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hw5 - 2.016 HW#5 Out Due 1 Expand the following for i=1,2,3...

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2.016 HW #5 Out: October 18, 2005 Due: October 25, 2005 1) Expand the following for i=1,2,3, j=1,2,3, and k=1,2,3, and write them in vector into a vector, and repeated letters imply summation. For example: a. b. u i = u 1 , u 2 , u 3 [ ] = r u notation. Recall for tensor notation, letters that appear only once imply expansion c. d. e. u i u i v i " u i " x i u i " v k " x i " ijk # u k # x j 2) Consider a submarine built very poorly, such that the axis of the propeller makes an angle " with the axis of the cylindrical submarine fuselage, as shown. If the propeller generates thrust, T, to drive the sub at steady-state speed, U, use inviscid flow theory (e.g. added mass) to find the trim angle, " . (Note: despite my misleading artwork, " # $ ) Assume the length of the sub is much greater than the radius, . ( R << L ) 3) The equation, " = a cos( kx # $ t ) , is for a plane progressive wave moving to the [left] [right]. Draw a picture of the wave at time t = 0 sec and time t = 1 sec, and label the distance and direction one of the peaks has traveled in that time.
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