UNIVERSITY COLLEGE LONDON EXAMINATION FOR INTERNAL STUDENTS MODULE CODE : ENGF0004 MODULE NAME : Modelling and Analysis II LEVEL: : Undergraduate DATE : 10-December-2020 EXAM START TIME : 09:00 EXAM PAPER LENGTH : 1 hour DEADLINE TO SUBMIT : 12:00 This paper is suitable for candidates who attended classes for this module in the following academic year(s): Year 2020/21 Answer papers must be submitted to the Turnitin Assignment on the ENGF0004 Moodle site by the end time of the exam. Late submissions will receive a mark of zero. Hall Instructions Standard Calculators Non-Standard Calculators TURN OVER ENGF0004 2020-21 Page 1 of 2 THIS EXAM PAPER CONSISTS OF TWO QUESTIONS OF EQUAL WEIGHTING ANSWER BOTH QUESTIONS Question 1 (50 marks) A football strikes a goalpost, and immediately afterwards at a time = 0, the top of the post is displaced horizontally by a distance k and moves with an initial velocity = . The subsequent horizontal displacement of the top of the post is described by the following equation: 2 2 + 20 + 0 2 = 0 where and 0 are positive constants. a) Show that the Laplace transform of the solution () is given by: () = ++20 2+20+0 2 . [6] b) By completing the square in the denominator, show that () has the following form: btCetx at cos.)( Providing = 0 , write down the values of the parameters , , and in terms of , 0 and . [14] c) Use the result in part (b) above to briefly explain why the top of the post will vibrate back and forth only if 0 < < 1 . [4] d) Sketch the solution () obtained in part (b) over the time interval = 0 to 2.5 seconds, for values = 1, = 1, and = 2. [8] e) Calculate the first three terms of the Taylor series expansion of () about the point = 0.5 seconds, using the values of the parameters in part (d). Sketch the result by adding to the sketch provided for part (d). [18] CONTINUED ENGF0004 2020-21 Page 2 of 2 Question 2 (50 marks) Use the method of separation of variables to solve the Laplace’s equation + = 0 in the rectangle 0 ≤ ≤ , 0 ≤ ≤ with the following boundary conditions (BCs): = 0, = 0; = , = 0; = 0, = 0; = , = () where () is some function satisfying the conditions ′(0) = 0 and ′() = 0. The problem is shown schematically below. The desired answer is (, ) satisfying the PDE given and all the BCs above. END OF PAPER
