Description
Question 1 (4 marks)
A particle of mass m is subject to a force,
F = −
2a
r
3
ˆr, (1)
in three dimensions (3D) with a > 0. Here, ˆr = r/r where r is the position vector of the particle with respect
to the origin and r = |r|.
(a) Explain why you expect the motion of the particle to be confined to a 2D plane. You are not expected
to give a mathematical proof, a concise qualitative sentence is sufficient.
(b) Starting from a Lagrangian describing the 2D motion in polar co-ordinates ϕ and r, show that the
motion of the particle is governed by an effective one-dimensional potential Veff(r).
(c) Discuss the particles motion as a function of the initial energy, the value of the constant a and any
other relevant factors.
(d) Show that when the 2D motion of the particle [see (a)] is described by polar co-ordinates r and ϕ, the
motion can be parameterized by the integral equation,
ϕ − ϕ0 =
Z r
r0
l
mr02
1
q
2
m [E − Veff(r
0)]
dr0
(2)
where ϕ0 and r0 relate to the particle’s initial conditions, E is the total mechanical energy and l is the
magnitude of the total angular momentum.
(e) (Optional) bonus question: Explicitly solve the integral above to obtain the orbit of the particle
and discuss the resulting expression. (This question is not marked/carries no points).
Question 2 (2 marks)
(a) Define and/or briefly discuss the terms:
(i) Limit cycle
(ii) Chaos
(iii) Poincare section/map
Your answer can involve an example illustrative sketch that highlights an important feature(s) if this
is useful.
(b) Consider the coupled equations of motion with b > 0,
x˙ = ax − by − C(x
2 + y
2
)x,
y˙ = bx + ay − C(x
2 + y
2
)y.
(3)
Introducing polar co-ordinates x = r cos θ and y = r sin θ the system can be equivalently described via
the equations of motion,
r˙ = ar − Cr3
,
˙θ = b.
(4)
Sketch a useful phase portrait of the system in the x − y plane for:
(i) a > 0 and C > 0
(ii) a < 0 and C > 0
(iii) a < 0 and C < 0
and indicate the important features including, e.g., limit cycles and fixed points. You should also
clearly state the radius or position of any limit cycles and fixed points, and comment on their stability
(provide evidence!).
Question 3 (4 marks)
A thin uniform disk of radius R and mass m is rigidly fixed to an axle as in Fig. 1. The axle is free to rotate
about its own central axis. The disk is oriented such that a normal vector from the disk’s surface makes an
angle θ with the axle.
For simplicity, assume in the following that the axle is massless and the the disk is
infinitely thin (i.e., it’s thickness is ignorable). For parts (a)-(c) you may assume the angular frequency of
the axle’s rotation is not fixed (i.e., the axle rotates freely).
(a) By identifying an appropriate set of body fixed axes, compute the principal moments of inertia of the
disk.
(b) Give an expression for the instantaneous angular velocity ~ω with respect to your set of body-fixed axes.
(c) Using your answers to (a) and (b) give an expression for the kinetic energy of the rotating disk.
(d) Assume that the axle is now driven to rotate at a fixed angular frequency Ω about its central axis.
Compute the magnitude of the applied torque that is required to preserve this motion.
Figure 1: A disk rigidly attached to an axle that can rotate.