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Homework Statement
In the beginning a point mass is rotating in a circle of radius [itex]L[/itex]. The spring is providing the centripetal force ([itex]\vec{F}=-k\vec{r}[/itex]) and the mass rotates with constant speed. At some point in time, a stick of radius [itex]a[/itex] ([itex]a<<L[/itex])lands near the center of the circle in such a way that center of the circle is tangent to the rim of the stick. Point mass is rotating about the stick and while doing so, it gets shorter.
Question is to find the time it takes for the mass to hit the stick
Solution in the book:
[itex]t=\sqrt{\frac{m}{k}}\cdot\frac{4a}{5L}[/itex]
Homework Equations
Equations of motion in cylindrical coords.
[tex]\vec{a}=\frac{a^2}{r}\hat{r}-\frac{a \sqrt{r^2-a^2}}{r}\hat{\theta}\\ \;
\\ \vec{r}\;'=\vec{r}-\vec{a} \\ \; \\ \vec{F}=-k\cdot\vec{r}\;' [/tex]
Vector [itex]\vec{a}[/itex] represents the radius of the stick, it's useful to have because I can write down the relation of the force to the new origin.
The Attempt at a Solution
Okay so first I tried approximating various things like angular momentum since [itex]a<<L[/itex] I assumed the angular momentum is conserved... also later on started using it to simplify some expressions like square roots and alike. As far as I can see, the energy is not conserved since part of the energy is "stuck" on the stick since it never has enough time to fully pull the mass. Other thing that occurred to me is to to look at part of the velocity vector perpendicular to the force. It yielded an equation, but none of these equations are simple equations of motion, I'm stuck with second order nonlinear ODE's.
Thanks for reading, Cheers.