Classical Mechanics - Home Exercise 7

Classical Mechanics - Home Exercise 7
1. A particle moves in a potential, V (r) = −C/(3r3 ).
(a) Given L, find the maximum value of the effective potential.
(b) Let the particle come in from infinity with speed v0 and impact parameter b. In terms of C, m, and v0 ,
what is the largest value of b (call it bmax ) for which the particle is captured by the potential? In other
words, what is the cross section for capture, πb2max , for this potential?
2. Consider a perfectly elastic scattering from an ellipsoid formed by rotating an ellipse about the z-axis:
x2
y2
z2
+
+
= 1.
A2
A2
B2
(1)
A 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.
(a) Since the ellipsoid is rotationally symmetric all the geometry can be worked out in the x - z plane. Make
a 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 θ. Calculate the di?erential
2
)
db
cross section. You may use the relationship | 12 d(b
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 /4
where 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) What is the total cross section of scattering from the elipsoid? Hint: do not do any calculations other than
those done in elementary geometry.
3. Find the shape of the trajectories of small oscillations of a point mass m on a plane, connected to three identical
springs whose other ends are fixed at the vertices of equilateral triangle (assume that the mass m moves inside
the triangle, the strength of each spring is k, and they are loose when the particle is in the equilibrium point).
Find the normal directions.
4. Two different pendulums (m1 6= m2 , l1 6= l2 , g = 1) are connected by a spring with energy 12 α(q1 − q2 )2 (where
α is a positive constant)
How do the characteristic frequencies behave as α → 0 and α → ∞? write the characteristic equation (i.e
det(B − λA) = 0), (where A and B stand for matrices of the kinetic and potential energies in the linearized
lagrangian L = 21 ~v T A~v − 12 ~qT B~q) and draw a graph of α(λ) in the (α, λ) plane.
2