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