School of Mathematics and Statistics MAST20029 Engineering Mathematics, Semester 2 2016 Assignment 1 and Cover Sheet Student Name Student Number Tutor's Name Tutorial Day/Time Submit your assignment to your tutor's MAST20029 assignment box before 5pm on Tuesday 23rd August. This assignment is worth 5% of your final MAST20029 mark. You must attach this cover sheet and a completed plagiarism sheet to your assignment. Note: • Full working must be shown in your analytical solutions. • Assignments must be neatly handwritten in blue or black pen. Diagrams can be drawn in pencil. • For the MATLAB questions, include a printout of all MATLAB code and outputs. This must be printed from within MATLAB, or must be a screen shot showing your work and the MATLAB Command window heading. You must include your name and student number in a comment in your code. 1. The moment of inertia of a solid body V about the x-axis is given by Ix = ZZZV µ(y2 + z2) dx dy dz where µ is the mass density per unit volume of V . Let V be the region a2 ≤ x2 + y2 + z2 ≤ b2; a > 0; b > 0 where µ is constant. (a) Calculate Ix. (b) Calculate the mass M of V . (c) Express Ix in terms of M. 2. Throughout history, many buildings have been designed with cycloidal arches. A cycloidal arch can be parametrised by: x(t) = t − sin t; y(t) = 1 − cos t; 0 ≤ t ≤ 2π: Let C be the curve formed by four such adjacent cycloidal arches. (a) Use MATLAB to sketch C. (b) Determine the length of C. (c) Check your answer to part (b) using the MATLAB symbolic toolbox. Page 1 of 23. Let S be that part of the surface z = 2 − x2 − y2; x ≥ 0; y ≥ 0; z ≥ 0 oriented by the upward unit normal. Let C be the closed curve around the boundary of S, oriented anticlockwise. Let F(x; y; z) = yzk be a force field. (a) Sketch S. (b) Calculate the flux of F across S. (c) Write down a parametrisation for each part of C, in the direction of an increasing parameter t. (d) Calculate the work done by F to move a particle along C. (e) Check your answer to part (b) using the MATLAB symbolic toolbox. Page 2 of 2