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BTaylor Group Title

Write a function that describes this situation (pics coming...) Two circles, radius 1, are drawn so their centers are 3 in. apart. Points marked on the circles rotate at a speed of 1 rotation every 4 seconds. Find a function that describes the distance (d) between the two marked points at any time (t).

  • 2 years ago
  • 2 years ago

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  1. BTaylor Group Title
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    I know it will be periodic, and can calculate for integral values of t (0,1,2,3,4,etc) but can't figure out how to find when t isn't a whole number.

    • 2 years ago
  2. BTaylor Group Title
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    @ParthKohli @agentx5 can you help?

    • 2 years ago
  3. agentx5 Group Title
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    Gear ratios? I still trying to understand this question the way it's worded, but if that's the case it's just a matter of the ratios of the radii, like you would for pulleys or gears.

    • 2 years ago
  4. agentx5 Group Title
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    Ah wait cycloids...

    • 2 years ago
  5. BTaylor Group Title
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    no, both are rotating clockwise.

    • 2 years ago
  6. agentx5 Group Title
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    http://www.daviddarling.info/images/cycloid.gif (prolate, nested function)

    • 2 years ago
  7. agentx5 Group Title
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    Which of course is going to make it look a whole lot like a sine function

    • 2 years ago
  8. BTaylor Group Title
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    yeah, but a cycloid only describes 1 point. I'm looking for distance between the 2 points.

    • 2 years ago
  9. agentx5 Group Title
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    I know that, which is why it'll end up being two of those together making it a sine function when I think about it visually. If they were touching it would be double amplitude and half period.

    • 2 years ago
  10. agentx5 Group Title
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    The tricky part is that they start out with an initial distance that also changes with respect to time

    • 2 years ago
  11. agentx5 Group Title
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    Or does it... Hmm, might just be additive.

    • 2 years ago
  12. BTaylor Group Title
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    this is what i've figured out: \[d(0)= 1\]\[d(1) = \sqrt{13}\]\[d(2) = 5\]\[d(3)= \sqrt{13}\]

    • 2 years ago
  13. agentx5 Group Title
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    |dw:1343667310281:dw|

    • 2 years ago
  14. agentx5 Group Title
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    They are pulleys, not gears in this case. |dw:1343667494092:dw|

    • 2 years ago
  15. BTaylor Group Title
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    at time t=.5, |dw:1343667478687:dw| so we get roughly a triangle:|dw:1343667609895:dw| by pyth. theorem, \[d=\sqrt{2+2.514} \approx 2.124\]

    • 2 years ago
  16. BTaylor Group Title
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    t(.5) = t(3.5) right?

    • 2 years ago
  17. agentx5 Group Title
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    Input your angular frequency and your initial condition (+1 inch), phase shift is zero here, for the formula see in this link. Your function should match my sketch above. When it does you can plug in any value for t and you'll have your distance at time t. Local max will be 5 every time, local min will be 1 every time. http://en.wikipedia.org/wiki/Sine_wave Also review: http://en.wikipedia.org/wiki/Angular_frequency

    • 2 years ago
  18. agentx5 Group Title
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    Standard form for this parametric is: \[y(t,x) = (A) \sin (\omega t - kx + \phi) + D\] In this case double A for amplitude Phi is zero D = 1

    • 2 years ago
  19. agentx5 Group Title
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    all you need to find is the angular frequency and see how the wave number will cancel out This formula can be used to construct any wave formula, but you'll need to figure out a few facts about the wave. At the moment I'm on a bit of a time crunch today so leave you to the number crunching :-D

    • 2 years ago
  20. BTaylor Group Title
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    @agentx5 @ganeshie8 I figured out d as a function of the angle: \[d(\theta) = \sqrt{(2 \sin \theta)^2 + (3 - 2 \cos \theta)^2}\]Tips on getting it as a function of time?

    • 2 years ago
  21. BTaylor Group Title
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    ah...got it: \[d(t) = \sqrt{(2 \sin (\frac{\pi x}{2}))^2 + (3-2 \cos (\frac{\pi x}{2}))^2}\]

    • 2 years ago
  22. agentx5 Group Title
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    Angular frequency relates the rate of the angular rotation per time (i.e.: degrees per second, or radians per second, or rpm). That's is how you shall have to relate time. I've tried finding something comparable to help you out but everything I've found has to do with energy, mechanical advantage, and torque, which doesn't help much here...

    • 2 years ago
  23. agentx5 Group Title
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    That actually looks plausible as an answer, but double check the phase shifts for sin and cosine. The net function should be similar to what I described initially, plot it on a calculator on on Wolf or something and see what it looks like.

    • 2 years ago
  24. BTaylor Group Title
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    https://www.desmos.com/calculator is my preferable graphing tool...and it works!

    • 2 years ago
  25. agentx5 Group Title
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    Highest distance should be when they're both "pointing" away, and shortest distance should be when they are "pointing" towards. I would test the horizontal and vertical cases as test angles to make sure your function works (that's 4 test cases). ;-D

    • 2 years ago
  26. BTaylor Group Title
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    @across is there an easy way to figure this out?

    • 2 years ago
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