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Math Help - Shortest Band around two cylinders

  1. #1
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    Shortest Band around two cylinders

    Hi,

    I have a question off the internet but I am rather stuck on the question however I am quite keen to get the answer. It is:

    2 cylinders have diamteters of 12 and 4cm respectively. Find the length of the shortest band that could be fixed around the cylinders to hold the two together.

    It would obviously be easy to obtain the answers if they were the same diameter, although, surely in this case, the 12cm cylinder would have the band ovlerlapping by a certain amount? I am not very sure how one would find this out?

    Thanks very much for any help in advance!
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  2. #2
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    Re: Shortest Band around two cylinders

    Quote Originally Posted by BobtheBob View Post
    Hi,

    I have a question off the internet but I am rather stuck on the question however I am quite keen to get the answer. It is:

    2 cylinders have diamteters of 12 and 4cm respectively. Find the length of the shortest band that could be fixed around the cylinders to hold the two together.

    It would obviously be easy to obtain the answers if they were the same diameter, although, surely in this case, the 12cm cylinder would have the band ovlerlapping by a certain amount? I am not very sure how one would find this out?

    Thanks very much for any help in advance!
    1. Draw a sketch.

    2. The band consists of 2 arcs (in red) and 2 straight line segments (in green)

    3. Let R = 6 cm and r = 2 cm (keep in mind these are radii). Then you can determine the angle \alpha:

    \cos(\alpha) = \frac{R-r}{R+r}

    With your values you'll get |\alpha| = 60^\circ

    4. Then the 2 arcs have the lengthes:

    arc_1=\frac{240^\circ}{360^\circ} \cdot 2 \pi R With yor value for R you'll get arc_1=8\pi

    arc_2=\frac{120^\circ}{360^\circ} \cdot 2 \pi r With yor value for r you'll get arc_2=\frac43 \pi

    5. The 2 green line segments can be evaluated using Pythagorean theorem:

    2 line = 2\sqrt{(R+r)^2-(R-r)^2} = 4\sqrt{rR} With yor values for r and R you'll get line = 8\sqrt{3}

    6. Sum up these lengthes.
    Attached Thumbnails Attached Thumbnails Shortest Band around two cylinders-bandumzylind.png  
    Thanks from BobtheBob
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  3. #3
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    Re: Shortest Band around two cylinders

    Oh of course! Right I get it now.
    Thanks so much for your help! (especially for going to the bother of drawing a diagram.)
    Thanks again!
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