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

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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] Page 1 of 3
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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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