Worksheet #8 – PHY102 (Spring 2010)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 often easierto work with the potential unless you dealing with problems such as thoseinvolving friction, where a potential does not exist. Visualizing the potentialcan be very helpful in developing physical insight into the trajectories. It isalso useful in understanding thermodynamic processes, which are statisticalin nature.For our purposes, we just need to know how to relate the force to thepotential, and that is via the equation:~F (x, y, z) = −Ã∂V∂x,∂V∂y,∂V∂z!. (1)Often it is easier to work in polar coordinates (r, θ, φ)—especially when weare dealing with central potentials, where V (r) does not depend on the angles(θ, φ).~F (r) = −∂V∂rˆr (for central potential). (2)Almost all that you do in undergrad physics (and most graduate-level physicscourses) is with central potentials.This week we study motion in two different central potentials: the gravi-tational potential generated by a fixed mass M:VG(r) = −GMr, (3)and the “Lennard-Jones” potential, which is an approximation to the inter-action potential between two atoms of an inert gas:VLJ(r) =Ar12−Br6. (4)1The constants A and B depend on the inert gas (e.g., they are different forHelium than for Xenon).Problem 1.(i) Make a plot of the Lennard-Jones potential. For the time being, use unitswhere A = B = 1.(ii) Find the value, r0, at which the Lennard-Jones Potential is a minimum.Evaluate VLJ(r0).(iii) By expanding around the minimum of the Lennard-Jones potential (usethe Series function and the Normal function), show that the LJ potentialcan be approximated near its minimum by a harmonic oscillator potential.(iv) Make a plot of the approximate potential used in part (iii) on top of acopy of your plot of the full L-J potential from part (i).(v) Optional challenge: find the frequency of oscillations in that approximatepotential, in units where A = B = M = 1, with M the mass of one of theatoms. The evaluate that frequency in standard (SI) units for a pair of Argonatoms, given that the LJ potential can be written asVLJ(r) = 4 ²³(σ/r)12− (σ/r)6´(5)where r = 0.36 nm and ² = 1.6 × 10−21J.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 asit approaches the sun (ignore all planets in this calculation!). Sun mass=1.991 × 1030kg, Sun radius = 6.96 × 108m. Assume that the ratio (mass ofcomet/mass of sun) → zero, so you can take the sun to be at rest.2For a few initial conditions, plot out the trajectory of the comet as it passesby the sun.Find some initial conditions that lead to the comet hitting the sun.Find some initial conditions that make the comet’s orbit a circle.In case you have not yet studied mechanics at the level needed for thisproblem—or have forgotten it—here is all you need to know to solve thisproblem:1. Because of angular momentum conservation, the motion lies in a plane.2. Use polar coordinates r and θ to describe the motion, with the sun atthe center of the coordinate system. (The usual rectangular coordinatesare given by x = r cos θ and y = r sin θ.)3. The angular momentumL = m r2˙θ (6)is a constant of the motion. (˙θ is classical mechanics shorthand fordrdt.)4. The kinetic energy isKE = (1/2) m v2(7)wherev2= ˙r2+ (r˙θ)2. (8)5. The total energyE = KE + V (9)is a constant of the motion.6. The initial conditions can be described by r, ˙r, θ, and˙θ at time t = 0.You can use the two constants of motion L and E to find the motion.(This is equivalent to~F = m~a, but much easier!)Further hints:3(1) The above equations of motion involve ˙r =drdtand˙θ =dθdt. You can takethe ratio of those:˙θ/ ˙r =dθdr, and thereby getdθdras a function of r, which canbe integrated to get θ as a function of r, which describes the orbit.(2) The obvious constants of motion are the total energy E and the totalangular momentum L as described above. But it is more convenient to user0= distance of closest approach, and v0= velocity at the point of closestapproach. It is easy to compute E and L in terms of those two quantities,since at the point of closest approach, ˙r = 0.(3) It may help to choose explict values for r0and v0before asking Mathe-matica to do the required integral, because it may otherwise like to give youa result containing imaginary numbers.(4) It may help to work in units where m and G M are set equal to
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