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Homework Statement
A particle moves along a closed trajectory in a central field of force where the particle's potential energy is U=kr2 (k is a positive constant, r is the distance of the particle from the center O of the field). Find the mass of the particle if it's minimum distance from O equals r1 and it's velocity at the point farthest from O equals v2.
Homework Equations
##rv=r^2\dot \theta=\text{constant}\equiv c_a##
##0.5mv^2+kr^2=0.5m(\dot r^2+r^2\dot \theta^2)+kr^2=\text{constant}\equiv c_b##
##\dot r = \dot \theta \frac{dr}{d\theta}##
##\vec F=-\nabla U=-2k\vec r=m(\ddot r-r\dot \theta^2)\hat r##
The Attempt at a Solution
My attempt ignored the force equation (the last of my "relevant equations").
What I did was to eliminate the time dependency in the energy equation to get a differential equation between r and θ, like this:
##c_b=kr^2+\frac{mc_a^2}{2r^4}\big ( \frac{dr}{d\theta} \big )^2+\frac{mc_a^2}{2r^2}##
which is a separable equation:
##d\theta=\frac{c_adr}{r^2\sqrt{\frac{2}{m}(c_b-kr^2)-\frac{c_a^2}{r^2}}}##
I haven't tried, but I don't think I can solve that integral. At any rate, there must be a simpler way. What I am doing will lead to the arbitrary path of an object in this field (the constants ca and cb are defined by the initial condition). I don't think it is necessary to solve for the entire trajectory like this... but I don't know what else to do.
[edited to include that k is a positive constant]
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