axis round a fixed cog-wheel of 24 teeth.
    How many times does the small cog-wheel make around the big one?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by kingman View Post
    Dear sir ,
    I would appreciate if anyone can show me the solution to the below question.
    thanks
    Kingman

    A cog-wheel of 12 teeth rotates on its axis round a fixed cog-wheel of 24 teeth.
    How many times does the small cog-wheel make around the big one?
    The answer is 24/12+1=3 rotations. This is because the smaller wheel 'rotates' once in the same direction due to the fact that its position is also changing. (i.e. if it were sliding instead of rolling, it would still rotate once as it goes around the bigger wheel).
    Last edited by alexmahone; April 18th 2011 at 01:03 AM.
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  3. #3
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    Thanks sir but can you please explain why you add 1 which is the part I cannot figure out either geometrically or algebraically.
    Also from this link 'http://en.wikipedia.org/wiki/Gear" I wonder this motion actually represent the proposed question which again I need help to undrstand the meaning of 'This is because the smaller wheel 'rotates' once in the same direction due to the fact that its position is also changing. (i.e. if it were sliding instead of rolling, it would still rotate once as it goes around the bigger wheel).

    thanks
    kingman
    Last edited by kingman; April 18th 2011 at 04:56 AM. Reason: add further imformation
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    Hi kingman,
    It's the old quarter question.Take two quarters heads up side by side.Carefuly rotate the one on the right around the left.The moving quarter makes two revolutions arond the other.One rev in rotation and one in translation Sliding as pointed out in other answer



    bjh
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by bjhopper View Post
    Hi kingman,
    It's the old quarter question.Take two quarters heads up side by side.Carefuly rotate the one on the right around the left.The moving quarter makes two revolutions arond the other.One rev in rotation and one in translation Sliding as pointed out in other answer



    bjh
    Does the fact that the small wheel cannot rotate around the big wheel, unlike the quarters, make a difference?
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  6. #6
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    cog and wheel

    Hi Alexmahone,
    The big gear is fixed and the small gear is loose and can be rotated around the large one.It rotates twice around but also translates once.


    bjh
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  7. #7
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    Thanks very much for the answer but can you please help me to visualise what do you mean by 'One rev in rotation and one in translation'
    thanks
    Kingman
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  8. #8
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    Quote Originally Posted by kingman View Post
    Thanks very much for the answer but can you please help me to visualise what do you mean by 'One rev in rotation and one in translation'
    thanks
    Kingman
    Not sure whether this will help:

    1. The smaller circle (blue) is going around the larger circle (black) in such a way that a fixed point on the smaller circle is always pointing to the center of the larger circle.

    2. When the smaller circle is at it's starting position the centers of the circles form the line c_0C_1.

    3. If you measure the angle between the line c_iC_i (that is the line connecting the actual centers of the circles when the smaller circle is at position i) and a parallel to c_0C_0 you see that the smaller circle performs a complete turn even though it seems to stay in afixed position.

    I tried to upload the image to illustrate my reasoning. Unfortunately it is impossible. As soon as I'm able to upload drawings again I'll provide you with the necessary image.
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    Last edited by earboth; April 19th 2011 at 03:56 AM.
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    Thanks sir for your patience and time .With regard to the difficulty in downloading images, I also had experienced the same situation of not able so downlaod image in any format except in word document.
    Therefore I would appreciate very mich if you can paste the image on Microsoft Word and have it uploaded.Thanks
    Kingman
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  10. #10
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    Hi kingman,
    There are two kinds of motion rotational and translational. A car wheel rotates and moves the center of wheel a distance. This distance is translational. When an object rotates about another it has both motions.If two gears are engaged you cannot translate in a locked position but you need to visualize that in this problem the center of the small gear moves around the large one one time


    bjh
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  11. #11
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    Quote Originally Posted by kingman View Post
    Thanks sir for your patience and time .With regard to the difficulty in downloading images, I also had experienced the same situation of not able so downlaod image in any format except in word document.
    Therefore I would appreciate very mich if you can paste the image on Microsoft Word and have it uploaded.Thanks
    Kingman
    I finally managed to upload the image in question - as a .pdf-file. Please go to my previous reply and open the attached pdf-file that you can see what I'm talking about.
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  12. #12
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    Thanks very much to everyone and I think finally I manage to visualise sth.
    this is my thought;
    1) the small circle makes 2 rotations along the circumference of the bigger circle.
    2) the centre of the small circle make one revolution about the centre of the big circle.
    which is what you called it a translation.
    Can anyone please confirm whether I am correct in putting up the about statements.

    Also I thought translation is always associated with a path that is a straight line but in this case it is a circular path and why we can called is a 'translation' which can be confusing.Please Explain.
    thanks very much
    Kingman
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  13. #13
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    Kingman google two dimenensional kinematics. I tried this and did not find it useful. Better would be translation and rotation in mechanics


    bjh
    Last edited by bjhopper; April 19th 2011 at 08:28 AM. Reason: improved answer
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