hw04soln - Ph-507 Homework 4(due Friday February 25.PROBLEM...

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Unformatted text preview: Ph-507. Homework 4 (due: Friday, February 25).PROBLEM 4-1 (4 pts)Episode II:A long time ago, in a galaxy far, far away...In 2D space, the gravitational potential energy shouldhave a logarithmic form. In particular, for a system of N particles,U(r1,r2, ...,rN) =γ2Xj6=kmjmklog|rj−rk|.Here the summation is performed over all pairs of diferent particles (a, b= 1, ..., N), and the double—counting iscancelled by the 1/2 factor.γis the "gravitational constant".a) What is the time-averaged kinetic energy of this system? (Hint: try to do "rescaling" of the coordinates, andfollow the logic of virial theorem);b) 2D astronomer Relpek has found that the period of the radial motion of a planet depends on its minimal andmaximal distances from the star (Rmin,Rmax), according to the following law:T=RαmaxFμRminRmax¶.What is the scaling exponentα? How much doesFvary whenRmin/Rmaxchanges from 1 (circular orbit), to 0.PROBLEM 4-2. (4 pts.)Two asteroidsof the same masscollide inelastically and form a single object. At the moment of the collision, theyare moving in the same direction. Originally, one of the asteroids had acircularorbit with periodT1, while the periodof the other wasT2. Find the periodTof the orbital motion of the newly formed object.PROBLEM 4-3 (3 pts)Consider a particle of massmscattered by the following attractive potential:U(r) =−krβ,a) Suppose you know the diferential cross-section, as a function of defection angleα:dσ/dΩ=f(α), for certainenergyE. Apply scaling ideas toFnd the cross-section for the same particle with an arbitrary energyE.b) Forβ≥2,Fnd is the cross-section of the "capture" process, as a function of the total energyE....
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This note was uploaded on 09/28/2008 for the course PHYS 507 taught by Professor Anon during the Spring '04 term at Cornell.

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hw04soln - Ph-507 Homework 4(due Friday February 25.PROBLEM...

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