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Homework Statement
A particle of mass m is free to slide on a thin rod. The rod rotates in a plane about one end at constant angular velocity ##\omega##. Show that the motion is given by ##r=Ae^{\gamma t}+Be^{+\gamma t}##, where ##\gamma## is a constant which you must find and A and B are arbitrary constants. Neglect gravity.
Show that for a particular choice of initial conditions [that is, r(t=0) and v(t=0)], it is possible to obtain a solution such that r decreases continually in time, but that for any other choice r will eventually increase. (Exclude cases where the bead hits the origin.)
Homework Equations
The Attempt at a Solution
I guess I have to use polar coordinates. In polar coordinates,
[tex]\textbf{a}=(\ddot{r}r\dot{\theta}^2)\hat{r}+(r\ddot{\theta}+2\dot{r}\dot{\theta}) \hat{ \theta }[/tex]
Here, ##a=0## and ##\ddot{\theta}=0##. Hence
[tex]0=(\ddot{r}r\omega^2)\hat{r}+2\dot{r}\omega\hat{\theta}[/tex]
This gives,
##\ddot{r}r\omega^2=0## and ##2\dot{r}\omega=0##.
Solving the first equation gives a solution of the form presented by the question and I get ##\gamma=\sqrt{\omega}## but the second equation gives ##\dot{r}=0##. This doesn't look right.
Any help is appreciated. Thanks!
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