midterm1sols - AE 352 Aerospace Dynamics II Fall 2008...

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AE 352: Aerospace Dynamics II, Fall 2008 Midterm Exam 1 Friday, October 3 11:00 - 11:50 am Closed book examination – no notes, calculators. Problem 1. In this problem, you will complete multiple steps to find the equations of motion for a spherical pendulum of mass m and length l . Figure 1: Spherical pendulum. (a) Find the angular velocity ω of the pendulum [20 pts]. Solution. ω = ˙ φ k - ˙ θ e φ where k = - cos θ e r + sin θ e θ . Thus ω = - ˙ φ cos θ e r + ˙ φ sin θ e θ - ˙ θ e φ (b) Find the velocity v of the mass [15 pts]. Solution. v = v rel + ω × r = 0 + ( - ˙ φ cos θ e r + ˙ φ sin θ e θ - ˙ θ e φ ) × l e r = ˙ θl e θ + ˙ φl sin θ e φ Page 1 of 3
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AE 352: Aerospace Dynamics II, Fall 2008 (c) Find the acceleration a of the mass [25 pts]. Solution. a = a rel + ω × ω × r + ˙ ω × r + 2 ω × v rel where a rel = 0 and v rel = 0 After performing the big cross product for ω × ω × r , we get: ω × ω × r = l ( - ˙ θ 2 - ˙ φ 2 sin 2 θ ) e r - ˙ φ 2 l sin θ cos θ e θ + ˙ θ ˙ φl cos θ e φ To find ˙ ω × r , first differentiate ω : ˙ ω = -
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