PHY 504
Problem Set #1
due 3 September 2010
1. A circular platform rotates in the horizontal plane about its center with frequency . In this problem, you may ignore motion in the vertical direction and the effects of gravity. (a) Write down a coordinate t
(f) Cross product is not associative (see (d), has no identity and no inverse.
2.
Since (Mi)jk = -ijk ,
[ M i , M j ] mn = ( M i ) mk ( M j ) kn ( M j ) mk ( M i ) kn
= imk jkn jmk ikn = imk jnk + jmk ink = ( ij mn in jm ) + ji mn jn mi = in jm jn mi
ijk
PHY 504
Problem Set #11
due 22 November 2010 1. Consider a system of N coupled oscillators like those in Figure 13.1, except that these oscillate transversely in the vertical direction only. Assume they are connected by springs of force constant k and tha
PHY 504
Problem Set #3
due 17 September 2010 1. Derivation 2.8. 2. A particle of mass m moves in a potential sin .
(a) Write down a Lagrangian and obtain Lagranges equation from it. (b) Solve the equation numerically using Mathematica. Make three plots, f
PHY 504
Problem Set #4
due 24 September 2010
1. Goldstein Exercise 2.12. Show that the Lagrangian at the end of the problem differs from a more familiar Lagrangian by a term which is a total time derivative. Assuming the result of Derivation 1.8, can you
PHY 504
Problem Set #5
due 1 October 2010
1. A particle of mass m moves in a uniform magnetic field of magnitude B. (a) Write down a Lagrangian describing this system and the resulting equations of motion. (b) What are the symmetries? What conserved quant
PHY 504
Problem Set #6
due 15 October 2010 1. For the attractive inverse-square potential
(a) Calculate the differential cross section (). (b) Calculate the capture cross section. This is defined to be the crosssectional area of an incoming beam of partic
PHY 504
Problem Set #7
due 22 October 2010
1. Derivation 4.14. Use these formulas to obtain expressions involving dot products for (a b) c, a (b c), and (a b) (c d). Give three reasons why vectors do not form a group under the cross product. 2. Derivation
PHY 504
Problem Set #8
due 29 October 2010 1. Problem 4.15. 2. A rotation about the z-axis by is followed by a rotation about the x ' axis by . What is the axis and angle of the resulting rotation?
3. A child riding on a platform rotating at rad/s drops
PHY 504
Problem Set #9
due 5 November 2010 1. Show that ijk is invariant under rotations, assuming that it transforms as a 3index tensor:
i + j + k + = Ai + i A j + j Ak + k ijk
2. A solid rectangular block has sides a, b, and c (a) Calculate the inerti
PHY 504
Problem Set #10
due 10 November 2009 1. The Lennard-Jones potential models the interaction between a pair of neutral atoms:
V ( r) =
AB r12 r 6
where the first term represents a van der Waals attraction at large distances and the second term appro
PHY 504
Problem Set #2
due 10 September 2010
1. A diatomic molecule consists of two atoms with unequal masses, bound by a linear force derived from the potential | | , where a is the equilibrium interatomic distance. Suppose that all external forces and t