Homework Assignment 7 Physics 302, Classical Mechanics

Fall 2010 A. V. Kotwal

Handed out: Monday, October 25, 2010
Due time:  Monday, November 1, 2010

Problems

1. Consider the Kepler problem. The energy conservation equation is given by

E = \frac{μ}{2} (\frac{dr}{dt} )^{2} + \frac{l^{2}}{2μr^{2}} − \frac{α}{r},

where α = GμM = Gm_{1}m_{2}, l is the angular momentum, and E is the total energy of the system.

(a) Prove the Kepler’s third law for elliptical orbits

T^{2} = \frac{4π^{2}}{G M} a^{3},

where M = m_{1} + m_{2} is the total mass, a = \frac{1}{2}(r_{min} + r_{max}) is the semi-major axis, and T is the period of the orbit.

(b) Show that the semi-minor axis is given by

b = a \sqrt{1 − ϵ^{2}},

where ϵ = \sqrt{1 + \frac{2El^{2}}{μα^{2}}} is the eccentricity of the orbit. We notice that when ϵ = 0, b = a, the orbit is a circle.

2. A uniform distribution of dust in the solar system adds to the gravitional attaction of the Sun on a planet an additional force

F⃗ = −mCr⃗

where m is the mass of the planet, C is a constant proportional to the gravitational constant and the density of the dust, and r⃗ is the radius vector from the Sun to the planet (both considered as points). This additional force is very small compared to the direct Sun-planet gravitational force.

(a) Calculate the period for a circular orbit of radius r_{0} of the planet in this combined field.

(b) Calculate the period of radial oscillations for slight disturbances from this circular orbit.

(c) Show that nearly circular orbits can be approximated by a precessing ellipse and find the precession frequency. Is the precession in the same or opposite direction to the orbital angular velocity?

3. A central force potential that is frequently encountered in nuclear physics is the rectangular well, defined by the potential

V = \{ \begin{matrix}−V_{0}, & r ≤ a, \\ 0, & r > a. \end{matrix}

Show that the scattering produced by such a potential in classical mechanics is identical with the refraction of light rays by a sphere of radius a and relative index of refraction

n = \sqrt{\frac{E + V_{0}}{E}} .

(This equivalence demonstrates why it was possible to explain refraction phenomena both by Huygens’s waves and by Newton’s mechanical corpuscles).

Show also that the differential cross section is

σ(θ) = \frac{n^{2}a^{2}(n cos \frac{θ}{2} − 1)(n − cos \frac{θ}{2})}{4 cos \frac{θ}{2}(1 + n^{2} − 2n cos \frac{θ}{2})}.

Calculate the total cross section explicitly. You should find that it has a value that is easily explained. If you use Mathematica to do the integral (which is OK), please turn in a page showing the relevant commands.

4. Consider the three spring, three mass system shown below. The center post is taken to be fixed to the table.

(a) Using coordinates referenced to equilibrium, write expressions for the kinetic and potential energies of this system.

(b) Derive the Lagrangian and Hamiltonian of this system.

(c) Derive the equations of motion.

(d) Find the eigenmodes and corresponding oscillation frequencies. Graph the eigenmodes.

5. An undamped linear oscillator with natural frequency ω_{0} is driven at frequency Ω by an external force with the following equation of the motion,

q̈ + ω_{0}^{2}q = f_{0} cos Ωt.

(a) For Ω ≠ ω_{0}, find the rate at which energy is pumped into the oscillator. Find the maximum energy of such an oscillator starting from rest, i.e. q(0) = q̇(0) = 0 at t = 0;

(b) For Ω = ω_{0}, by going into the complex variables, i.e. replacing cos ω_{0}t by e^{iω_{0}t} and setting q(t) = α(t)e^{iω_{0}t}, find the real solution for the driven oscillator;

(c) For an oscillator starting from rest, i.e. q(0) = q̇(0) = 0 at t = 0, and Ω = ω_{0}, find the rate at which energy is pumped into the oscillator and find the energy of the oscillator, averaged over a period, as a function of time.

6. A uniform rod with mass m slides with its ends inside a smooth vertical circle of radius R. The rod subtends an angle of Φ at the center of the circle.

(a) Graph this system and write down the Lagrangian and Hamiltonian; (b) Derive the rod’s equation of the motion for a small oscillation angle.

(c) Derive the equation of the motion for a simple pendulum with mass m and length l.

(d) Suppose that the rod and simple pendulum have the same oscillation frequency, if the radius of the circle equals to the length of the pendulum, what is the value of the angle Φ.