Phys 325 Homework 3A Name: ____________________________Due: Feb 12, 2015 by 1pm (lecture or 325 box)1. (20 points) A particle of mass m moves in 3-d subject to a force field F = - A ˆr / r4 .Thus it is attracted to the origin with a strength that increases rapidly as the particleapproaches the origin…. much more so than Newtonian gravity.•What is the potential energy function U(r) for this force field?•Given that the particle has angular momentum L ≠ 0, find an expression for theeffective potential Ueff(r;L) governing r(t) such that 12m&r2+ Ueff(r; L) = E. •Sketch atypical plot of Ueff(r) . •From this, determine the kinds of motion r(t) the particle couldhave for different total energies E and different starting positions. What is required fortrajectories to escape? For trajectories to be trapped?2. (10 points) Newton's law ofgravitational force on a particle can bewritten in the form rF =Gm1m2d2ˆt, where ˆtis the unit vector towards the other particle.Now consider two particles, one ofconventional mass m1 = µ >0, another of a highly hypothetical mass m2 = -µ . The twoparticles are separated by a distance d. All of the types of mass for the second particleare equal and negative, the passive gravitational, the active gravitational and the inertial.•What is the acceleration ( magnitude and direction, leftwards or rightwards? ) of eachparticle?•Show that total momentum is conserved.3. (20 points) Newton's gravity in a 2-d universe. Here we hypothesize that it would begoverned by a Poisson equation. ∇2Φ(rr ) = 2πGρ(rr )[ Here ρ is mass per area and G must have different units from what we are familiarwith.] • What is the potential Φ and gravitational acceleration g at a distance r from apoint mass M? Note that in 2-d ∇2Φ(rr ) ≡1r∂∂r[r∂∂rΦ(r,φ)]+1r2∂2∂φ2Φ(r,φ)Hint1: assume Φ is a function only of r and solve the above PDE for points r ≠ 0 where ρ = 0.This will introduce some constants of integration. Hint2: integrate both sides of ∇2Φ(rr ) = 2πGρ(rr ) over a finite circular area; on the right side you get 2πG times the massinterior to the area. On the left side you can use the divergence theorem to get the gradient ofΦ. This will help you determine a constant of integration. Assume Φ is a function only of r.Phys 325 Homework 3B Name: ____________________________Due: Feb 12, 2015 by 2pm (lecture or 325 box)1. (20 points) Calculate the gravitational potential Φ due to a uniform thin rod of lengthL and mass M at a distance y from the center of the rod and in a direction perpendicularto the rod. Check to make sure it reduces to the correct limit -GM/y when y >> L.Hint: You may need the following integral: dxx2+ a2∫= log(x + x2+ a2)2. ( 20 points) It is proposed to dismantle a uniform spherical asteroid of mass M andradius R. How much energy is required to disperse its matter to infinity against its self-gravity? Do this by imagining successively removing shells at radius r with thickness dr,starting with the one at r=R and going down to the center. The shell at r, just beforebeing removed, has a potential energy per mass of -G m(r) /r where m(r) is the massinterior to the shell. Do the integral and show that the total energy needed is
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