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Thread: Motion of a Plate

  1. #1
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    Motion of a Plate

    Hey All

    Im trying to solve this problem, but I keep getting the wrong answer? Im using the relative motion formulas and $\displaystyle V=(Ang Vel)*r$ and the $\displaystyle i$ and $\displaystyle j$ components of the problem. Could some explain there process of how they would solve this problem so I can learn how I should do this in future and see where I have gone wrong.

    Cheers

    Edward Fitzgerald
    Attached Thumbnails Attached Thumbnails Motion of a Plate-aq_04.jpg  
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  2. #2
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    I would write $\displaystyle \theta$ for the angle between AB and the horizontal (given as 56 degrees in the diagram), and similarly $\displaystyle \phi,\;\psi$ for the angles that BD and DE make with the horizontal. Also, write x for the length of DE. Then the horizontal and vertical distances from A to E remain constant as the system moves. Therefore

    $\displaystyle 55\cos\theta+135\cos\phi+x\cos\psi = \text{const.},$
    $\displaystyle 55\sin\theta-135\sin\phi+x\sin\psi = \text{const.}$

    Differentiate these equations with respect to time:

    $\displaystyle 55\sin\theta.\dot{\theta}+135\sin\phi.\dot{\phi}+x \sin\psi.\dot{\psi} = 0, $
    $\displaystyle 55\cos\theta.\dot{\theta}-135\cos\phi.\dot{\phi}+x\cos\psi.\dot{\psi} = 0.$

    Now plug in the values $\displaystyle \theta=56,\; \dot{\theta}=2.1,\;\phi=35,\;\psi=32.$ You can eliminate $\displaystyle x\dot{\psi}$ between the two equations, and you're left with an equation for $\displaystyle \dot{\phi}$ (which is the angular velocity of the plate, in a clockwise direction).
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  3. #3
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    Wrong Answer?

    Thanks Heaps Opalg! I been using the working you provided to solve the question, but I continue to get the wrong answer? I get -0.0939 rad/s for an answer. I would think the answer would be positive and possibly larger, but Im not 100% sure. Could you please help I really wont to master this process!

    Edward
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  4. #4
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    Quote Originally Posted by edeffect View Post
    Thanks Heaps Opalg! I been using the working you provided to solve the question, but I continue to get the wrong answer? I get -0.0939 rad/s for an answer. I would think the answer would be positive and possibly larger, but Im not 100% sure. Could you please help I really wont to master this process!

    Edward
    The numbers I get are

    $\displaystyle 95.7538 + 77.4328\dot{\phi} + 0.5299x\dot{\psi} = 0,$
    $\displaystyle 64.5868 - 110.5855\dot{\phi} + 0.8480x\dot{\psi} = 0.$

    Multiply the first equation by 0.8480 and the second one by 0.5299, and subtract, to get

    $\displaystyle \dot{\phi} = \frac{34.2245 - 81.2038}{65.6667 + 58.5993} \approx -0.378.$

    Arithmetic is not my strong point, so I don't guarantee those numbers. At least we both agree that the answer is negative. Remember that $\displaystyle \phi$ is measured in such a way that $\displaystyle \dot{\phi}$ is positive when the plate is rotating in a clockwise direction. It seems to me quite plausible from the picture that $\displaystyle \phi$ should be decreasing as $\displaystyle \theta$ increases.
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