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Math Help - Three wheels rotating at different speeds.

  1. #1
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    Three wheels rotating at different speeds.

    Hello, everyone!

    Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

    Is that true?
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  2. #2
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    Quote Originally Posted by augmata View Post
    Hello, everyone!

    Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

    Is that true?
    Let  \dot{\theta_1} denote the rotational speed of wheel 1.

    Let  \dot{\theta_2} denote the rotational speed of wheel 2.

    Let  \dot{\theta_3} denote the rotational speed of wheel 3.

    Hence!

     \dot{\theta_1} = \theta_1 t

     \dot{\theta_2} = \theta_2 t

     \dot{\theta_3} = \theta_3 t

    Now:

     \dot{\theta_2}=2\dot{\theta_1}

    Hence:

     \theta_2t=2\theta_1t

     \theta_2=2\theta_1

     \dot{\theta_3}=\frac{1}{\pi}\dot{\theta_1}

    Hence:

     \theta_3t=\frac{1}{\pi}\theta_1t

     \theta_3=\frac{1}{\pi}\theta_1

    If they ever meet up again, then  \theta_1 =\theta_2=\theta_3

    This implies that

     \theta_1= \theta_2

     \theta_1 = 2\theta_1

    It also implies that:

     \theta_1= \theta_3

     \theta_1=  \frac{1}{\pi}\theta_1

    How can a wheel pass through an angle, half that angle, and one ' \pi^{th} ' of that angle simultaneously?
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  3. #3
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    Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.

    Is that true?
    It is true.

    The first 2 wheels will line up whenever the first wheel reaches the starting point and not otherwise. So the first 2 wheels will line up only every time the first wheel has undergone a whole rotation. in this time, the third wheel will have undergone 1/\pi rotations. Adding an integer number of 1/\pi rotations will never produce a whole number because 1/\pi is irrational. Therefore the 3 points will never line up again.
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