Attempt THREE questions. You will not be penalised if you attempt additional questions.

Transcription

1 UNIVERITY OF EAT ANGLIA chool of Mathematics Main eries UG Examination 07 8 MATHEMATIC FOR CIENTIT C MTHB5007B Time allowed: Hours Attempt THREE questions. You will not be penalised if you attempt additional questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHB5007B Module Contact: Dr Emilian Parau, MTH Copyright of the University of East Anglia Version:

2 - -. Consider the matrix A = (i) Compute A, A 3 and A 4. [7 marks] (ii) Write down a formula for A k where k is any integer. [3 marks] (iii) olve the following system, by carrying out elementary row operations, or otherwise, A x y z = where a is a real number. For which value of a is the system consistent? 4 a, [0 marks]. Let A and B be the matrices 0 A = 3, B = (i) Calculate det(a) and det(b) and check whether det(a + B) = det(a) + det(b). or not. [0 marks] (ii) Compute the inverse A. Then solve the matrix equation AX = B for the unknown matrix X. [0 marks] MTHB5007B Version:

3 Consider a metal bar, which occupies 0 < x <. The initial temperature in the bar is θ(x, 0) = sin(πx) + 3 sin(3πx), for 0 < x <. () The temperature θ(x, t) is governed by the heat equation θ t = 0 θ, for 0 < x < and t > 0. () x The ends of the bar are kept at zero temperature, so θ(x, t) satisfies the boundary conditions θ(0, t) = 0, θ(, t) = 0. (3) (i) Use the method of separation of variables to find the solution θ(x, t) satisfying the equations (), () and (3). [ marks] (ii) If the boundary conditions (3) are replaced by θ(0, t) =, θ(, t) =, (4) verify by substitution, or otherwise, that a time-independent solution satisfying the equations () and (4) is θ(x, t) = + x. [3 marks] (iii) Using the results from (i) and (ii) and the separation of variables, find a solution θ(x, t) satisfying the equations (), (4) and θ(x, 0) = sin(πx) + 3 sin(3πx) + + x, for 0 < x <. Explain how the solution behaves as t in this case. [5 marks] MTHB5007B PLEAE TURN OVER Version:

4 In terms of Cartesian coordinates (x, x, x 3 ), a cuboid of elastic material occupies 0 < x < π, 0 < x <, 0 < x 3 <. It undergoes a small amplitude deformation in response to body and surface forces. The body force and the components of the stress tensor in the deformed state are given respectively by x x 3 cos x x sin x x x 3 b = x x 3, σ = x sin x x cos x x x 3. 0 x x 3 x x 3 (i) how that the equilibrium equations σ ij x j + b i = 0 are satisfied in the deformed configuration specified above. [5 marks] (ii) (a) Write down the formula for the traction t at a surface with normal ˆn. (b) What is the unit normal to the face x 3 =? Confirm that the traction at the point ( π,, ) on this face is given by π t =. (c) Compute the normal stress and shear stress components at this point. Verify that the shear stress magnitude is + π. [9 marks] (iii) (a) Write down an integral expression involving σ for the total surface force F on a face of an elastic body. (b) What is the unit normal to the face x =? What are the bounds on x and x 3 on this face? (c) Find the total surface force on the face x = of the body. [6 marks] END OF PAPER MTHB5007B Version:

5 MTHB5007B Mathematics for cientists C Exam Feedback 07/8 Question This question was answered well by the majority of students. Almost everybody found the right solution for (i). For (ii) some marks were lost for having the wrong (or inexistent) infinite solutions in the case a = 4. In (iii) marks were lost when the formulas for A k was not the correct one a = k, a = 0 etc. Question This was the question with the highest average. While the vast majority of students obtained the correct results for determinants at (i), some minor computational errors resulted in marks being deducted on some scripts. Also, the majority have used correctly the Gaussian elimination to obtain A, with a minority using the cofactors. ome marks were lost for the last part, where the solution for AX = B was given wrongly as X = BA, instead of the correct one X = A B. Question 3 This was the question with the lowest average, being tried by 35 out 55 students. Most marks were obtained by starting correctly the separation of variables, but then continuing wrongly when imposing boundary conditions. Also, the equation for t was not solved correctly by a number of students. Also, at (ii) very few have done the computation of θ t and θ xx to check the solution. Just 3 students have solved the last part. Question 4 This question was not attempted by that many students, but most of those who did scored reasonably well. Part (i) was mostly fine, though some students made the mistake of just showing that ( σ + σ + σ ) ( 3 σ + + σ + σ ) ( 3 σ3 + + σ 3 + σ ) 33 + b + b + b 3 = 0. x x x 3 x x x 3 x x x 3 What you actually need to show is that three desperate equations (one for each value of i) are all satisfied. The summation convention only operates on the repented index j, so the terms with different values of i are not added together. In part (ii), (a) and (b) were generally fine. (c) caused more difficulties though, presumably because students were unable to recall the formulæ. The normal stress is the scalar t n = t ˆn. The shear stress the the vector t s = t τ n ˆn. The shear stress magnitude τ s can be calculated either as t s or via the formula τ s = t t n. In part (iii), (a) and (b) were generally fine, though some students gave the formula for the moment/couple/torque rather than the force. Most of those who got this far were able to evaluate the integral in (c) to obtain the vector answer F = (, 0, π/3). However, remember that for an integral of a vector quantity like this, you need to integrate each component individually, not add them all together: ( ) ( ) ( ) (F ê + F ê + F 3 ê 3 ) da = F da ê + F da ê + F 3 da ê 3 (F + F + F 3 ) da.