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Math Help - Rolling Wheel - Parametric Paths

  1. #1
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    Rolling Wheel - Parametric Paths

    Need help with this probelm not with the maths involved but the theory,

    Wheel radius R rolls to the right along straight line with speed v, the path of a point on the rim is given by;

    c(t)= ( vt - Rsin((vt)/R) , R-Rcos((vt)/R) )

    When is the velocity vector of this point horizontal and what is the speed at this point?

    Trying To work out when velocity is horizontal?
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  2. #2
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    Hello, monster!

    I'll get you started . . .


    Wheel radius R rolls to the right along straight line with speed v.

    The path of a point on the rim is given by: .  c(t) \:=\:\begin{Bmatrix} x &=& vt - R\sin\left(\frac{v}{R}t\right) \\ \\[-4mm] y &=& R-R\cos\left(\frac{v}{r}t\right) \end{Bmatrix}

    When is the velocity vector of this point horizontal and what is the speed at this point?

    We have: . \begin{array}{ccc}\dfrac{dx}{dt} &=& v - v\cos\left(\frac{v}{R}t\right) \\ \\[-3mm] \dfrac{dy}{dt} &=& v\sin\left(\frac{v}{R}t\right) \end{array}\quad\hdots\quad \text{and the velocity is: }\:\frac{dy}{dx} \:=\:\frac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}

    . . Hence: . v(t) \;=\;\frac{\sin\left(\frac{v}{R}t\right)}<br />
{1-\cos\left(\frac{v}{R}t\right)}


    If the velocity is horizontal, then: . v(t) \,=\,0 \quad\Rightarrow\quad \sin\left(\tfrac{v}{R}t\right) \:=\:0

    Then: . \frac{v}{R}\,t \:=\:\pi n \:\text{ for }n \in I

    . . Hence: . t \:=\:\frac{R\pi}{v}\,n


    You'll have to fine-tune this result.
    Some values of t produce an undefined form for v(t).

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