Description
Question 1 (3 marks)
Consider a half-cylinder of radius r0 and mass m that is allowed to roll on a horizontal plane, as in Fig. [].
Take the angle ϕ to correspond to the deviation of the upper flat plane from horizontal.
(a) Compute the moment of inertia tensor with respect to the half-cylinder’s center-of-mass.
(b) Adapt your result from (a) to show that the Lagrangian for the half-cylinder can be written as,
L =
1
2
3
2
−
8
3π
cos(ϕ)
mr2
0ϕ˙
2 +
4mgr0
3π
cos(ϕ). (1)
Hint: You might find it easiest to use the contact point of the half-cylinder with the horizontal plane
as your point of reference.
(c) Show that when the half-cylinder is near equilibrium its motion is equivalent to that of a pendulum of
effective length
l =
9π
8
− 2
r0. (2)
Question 2 (3 marks)
Consider a system of 3 masses, two with mass m and one with mass 2m constrained to slide (frictionlessly)
about a ring of radius r0. All three masses are linked by 3 identical springs with associated spring constant
k.
(a) Compute the normal modes of the system and comment briefly on the nature of each mode.
(b) Assuming the three masses are initially prepared near their equilibrium condition. The heavier mass is
perturbed slightly away from this configuration while the others are held in place. Assuming the masses
are simultaneously released, solve for the ensuing motion. Which of the normal modes participate?
Figure 1: The semi-cylinder of Q1.
Question 3 (4 marks)
The remainder of this assignment will be focused on the paper Appl. Phys. B 66, 181 (1998) by Daniel James.
This work discusses the formation of crystals of trapped ions, which is a platform that is being pursued to
realize a quantum computer by many prominent academic research groups (Maryland, Innsbruck), national
labs (NIST) and companies (Honeywell, IonQ).
(a) Read up to the end of Sec. 2 of the manuscript. In just a few sentences, summarize the key points of
the work.
(b) Explain what the terms in Eq. (1) are describing.
(c) Derive Eq. (5).
(d) Look at the result of Table 1 for the equilibrium positions of the ions. Focusing only on N even, can
you explain the trends in:
(i) the spacing between the two central ions,
(ii) the position of the outermost ions.