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Pagan Harpoon
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Homework Statement
A point particle of mass m is released at point A on the rim of a half cylinder of radius R, it is hit sharply such that it acquires initial velocity v_{0} directly downwards. There is nonzero friction between the particle and the cylinder, Mu. The particle's motion is always in a plane perpendicular to the central axis of the cylinder. Calculate, in terms of the given unknowns, the minimum value of v_{0} that allows the particle to reach the point B on the other side of the cylinder. I have made a diagram,
Homework Equations

The Attempt at a Solution
Looking at the forces acting on the particle, it's pretty clear that, at any particular point, the component of weight that acts along the tangent line to the surface of the half cylinder is [tex]mgsin\theta[/tex]. Also, the normal force is [tex]mgcos\theta+mR\dot{\theta}^2[/tex], so the total tangential force is [tex]mgsin\theta\mu(mgcos\theta+mR\dot{\theta}^2)[/tex]. Now, my (apparently bad) idea was to just integrate that across [tex]\theta[/tex] from [tex]\theta=\pi/2[/tex] to [tex]\theta=\pi/2[/tex]. I hoped that that would provide an expression for the work done on the particle (the energy that leaks out of the system) in terms of the various unknowns, including v_{0} and I would tune v_{0} such that the initial kinetic energy is equal to the energy that it loses to friction. However, integrating that involves integrating a [tex]\dot{\theta}[/tex] term, it would seem to me that in order to do that, I would need to have [tex]\dot{\theta}[/tex] (and then easily v) as a function of [tex]\theta[/tex], which would enable me to just set it equal to 0 at [tex]\theta=\pi/2[/tex] and the problem would be solved.
Any help is greatly appreciated.
There is also the option of setting [tex]mgsin\theta\mu(mgcos\theta+mR\dot{\theta}^2)=mR\ddot{\theta}[/tex] and trying to solve that differential equation, but I'm pretty sure that that is a bit beyond my current ability, and presenting nasty differential equations is not the usual style of my mechanics homework, so I suspect that there is something better to do.
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