Uthm Final Exam

CONFIDENTIAL

UNIVERSITI TUN HUSSEIN ONN MALAYSIA FINAL EXAMINATION SEMESTER I SESSION 2011/2012
COURSE NAME COURSE CODE PROGRAMME : : : ENGINEERING MATHEMATICS IV BWM 30603/BSM 3913 1 BEE 2 BDD/BEE/BFF 3 BDD/BEE/BFF 4 BDD/ BFF JANUARY 2012 3 HOURS ANSWER ALL QUESTIONS IN PART A AND TWO (2) QUESTIONS IN PART B. ALL CALCULATIONS AND ANSWERS MUST BE IN THREE (3) DECIMAL PLACES.

EXAMINATION DATE : DURATION INSTRUCTION : :

THIS EXAMINATION PAPER CONSISTS OF SEVEN (7) PAGES

CONFIDENTIAL

BWM 30603 /BSM 3913

PART A Q1 (a) Consider the heat conduction equation

 2 T ( x, t )   2 T ( x, t ), t x

0  x  10, t  0 ,

where  is thermal diffusity  10, since   c 2 . Given the boundary conditions,
T (0, t )  0, T (10, t )  100

and initial condition,
T ( x,0)  x 2 .

By using explicit finite-difference method, find T ( x,0.055) and T ( x,0.11) with 5 grid intervals on the x coordinate. (10 marks) (b) Let y( x, t ) denotes displacement of a vibrating string. If T is the tension in the string,  is the weight per unit length, and g is acceleration due to gravity, then y satisfies the equation

 2 y Tg  2 y  , 0 x  2, t 0. t 2  x 2
Suppose a particular string is 2 m long and is fixed at both ends. By taking T  1.5 N,   0.01 kg/m and g  10 m/s2, use the finite-difference method to solve for y up to level 2 only. The initial conditions are

 x 0  x 1  2,  y ( x , 0)   2  x , 1 x  2  2 

and

y ( x , 0)  x( x  2). t

Performed all calculations with x  0.5 m and t  0.01 s. (15 marks)

Q2

The steady state temperature distribution of heated rod follows the one-dimensional form of Poisson’s equation d 2T  Q( x)  0 . dx 2 Solve the above equation for a 6 cm rod with boundary conditions of T (0, t )  10 and T (6, t )  50 and a uniform heat source Q( x)  40 with 3 equal-size elements of length by using finite-element method with linear approximation. (25 marks) 2

BWM 30603 /BSM 3913

PART B Q3 (a) Given f  x   7e x sin  x  