Question

# Consider a bead that is threaded on a rigid circular hoop of radius R lying in the xy plane with its center at O, and use the angle φ of two-dimensional polar coordinates as the one generalized coordinate to describe the bead's position. Write down the equations that give the Cartesian coordinates (x, y) in terms of φ and the equation that gives the generalized coordinate φ in terms of (x, y).

Solution

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If the point on the circle is described by $R$ and $\varphi$, then the usual transformations can be used to describe the position in cartesian coordinates:

$\begin{equation} x=R\cos\varphi \end{equation}$

$\begin{equation} y=R\sin\varphi \end{equation}$

$\begin{equation} \dfrac{y}{x}=\tan\varphi\iff\varphi=\arctan\left(\dfrac{y}{x}\right) \end{equation}$

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