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## Homework Statement

Consider a bead sliding without friction on a circular hoop of wire rotating at constant [tex]\Omega[/tex], where [tex]\phi[/tex] is the angle between the bottom of the hoop and the bead. Find the equation of motion of the bead.

[tex]\hat{\Omega}=\hat{z}[/tex]

## Homework Equations

[tex]m\ddot{\vec{r}}=\vec{F}+2m(\dot{\vec{r}} \times \vec{\Omega})+m(\vec{\Omega} \times \vec{r}) \times \vec{\Omega}[/tex]

## The Attempt at a Solution

I started by taking the time derivative (first and second) of [tex]\hat{r}[/tex] to get an expression for the above equation in terms of [tex]\hat{x}[/tex], [tex]\hat{y}[/tex],and [tex]\hat{z}[/tex], but after separating the differential equations for each of those directions I have complicated differential equations that I can't solve. For example for the [tex]\hat{x}[/tex] direction, I had

[tex]mr(\ddot{\phi}cos \phi-\dot{\phi} sin\phi)=F_x + m\Omega^2 sin \phi[/tex]

Where [tex]F_x[/tex] is the normal force in the [tex]\hat{x}[/tex] direction.

Is that the right approach, and if so do you have any idea what I might have done wrong or is more information (steps) required?

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