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Two-Body Central-Force Problems

Two-Body Central-Force Problems. Hasbun Ch 8 Taylor Ch 8 Marion & Thornton Ch 8. Central Force Examples. Gravitational Force. Coulomb Force. Also a conservative force, described by the potential energy function:.

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Two-Body Central-Force Problems

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  1. Two-Body Central-Force Problems Hasbun Ch 8 Taylor Ch 8 Marion & Thornton Ch 8

  2. Central Force Examples Gravitational Force Coulomb Force Also a conservative force, described by the potential energy function: • Conservative force, so the force can be derived from a potential energy, U(r1, r2) r = r1 - r2 r1 r2 O

  3. White Boards • Write the Lagrangian for a two-body central-force system. • The Problem: find the possible motions of these two bodies

  4. Center of Mass and Relative Coordinates • What generalized coordinates should we use to solve the two-body central force problem? • Relative position, r • Position of center of mass, R, where The total momentum of the system is: B/c total momentum is constant, CM frame is an inertial reference frame. CM R r1 r2 O

  5. White Boards • Write r1and r2 in terms of R and r. • Write the kinetic energy T in terms of R and r. • Write the Lagrangian L in terms of R and r.

  6. White Boards • Write the equations of motion from the Lagrangian(one equation for R, one equation for r)

  7. Center of Mass Frame • If we choose the center-of-mass frame as our reference frame (we can do this b/c it’s an intertial frame), thenThe center of mass part of L is zero, so the Lagrangian becomes: Note this is the Lagrangian for a single particle moving in a central potential!

  8. Conservation of Angular Momentum • Angular momentum of the two particles is conserved. • Rewrite this in the COM frame Because the direction of angular momentum is constant, the motion remains in a fixed plane, which we can take to be the x-y plane.Always a 2D problem in COM frame

  9. White Boards • The easiest coordinates for this 2-D problem are polar coordinates. • Remember that the velocity in polar coordinates is given by • Write the Lagrangian in polar coordinates, and find the equations of motion.

  10. Equations of Motion • Angular Equation • Radial Equation Centrifugal Force

  11. White Board • Use the phi equation to eliminate phi-dot from the radial equation

  12. White Board • Use the phi equation to eliminate phi-dot from the radial equation

  13. Bound & Unbound Orbits At rmin, the energy is equal to Ueff

  14. Matlab Problem • Write down the actual and effective potential energies for a comet (or planet) moving in the gravitational field of the sun. Plot the 3 potential energies involved (U, Ucf, Ueff) and use the graph of Ueff vs r to describe the motion as r goes from large to small values. • Use • G=1 • m1=m2=l=1

  15. Matlab Problem

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