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Spring 2010 ME-210 Applied Math for ME Section 1 & 3 Homework Assignment 9 Due: Monday, 17 May 2010 IMPORTANT NOTES: Neatness will be graded!...

Although you have found the same result in part a and b, this does not necessarily mean that
this line integral is path independent. Show that the line integral you were asked to solve in
part a and b is actually path dependent.
Spring 2010 ME-210 Applied Math for ME Section 1 & 3 Homework Assignment 9 Due: Monday, 17 May 2010 IMPORTANT NOTES: Neatness will be graded! Show all your calculation steps and state any conclusions clearly. Matrices are designated by either bold letters (e.g. M ), as used in the textbook, or by letters in brackets (e.g. [M]). In either case, an upper case letter is used for matrices with number of columns greater than 1. Lower case letters are used for column vectors, such as a or [a]. You should use brackets for all matrices unless you submit a computer print-out. Pr.1. Suppose that you are an ant walking over an ice cream cone as shown on below figure. Cone is a right angle triangle and the ice cream is a half circle attached to the short side of the cone. a) You start walking from point O, walk over line C 1 and reach point A. Then you continue walking over curve C 2 and reach point B. During this journey, you use your six tiny legs to walk over the ice cream cone. However, gravitational force and a light summer breeze work against you. As a result, you are affected by a net force of ܨ Ԧ ሺݔǡ ݕሻ ൌ ݔଓƸ െ ݉݃ଔƸ ൅ ݕ݇ C 3 C 2 C 1 B A x y O R R R 30° ·
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As an ant who is interested in physics, you know that the line integral නܨ Ԧ ሺݎԦሻ ȉ ݀ݎԦ ൌ න ܨ Ԧ ሺݎԦሺݐሻሻ ȉ ݀ݎԦ ݀ݐ ݀ݐ gives the net change of your kinetic energy over a curve C. Find your net change of kinetic energy when you reach point B under the effect of net force ܨ Ԧ ሺݔǡ ݕሻ . To do this, - You must first represent curves ± ሬԦ and ± ሬԦ parametrically such as ± ሬԦ ሺ²ሻ , ± ሬԦ ሺ³ሻ . - Then, you must write the net force ܨ Ԧ ሺݔǡ ݕሻ in terms of the variables of the parametrically represented curves such as ܨ Ԧ ൫± ሬԦ ሺݐሻ൯ and ܨ Ԧ ൫± ሬԦ ሺݐሻ൯ . - Then, differentiate parametric curves once with respect to their parameters to evaluate the line integral. Apply this procedure for curves ± ሬԦ and ± ሬԦ and show your calculations clearly and neatly. b) Now suppose that you didn’t go over curves ± ሬԦ and ± ሬԦ but went over curve ± ሬԦ from point O to point B. Again applying the steps above, find the net change of your kinetic energy. You are going to end up with the same net kinetic energy change that you have found in part a. c) Although you have found the same result in part a and b, this does not necessarily mean that this line integral is path independent. Show that the line integral you were asked to solve in part a and b is actually path dependent.
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