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Thread: cycloid generated by a circle

  1. #1
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    cycloid generated by a circle

    See attachment for figure.
    Attached Thumbnails Attached Thumbnails cycloid generated by a circle-untitled.jpg  
    Last edited by rmpatel5; Nov 29th 2008 at 09:04 AM.
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  2. #2
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    Hello,

    Consider the line from A to the y-axis. Its length is $\displaystyle \pi a$

    It is also equal to $\displaystyle x+s+AP$
    But $\displaystyle x=a(t-sin(t))$

    So we have $\displaystyle \pi a=a(t-\sin(t))+s+AP \Leftrightarrow \boxed{s=\pi a-a(t-\sin(t))-AP}$

    Consider the triangle ABP. It's a right angle triangle.
    Length of BP is a.
    Angle ABP is pi-t.
    So since sin(ABP)=AP/BP, we get :
    $\displaystyle \sin(\pi-t)=\frac{AP}{a}$

    by simple trigonometry, $\displaystyle \sin(\pi-t)=\sin(t)$
    Hence $\displaystyle \boxed{AP=a \sin(t)}$

    Finally :
    $\displaystyle \begin{aligned}
    s&=\pi a-a(t-\sin(t))-AP \\
    &=\pi a-at+a \sin(t)-a \sin(t) \\
    &=\boxed{a(\pi-t)} \end{aligned}$
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