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Math Help - complicated geometry in mechanics

  1. #1
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    complicated geometry in mechanics


    I have to determine the distance along the Z-axis from O to where the upper line crosses the Z-axis and makes angle \beta between the Z-axis and the line.
    The lower line which length equals L starts at the origin and makes an angle \alpha between the Z-axis and the line itself (so the angle between the line and the X-axis is \frac{1}{2}\pi-\alpha)


    This distance along the Z-axis equals (Lcos \alpha + Lsin \alphacotan \beta)ez
    ez is the unit vector and there should be an arrow above the letter e.
    I don't understand why you can express this distance like this.



    Because I'm not certain if the rest of you can see this picture, since it's posted on a secured website (login), I have posted the same picture as an attachment.
    Attached Thumbnails Attached Thumbnails complicated geometry in mechanics-20080025_1_2.jpg  
    Last edited by Sebastian de Vries; January 16th 2009 at 12:07 PM.
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  2. #2
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    Hi

    See drawing attached

    z = z_0 + z_1

    z_0 = L\: cos \alpha

    tan \beta = \frac{L \:sin \alpha}{z_1}

    Therefore z = L \:cos \alpha + \frac{L \:sin \alpha}{tan \beta}

    ImageShack - Image Hosting :: drawingue6.jpg
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  3. #3
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    Great, I understand it.

    Quote Originally Posted by running-gag View Post
    Hi

    See drawing attached

    z = z_0 + z_1

    z_0 = L\: cos \alpha
    This part I understood.

    tan \beta = \frac{L \:sin \alpha}{z_1}

    Therefore z = L \:cos \alpha + \frac{L \:sin \alpha}{tan \beta}
    This was my problem because I was confused about that cotan, I tried to solve it by using the geometric circle.
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  4. #4
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    OK

    Cotangent is only the inverse of tangent

    cotan \theta = \frac{cos \theta}{sin \theta} = \frac {adjacent \:side}{opposite \:side}

    List of trigonometric identities - Wikipedia, the free encyclopedia
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  5. #5
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    That wasn't the problem.
    I'll attach another file so you can see which figure I used, then you might understand my confusion.
    It's a words-file but it just contains 1 picture (copied from a PDF-file).
    I assume you can open this file, otherwise just let me know and I'll try to use Imageschack.
    As you can see an arbitrary line which makes an arbitrary angle crosses Y=1 (cartesian graph with X-axis and Y-axis, unit circle with a diameter of 1) at X=cotan angle.
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