MAST20029 Engineering Mathematics, Semester Summer 2022
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School of Mathematics and Statistics
MAST20029 Engineering Mathematics, Semester Summer 2022
Assignment 1 and Cover Sheet
Design of machine components. Summer time is an excellent opportunity to get active and have some fun in the water. This assignment is inspired by the design and construction of water bikes. The bike components are made of composite materials which are a combination of multiple materials with different physical and chemical properties. Composite materials have enhanced functionalities and are typically used to optimize strength, weight etc. Mathematically, the varying properties within the one material as you move from one location in the body to another means that the body density is, in general, a function of x, y and z and not simply a constant.
Image credit: https://www.redballoon.com.au/product/mandurah-water-bike-family-hire---60-minutes/EBB017-M.html
https://manta5.com/; https://blueride.com.au/
https://sydneybykayak.com.au/waterbike-tour-of-lavender-bay/
Question 1
A bike component A of density g(x, y) is to be 3D printed. The mass of A is given by mA! ,
where and RA is the region defined by
and
such that a > 0.
(a) Sketch the region RA. In your sketch, identify the coordinates of any points of intersection with the y axes.
(b) Use vertical strips to express ! as a sum of double integrals. Do not evaluate the integrals. [To get full marks, you need to show all working including any sketches that support your answer.]
(c) Write down a possible alternative expression for ! , such that ! is expressed as a difference of two double integrals, where each integral is over a region with no hole. [Hint: examine region R relative to the integrand g(x, y).]
Consider a second bike component B of mass mB given by
where the density and RB is the region bounded by the x -axis, x = π and the curve = #, ≥ 0.
(d) Sketch the region $ .
(e) Use the vertical strip method to evaluate $ . Use Matlab to check your answer.
Question 2
You are designing a solid component for a prototype water bike. lies between the surfaces
and
(a) Sketch for the case = 2.
(b) Consider the simple case when = 1. Using cylindrical coordinates, compute . , the moment of inertia of with respect to the axis of rotation, here the z-axis. Assume . is given by the integral.
Use Matlab to check your answer.
Question 3
Consider the following velocity vector field
(a) Show that V(x, y, z) is conservative.
(b) Find a scalar function Φ(x, y, z), such that V = ∇Φ.
(c) Consider a path described by the parametrization
i. Write down the coordinates of the endpoints of C. Use Matlab to sketch the path C.
ii. Determine the work done by V to move a particle fluid along C.
iii. Find the divergence of V at the endpoints of C. For each endpoint, state whether it is a source or sink.
iv. Is there local rotation at the endpoints of C? Justify your answer.
2022-01-13