I have been trying to solve the following question but i am very unsure about my solution, can someone help me with it?
Consider a bead of mass $m$ that slides along a circular rigid wire hoop of radius $R$, which is constrained to rotate at a constant angular velocity $\omega$ about its vertical axis. We also assume that there is a friction force, of friction coefficient $\mu > 0$, that opposes to the motion. The equation of the bead is \begin{equation} mR\ddot{\theta} = -mg\sin\theta - \mu\dot{\theta} + mR\omega^2\sin\theta\cos\theta, \end{equation} where $-mg\sin\theta$ is the weight force with respect to a upward vertical axis, $-\mu\dot{\theta}$ is the friction force and $mR\omega^2\sin\theta\cos\theta$ is the centrifugal force.
- Write the equation of motion in such a way that it depends on only 2 parameters
- Assume that $mR \ll 1$ and write the corresponding equation. Give a physical interpretation of the considered approximation.
- Study the dynamics of the equation determined at the point ii., determining the equilibria and the phase portrait, and study the bifurcation of the system. (Use the fact that such an equation depends on only one parameter).
My solution: To make the equation of motion depend on only two parameters, we can introduce the dimensionless time $\tau = \omega t$ and divide through by $mR\omega^2$: \begin{align*} &\frac{d^2\theta}{d\tau^2} = -\frac{g}{R\omega^2}\sin\theta - \frac{\mu}{\omega^2 mR}\frac{d\theta}{d\tau} + \sin\theta\cos\theta \\ &\frac{d^2\theta}{d\tau^2} + \frac{\mu}{\omega^2 mR}\frac{d\theta}{d\tau} + \frac{g}{R\omega^2}\sin\theta = \sin\theta\cos\theta \end{align*}
This equation now depends only on the dimensionless parameters $\frac{\mu}{\omega^2 mR}$ and $\frac{g}{R\omega^2}$.
Assuming $mR \ll 1$ we can neglect the centrifugal force term This simplifies the equation to:
$$\frac{d^2\theta}{d\tau^2} + \frac{\mu}{\omega^2 mR}\frac{d\theta}{d\tau} + \gamma\sin\theta = 0$$
where $\gamma = \frac{g}{R\omega^2}$
- Is the dimensionless time parameter correct?
- How do i find out the dynamic of this system? how to find the equilibriums of it?
