midterm2sols - AE 352 Aerospace Dynamics II Fall 2008...

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: