ps2 - r to obtain a single first-order equation for . 2. A...

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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 ݎؠݎ ଵଶ and θ in terms of E and L . Eliminate from the equation for
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Unformatted text preview: r to obtain a single first-order equation for . 2. A single particle moves in the xy-plane, in a general central potential V ( r ). (a) Using Newtons 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 and . Show that the equations found in (a) follow from those in (b). (c) Obtain integral expressions for and . (d) Evaluate for the potential . 3. Mathematica problem: (a) Use the command ParametricPlot to plot a circle parameterized by . The option AspectRatio->Automatic may be useful. (b) Plot a trajectory for the solution found in 2(d)....
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