Math 402 Problem Set II
1
.
A molecule consisting of three atoms of mass m
1
= M, m
2
=m
3
= m where m << M is oriented as shown at
time t = 0.
r
1
(t=0) = 0
r
2
(t=0) = dsin
α
x^
+ dcos
α
z^
r
3
(t=0) =
dsin
α
x^
+ dcos
α
z^
You can use Mathcad for
this problem. See B. Hale
for help.
a)
Assume that M>>m so that the center of mass of the molecule is given
by the position of particle 1, and that the velocity of particle 1 and the Euler angles
ψ
(t),
θ
(t), and
φ
(t) of the
molecule are given by:
d
r
1
(t)/dt
=
3v
x^
 v
y^
+ 2v
z^
where v is a constant (no t dependence)
φ
(t)
=
[v/d]t
θ
(t)
=
[3v/d]t
ψ
(t)
=
ε
t
(
For question
1
ε
=0
)
Write out the Euler angle rotation matrix,
R
(
ψ
(t),
θ
(t),
φ
(t) ) as a function of v, d and t.
Note that the molecule
rotates as it moves and the primed coordinate system (
x
^‘
,
y
^
‘ and
z^
‘) is fixed on the molecule. The orientation of
the primed coordinate system is prescribed by
R
. The unprimed coordinate system
x
^
,
y
^
and
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This note was uploaded on 12/20/2010 for the course PHYS 402 taught by Professor J. during the Fall '09 term at Missouri S&T.
 Fall '09
 J.
 Mass

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