Worksheet #8 - PHY102 (Spring 2003)Motion in a potentialAlthough you first learn about Newton’s second law~F = m~a and thedynamics that results from it, much of the discussion in the more advancedphysics texts is in terms of “potentials” V (~r). A particle undergoes motion“in a potential”. Note that V is a scalar, while~F is a vector. It is ofteneasier to work with the potential unless you are forced to work with the force.Actually even motion in a potential is carried out using Newton’s secondlaw. However visualizing the potential is very helpful in developing physicalinsight into the trajectories. It is also useful in understanding thermodynamicprocesses, which are statistical in nature. Anyway for our purposes, we justneed to know how to relate the the force to the potential, and that is via theequation:~F (x, y, z) = −(∂V∂x,∂V∂y,∂V∂z) (1)Often it is easier to work in polar co-ordinates (r, θ, φ). If we work withcentral potentials, V (r) which do not depend on the angles (θ, φ) things aresimple,~F (r) = −∂V∂rˆr − for central potentials. (2)Almost all that you do in undergrad. physics (and most of postgrad. physicscourses) is with central potentials.This week we study motion in two different central potentials: The grav-itational potential near a mass M:VG(r) = −GMr. (3)The “Lennard-Jones” potential between two inert gas atoms:VLJ(r) =Ar12−Br6. (4)The constants A and B depend on the inert gas (e.g. they are different forHelium than for Xenon).1Assignment 8. - Hand in by Monday Mar. 19Problem 1.(i) Make a plot of the Lennard-Jones potential.(ii) Find the value, r0, at which the Lennard-Jones Potential is a minimum.Evaluate VLJ(r0). What is the physical meaning of VLJ(r0).(iii) By expanding around the minimum of the Lennard-Jones potential(usethe “Series” function), show that, at low kinetic energies, two inert gas atomsundergo simple harmonic motion with respect to each other. For what ki-netic energies would you expect this to be true (compare the kinetic energywith the “depth” of the potential well).Problem 2.(i) Make a plot of the gravitational potential energy.(ii) Write a piece of Mathematica code to study the motion of a comet as itapproaches the sun(ignore the planets in this calculation). Sun mass= 1.9911030kg, Sun radius = 6.96 108m, Assume that the ratio (mass of comet/massof sun) → zero. For a few initial conditions, plot out the trajectory of thecomet as it passes by the sun (e.g. try to find initial conditions that leadto trapping of the comet). Find a set of initial conditions which makes thecomet’s orbit a
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