Part IB Physics B
Classical Dynamics and Fluids — Examples 2 — 2009
Problem grading:
(A) Problems that can be answered directly by quoting the lectured material or by
straightforward calculation.
(B) Problems that require some algebraic formulation and manipulation as well as
calculation.
(C) Problems which are either harder or longer than (B) problems. You should feel a
sense of achievement in completing these.
Normal Modes
1. (B) When a diatomic molecule is adsorbed onto the surface of a metal the frequency of
its internal vibrational mode is changed. If we consider a horizontal surface with the
axis of the molecule vertical, a simple model which might describe this phenomenon is
as follows. The molecule consists of two point masses, m, separated by a light spring
of spring constant
k
. The mass nearer to the metal is attached to a fixed point (the
metal surface) with a light spring of constant
K
. The two springs are collinear and
the motion of the masses is regarded as confined to the line of the springs. Find the
normal modes and how their frequencies vary with the ratio of
K
to
k
. Sketch the
results and comment on their physical significance for large and small
K
.
[ANS:
mω
2
/k
= 1 +
1
2
K/k
±
q
1 +
1
4
K
2
/k
2
.]
2. (B) A uniform rod of length
a
hangs vertically on the end of an inelastic string of
length
a
, the string being attached to the upper end of the rod. What are the
frequencies of the normal modes of oscillation in a vertical plane?
[ANS:
ω
2
= (5
±
√
19)
g/a
.]
3. (C) A simple model of a jet engine comprises three identical thin rigid discs mounted
equidistant on a uniform light shaft. Describe the normal modes of oscillation of the
system.
A small object entering the engine produces an abrupt change
ΔΩ
in the angular
velocity
ω
of the first disk. Obtain an expression for the maximum resultant angle of
twist of the shaft between the discs, given that the angular frequency of the lowest
vibrational normal mode is
Ω
.
ANS:
≤
(1 + 1
/
√
3)
ΔΩ/
2
Ω
 an upper bound from the difference between two non
harmonically related sine waves.
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2
4. (B)
M
m
x
x
2
1
k
k
k
(b)
(a)
m
m
x
x
2
1
k
K
k
(a) Find the frequencies of the normal modes of the twomass system shown in (a)
above, and sketch how they vary with
K/k
, the ratio of the spring constant of the
centre spring to the outer ones. Describe how the normal modes vary with
K/k
.
(b) Find the frequencies of the normal modes of the twomass system shown in (b)
above, and sketch how they vary with
M/m
(show the ratio
M/m
varying from 1
to
∞
). Annotate your sketch with pictures showing the normal modes associated
with each eigenvalue at
M
≈
m
and
M
m
.
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 Spring '07
 SFGull
 Orders of magnitude, Normal mode, normal modes, incompressible fluid

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