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Unformatted text preview: 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 , , 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,,z ), determine the Lagrangian function for this system. [20 pts] Page 1 of 3 AE 352: Aerospace Dynamics II, Fall 2008 Solution. The magnitude squared velocity of the particle is...
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This note was uploaded on 09/16/2009 for the course AE ae352 taught by Professor Sri during the Spring '09 term at University of Illinois at Urbana–Champaign.
- Spring '09