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Math Help - Practice

  1. #1
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    Practice

    A particle of mass m moves under the influence of a central force whose magnitude is given by f(r) = kr, where k > 0. (The particle is thus repelled by the force center.) At some instant t_{0}, the particle is observed to be at ( r=r_{0} \theta=\theta_{0}) and moving with a speed v_{0} in a direction that makes an angle \gamma with the polar axis (i.e., the positive x-axis). Assume \gamma is not equal to { 0,\pi,2\pi} and v_{0}^2 is not equal to \frac{k}{m}r_{0}^2.

    (a) Show that the polar equation for the orbit of the particle has the form \frac{\alpha}{r^2}=1+\varepsilon\sin[{2(\theta-\theta_{0})}]
    (b) Determine the values of the constants \alpha and \varepsilon in terms of the parameters m, k, E, and L.
    (c) Determine the values of the constants \alpha and \varepsilon in terms of the parameters m, k, r_{0},v_{0}, and \gamma.
    (d) Show that the orbit equation, in Cartesian coordinates, is a quadratic form A x^2 + Bxy + C y^2 + Dx + Fy + G = 0 and determine explicitly the constants A, B, C, D, F, and G in
    terms of \alpha,\varepsilon,\theta_{0}
    (e) The orbit of the particle is thus a conic section! Determine the type of conic section.
    (f) Show that the conic section (whose precise nature you determined in part (e)) is one whose center is at the origin and whose axes are rotated by an angle \beta (with respect to the polar axis)
    where the value of \beta is given by the equation \tan(2\beta)=-\cot(\theta_{0})
    Thus, the orientation of the orbit (w.r.t. a set of fixed coordinate axes) depends only on \theta_{0}!
    (g) Prove that the shape of the orbit is independent of \theta_{0}(i.e., \theta_{0}only affects the orientation).
    (h) Find the distance of closest approach to the center of force (i.e. r_{min}).
    (i) Determine the eccentricity e of the orbit.
    (j) How do parts (a)-(i) change if the restriction ( v_{0}^2 is not equal to \frac{k}{m}r_{0}^2) is removed?
    Last edited by zorop; July 20th 2009 at 12:43 AM.
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  2. #2
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    \frac{\alpha}{r^2}=1+\theta*\sin2{2*(\theta-\theta_{0})}
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  3. #3
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    \frac{\alpha}{r^2}=1+\epsilon\sin[{2(\theta-\theta_{0})}]
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