# midterm2 - a ji coeﬃcients[15 pts(c Using cylindrical...

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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] (b) Using cylindrical coordinates ( r , φ , z ), write down the constraint equation in diﬀerential form, and determine the
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Unformatted text preview: a ji coeﬃcients. [15 pts] (c) Using cylindrical coordinates ( r,φ,z ), determine the Lagrangian function for this system. [20 pts] (d) Write down the Lagrangian equations of motion for this system, using the form which includes Lagrange multipliers. [20 pts] (e) Solve for λ in terms of r, ˙ r, ˙ φ . [20 pts] (f) Solve for ˙ r in terms of r , and ˙ φ . (Hint: what is R r ˙ r 2 dr ?) [20 pts] Page 1 of 1...
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