Mathematics 317 Midterm#1 _2009_1 - Mathematics 317...

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Mathematics 317 Midterm #1: Monday, October 5, 2009 1. Do only two questions. Each question is of equal value. If you attempt more than two questions, only the first two will be marked. 2. No notes, books or calculators are permitted. A data sheet is included separately. 3. Time Limit: 50 minutes 1. Let ( t r be the position vector of a particle at time t . Let . , , r v a v r v & & & & = = = = = dt ds v (a) Show that the velocity of the particle satisfies the equation . T v v = (b) Show that the acceleration of the particle satisfies the equation N T v a 2 v v κ + = = & & where κ is the curvature, at time t , of the path traced out by the particle. (c) Let . ) ( r r A × = & t Find a condition for A (t) so that the motion is planar. 2. A particle’s position at time
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Unformatted text preview: t is given by . cos sin ) ( 2 k j i r t t t t + + = (a) Set up an integral that gives the distance travelled by the particle in the time interval . π ≤ ≤ t (b) Find the curvature of the particle’s path at t = 0. (c) Find the tangential and normal components of the particle’s acceleration at t = 0. 3. Consider the curve given by the pair of equations . 2 , 1 2 2 = + + = + z y x y x (a) Find a parametrization for this curve. Hint: Consider the first equation of the pair. (b) Find the curvature of the curve at the point (0,1,1). (c) Show that the curve is planar. Can you say anything about its shape?...
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