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ps2 - r to obtain a single first-order equation for...

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PHY 504 Problem Set #2 due 10 September 2010 1. A diatomic molecule consists of two atoms with unequal masses, bound by a linear force derived from the potential ܸ ሺݎ ଵଶ ሻ ൌ ݇ሺ|࢘ – ࢘ ߠ ሺݎሻ | െ ܽሻ , where a is the equilibrium interatomic distance. Suppose that all external forces and torques on the molecule vanish. (a) Show that the kinetic energy T can be expressed as the sum of two terms: the center-of-mass kinetic energy and a second term depending only on r 12 . Do the same for the angular momentum L . (b) Assume that the molecule moves entirely in the xy -plane. Working in the CM frame, write the total energy E and angular momentum L in polar coordinates. (c) Obtain first-order equations for ݎ ؠ ݎ ଵଶ and θ in terms of E and L . Eliminate θ
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Unformatted text preview: r to obtain a single first-order equation for ݎሺݐሻ . 2. A single particle moves in the xy-plane, in a general central potential V ( r ). (a) Using Newton’s 2 nd Law, write down the equations of motion for the particle in polar coordinates. (b) As in Problem 1(c), obtain first-order equations for ݎሺݐሻ and ߠሺݐሻ. Show that the equations found in (a) follow from those in (b). (c) Obtain integral expressions for ݐሺݎሻ and ݐሺ ሻ. (d) Evaluate ݎሺݐሻ for the potential ܸ ൌ ଵ ଶ ݇ݎ ଶ . 3. Mathematica problem: (a) Use the command ParametricPlot to plot a circle parameterized by θ . The option AspectRatio->Automatic may be useful. (b) Plot a trajectory for the solution found in 2(d)....
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