These are problems from the January 2024 qualifying exam.  The address topics discussed in Physics 513.  Problems 1 - 2 are from part 1 of the exam and problem 3 -4 are from part 2 of the exam.

Problem 1:

Two objects are dropped from the top of a cliff of height H.  The second object is dropped when the first has traveled a distance D.  At the instant when the first object reaches the bottom, what is the distance of the second object above the ground in terms of H and D?

Problem 2:

In reference frame S a firecracker explodes and a second firecracker explodes a distance Δx = c*25 ns away and 52 ns later.  In another inertial reference frame S', moving with velocity v i with respect to S, the two explosions are measured to occur a distance d' = c*42 ns apart in space.  How much time passes between the explosions in frame S'?

Problem 3:

Consider a potential barrier with height U and width a.
(a)  Which will tunnel more easily through the potential barrier, an electron or a proton?   Why?  Assume each particle's kinetic energy is smaller than U.
(b)   If the incoming particle's kinetic energy is 32.0 eV, U = 41 eV, and a = 0.25 nm, find the probability that the particle will tunnel through the barrier, both for the proton and the electron.

Problem 4:

(a)  For the system shown, consider only motion in the vertical plane.
Write the Lagrangian for small displacements from equilibrium of the system in the form

where T and K are matrices.  Clearly identify (write out) T and K.
If l₁ = l₂ = l, and m₁ = 21 and m₂ = 4 (everything in SI units), find the frequencies of the normal modes of this system?   Briefly describe these modes.