AE 352: Aerospace Dynamics II, Fall 2008
Midterm Exam 2
Friday, November 7
11:00  11:50 am
Closed book examination – no notes, calculators.
Problem 1.
A particle of mass
m
in a uniform gravitational field is constrained to
lie on a parabolic surface given by
x
2
+
y
2
=
az
.
Figure 1: Parabolic surface.
(a) What are the degrees of freedom of this system before and after the constraint?
[5 pts]
Solution.
There are 3 DOFs before the holonomic constraint, and there are 2
DOFs after the constraint is applied.
(b) Using cylindrical coordinates (
r
,
φ
,
z
), write down the constraint equation in
differential form, and determine the
a
ji
coefficients. [15 pts]
Solution.
The constraint equation
x
2
+
y
2
=
az
becomes
r
2

az
= 0
which becomes
2
rdr

adz
= 0
Thus,
a
11
= 2
r
,
a
12
= 0,
a
13
=

a
, and
a
1
t
= 0.
(c) Using cylindrical coordinates (
r, φ, z
), determine the Lagrangian function for
this system. [20 pts]
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AE 352: Aerospace Dynamics II, Fall 2008
Solution.
The magnitude squared velocity of the particle is
v
2
= ˙
r
2
+
r
2
˙
φ
2
+ ˙
z
2
Thus the kinetic energy is:
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 Spring '09
 SRI
 Energy, Kinetic Energy, Potential Energy, Lagrangian mechanics, Aerospace Dynamics II

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