Reference no: EM133697772
Engineering Mathematics
Description
Select one of Workshops 6-11 to write a report on.
Please carefully read the below requirements.
Formatting requirements:
• Use a simplistic style, without decorative text/borders etc.
• All mathematics is to be formatted either in Microsoft Word equation editor, or a non- Microsoft Word equivalent such as LaTeX.
• Write in a self-contained manner. Do not include phases like 'I chose extension 1 from workshop 5'.
• Pictures are to be drawn electronically
Content requirements:
The report should be succinct and minimal, that is, only include highly relevant material. Include exactly the following headings
Introduction - Describe the problem, including any relevant background information.
Analysis - Provide the derivation of the mathematical formulation of the problem. In the process of doing so, describe any assumptions you are making, introduce the variables you have chosen, and provide any known constants that you are using.
Solutions - Solve the mathematical formulation, showing each step.
Interpretation - Interpret the solution of the mathematical formulation in the context of the original problem. Make sure to include relevant units.
Discussion - Explore and discuss the extension problems described in the workshop document. Other extensions may be acceptable, but discuss with your tutor beforehand.
References - Include any references you have used, such as sources for your known constants, or for interesting features of your problem.
Note that the Assignment Grading Rubric (available on Canvas/Modules) indicates that weightings for each of the sections, and what is expected.
Length requirements:
A maximum length is needed on these reports to not overburden our markers. Please adhere to the below maximum lengths, anything significantly different will attract penalty.
• Introduction and Analysis sections combined are under 1.5 pages
• Solutions and Interpretation sections combined are under 1.5 pages
• Discussion section is under 1.5 pages
• References are unrestricted
Workshop 6
Problem 1
A ball rolls of a table. What is the horizontal distance that it travels before it hits the ground?
Problem 2
If the ball loses 20% of its energy every time it bounces, calculate the distance
d1 between the first bounce and the second bounce.
Problem 3
In the previous problem you probably assumed that the energy loss is "symmet- ric" in the x and y direction, meaning that the kinetic energy from the velocity in the x direction and the kinetic energy from the velocity in the y direction each decrease by 20% every time the ball bounces. Is this a reasonable assumption? That is, is it possible for the kinetic energies in the different axes to decrease by different factors? If this is a reasonable assumption, justify it. If not, what is a better model for the energy loss?
Extension Problems
For the assignment, choose one of the following:
How far does the ball go in total (including all of the bounces)? Is its range finite or infinite? If the range is finite, calculate it.
What is the aerial distance that the ball travels on the nth bounce?
Let's say that the ball is "rolling" when its bounce height is less than 1 millimetre. How many times does the ball bounce before it rolls?
Workshop 7
Fluid dynamics models the flow of liquids, gases, and many other interesting phenomena. For this workshop we will consider a space R3 filled with a flowing "mathematical" liquid/gas and a small cube B ⊂ R3 called a control volume. The density of the fluid is given by a function ρ(x, y, z, t).
Problem 1
By analysing the mass flowing into and out of B, derive the continuity equation, which is an expression that ensures conservation of mass.
Problem 2
Simplify the continuity equation by assuming that the fluid is incompressible. Make an observation about the velocity of the fluid moving through the faces of B.
Problem 3
Simplify the continuity equation by assuming that the fluid has a constant velocity (a steady flow).
Extension Problems
This is a difficult derivation, so very little extension is required. For your as- signment focus on discussing in detail the assumptions that we've made. Below are some suggestions on what to talk about; addressing two of these is a good level of extension for your assignment.
If density remains constant, the mass inside the region B can be found as m = ρ vol(B). Investigate how this mass can be found for non-constant density, that is, a heterogeneous fluid. Write this expression as a functions of time, and differentiate with respect to time to find an expression for the change in mass over time.
Incompressible fluids don't really exist in the real world. Explain how a truly incompressible fluid would violate the laws of special relativity.
When we wrote down the mass flow through a face of the cube, we made an approximation. Explain whether this approximation would still be reasonable if the velocity vector field v was not differentiable continuous.
Workshop 8
The field of optimisation is dedicated to solving problems of the form: Find the best solution, according to some given criterion. A mathematical opti- misation problem underlies any real-world problem that involves a concept of efficiency.
A community of five outback homesteads wants to run landlines between themselves to allow communication. They decide a straight-line trunk cable with perpendicular lines to each of the homesteads is easiest. Relative to the westernmost homestead the second is 5km east and 15km north, the third 20 km east and 10 km south, the fourth 30 km east and 10 km north, and the final homestead is 50km due east.
Problem 1
Find the location of the trunk that minimizes the total amount of cable that needs to be laid. Also find the total length of the cable for this setup.
Extension Problems
This problem has a fair amount of computer work involved and extension work is required for the assignment. For the assignment, investigate all of the following:
Demonstrate, using visualisations, that it is not enough to consider one fixed pair of initial/final homesteads
Explore why the solution you found may not be the best possible (nei- ther mathematically nor physically). Can you find (or describe) a better solution?
Suppose that, due to environmental conditions, the slope and intercept of the trunk was constrained to be 5mT + c ≥ 3. Solve this inequality- constrained optimisation problem (via MATLAB, Python, Mathematica, Wolfram) and comment on the relationship to the original solution.
Workshop 9
For this problem, there is 150 grams of rubber to be split and formed into two rubber wheels. Wheel 1 is a solid disc with thickness 1cm. Wheel 2 is a ring (cylinder) with a 1cm×1cm square cross section of solid rubber (see Figure below).
Problem 1
Find parametric equations which relate the radii of the wheels to the mass of one of the wheels.
Problem 2
The kinetic energy of a rolling object is 1/2mv2 + 1/2Iω2 where m is mass, v is translational velocity, I is the moment of inertia and ω is the angular velocity.
Consider releasing the two wheels at the top of a hill (of appropriate height). Use the kinetic energy formula to determine which wheel will reach the bottom first.
Problem 3
Introduce a third wheel, made from the same lump of rubber. The third wheel is a hollow tube with circular cross section of diameter 2cm with 0.5cm thick walls.
Extension Problems
Investigate changing the rubber mass given to wheel 1, 2 and 3, and if/how this changes the result of problem 2. Demonstrate your observations with a plot.
Workshop 10
The ideal gas law PV = nRT is used to approximate the behaviour of gas in a system. Here P is pressure, V is volume, n is the amount of gas, R is an ideal gas constant and T is temperature. This workshop will examine a cylinder in a four-stroke engine and approximate the power output as combustion happens.
Problem 1
Use the ideal gas law to model the changes in pressure and volume as the engine undergoes it's four strokes.
Problem 2
For a piston head with area A, the force applied from inside the cylinder is F = PA. Formulate the total work done on the piston head as a path integral. Use Green's Theorem to simplify the path integral to a double integral.
Extension Problems
Investigate both of the following:
Find (approximate) manufacturer values for the various assumptions made throughout the calculations and produce a power value for your model,
Show that the path integral and the double integral evaluate to the same answer by explicitly computing the line integral.
Workshop 11
Problem 1
Model a tornado as particles rotating around the z-axis. By considering the linear velocity vector field v→, find the relationship between curl (v→) and angular velocity ω.
Problem 2
A desert island in the shape of the top quarter of a sphere is covered in mini- tornadoes. Explain why you can measure the combined effect of the tornadoes without venturing into the centre of the island.
Extension Problems
Investigate both of the following:
Explain Stokes Theorem in the context of Problem 2 with mathematical exposition.
Parameterise the surface in Problem 2 and convert the vector field integral of Stokes Theorem into a standard surface integral via the parameterisa- tion.
Optional replacement for Problem 2: Replace tornadoes by a magnetic field produced by current flowing in a wire. Explain why the magnetic field only depends on the electric field on the surface of a conductor.