Massachusetts Institute of Technology Department of Physics Quiz 2
Massachusetts Institute of Technology Department of Physics
Course: 8.09 Classical Mechanics Term: Fall 2004
Quiz 2 November 10, 2004
Instructions
• Do not start until you are told to do so. • Solve all problems. • Put your name and recitation section number on the covers of all notebooks you are using. • Show all work neatly in the blue book, label the problem you are working on. • Mark the final answers. • Books and notes are not to be used. Calculators are unnecessary. 2
Useful Formulae
Newton and Basic Kinematics:
F = p ˙ = m a for v c � t� =t v = v0 + dt a t� =0 � � t�=t t�� =t� r = r0 + v0t + dt dt F (t )/m t�=0 t�� =0
Gravitational Law:
−Gm1m2 F = 2 rˆ12 r12 Lagrangian and Hamiltonian: ∂q L(q, q˙)= T − U; H(p, q)= T + U = p − L ∂t Hamilton Equation of Motion: ∂H ∂H = −p˙; =˙q ∂q ∂p Poisson Brackets: ∂g ∂f ∂g ∂f [g, f]= − ∂q ∂p ∂p ∂q
Euler-Lagrange (without and with constraints):
∂L d ∂L ∂L d ∂L ∂g − =0; − + λ =0 ∂x dt ∂x˙ ∂x dt ∂x˙ ∂x Polar Coordinates:
x = r sin θ cos φ; y = r sin θ sin φ; z = r cos θ
Orbit Equation: µ 1 u + u = − F (u)with u = 2u2 r Effective Potential: 2 V (r)= U(r)+ 2µr2 3
Keplerian Orbits: k α U (r)= − ; = ε cos θ +1 r r � 2 2E2 4π2µ α = ; ε = 1+ ; τ 2 = a3 µk µk2 k α α rmin = a(1 − ε)= ; rmax = a(1 + ε)= 1+ε 1 − ε Polar Coordinates:
x = r sin θ cos φ; y = r sin θ sin φ; z = r cos θ
Inelastic Scattering (coefficient of restitution):
|v2 − v1| = |u2 − u1|
Scattering: � � � � dσ dσ b � db � σ(θ)= = = � � dΩ dφ sin θdθ sin θ � dθ � sin θ π θ tan ψ = ; ζ = − ; cos θ +(m1/m2) 2 2
Acceleration in accelerated frame: ˙ a = g − V − ( ω˙ × r) − 2( ω × v) − ( ω × ( ω × r)) ; 4
Problem 1: Poisson Brackets (30 points)
Consider an arbitrary, well behaved, function of coordinates and momenta f(q(t),p(t)). a) Show that the function f can be expressed in terms of p and q at t =0 as
t t2 t3 f(q(t),p(t)) = f + [f,H]+ [[f,H],H]+ [[[f,H],H],H]+ 1! 2! 3! Where f = f(q(0),p(0)) and H = H(q(0),p(0)) is a Hamilton function. Assume that the series converges. b) Use this formula to calculate p(t), q(t), p2(t), q2(t) for a free moving particle of mass m. c) Use this formula to calculate p(t), q(t) for a one dimensional harmonic oscillator with frequency ω. 5
Problem 2: Orbital motion (35 points)
Consider the motion of mass µ in a central force field. Force depends on the radial distance r as k F (r)= − e−ar r2 where k>0and a> 0. You are being asked to study the existence and stability of the circular orbits. a) Find the condition for the existence of the circular orbit(s) for motion for a given angular momentum . Express the condition in terms of the dimensionless parameter x = ar and the dimensionless combination of , µ, k and a and their powers. b) How many possible solutions can equation a) have? Give the range of parame- ters which determine the number of possible solutions. You can use any method or argument to show that. There is no obvious analytical solution to a) but you can use e.g. graphical method. c) Assuming that the circular orbit(s) is(are) possible, establish if it(they) are stable or not. The stability condition derived in class was:
3g(r0) + g (r0) > 0 r0 where F (r)= −µg(r). 6
Problem 3: Elastic Collision (35 points)
Consider a perfectly elastic scattering from an ellipsoid formed by rotating an ellipse about the z-axis: x2 y 2 z 2 + + =1 A2 A2 B2 . An beam of point-like particles is traveling along the z-axis and scatters off the ellipsoid. In this problem you are asked to calculate the differential cross-section σ(θ), compare it to the cross section of scattering from hard sphere and to find the cross section in the very forward and backward directions. a) Since the ellipsoid is rotationally symmetric all the geometry can be “worked out” in the x − z plane. Make the sketch of the ellipse and indicate the scattering angle and the impact parameter. Write the relationship between the scattering angle θ and the slope of the ellipse at the impact point dx/dz. b) Write the relationship between the impact parameter b and the scattering angle | 1 d(b2) | | db | θ. Calculate the differential cross section. You may use relationship 2 dθ = b dθ to slightly simplify the calculations. c) Set A = B = R and verify that the differential cross section does not depend on θ and it is equal to R2/4where R is the radius of the sphere. d) Consider an ellipsoid that is “short” in the z direction. Assume that A is kept constant and B = αA where α<1. Write the cross section in the “forward direction” at θ = 0 in terms of α and A. Is it smaller or larger than that of a hard sphere ? e) Use the “short” ellipsoid to calculate cross section in the backward direction, θ ≈ π. Hint: you can calculate the cross section at θ = π − and use approximations π− ≈ π− ≈ 2 cos( 2 ) 2 and tan( 2 ) . Cross section at θ = π corresponds to a limit where =0. f) What is the total cross section of scattering from the elipsoid? Hint: do not do any calculations other than those done in elementary geometry.