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48221 ENGINEERING COMPUTATIONS

Spring 2021

Instructions for Assessment Task Two

The Numerical Methods Quiz is a take home quiz that replaces what would normally be an in-class test. This quiz includes four (4) questions that students must complete by writing Matlab code.

Submission overview: A higher-level summary of the requirements for this assessment task is presented in the table below.

Submission requirements: To complete this task, students should develop one (1) Matlab script for questions one, two and three. For question four it will be more convenient for students to use a combination of functions and a single driver script to solve this problem. Once the attempt for each of the four questions is complete, students should place their Matlab scripts into a .zip file and submit this zip file to Canvas.

Marking Criteria Assessment Task Two

Question One [7.5 marks]

Hooke’s Law states that when a force is applied to a spring constructed of uniform material, the length of the spring is a linear function of that force. We can write the linear function

where F(l) represents the force required to stretch the spring l units, the constant E represents the length of the spring with no force applied and the constant k is the spring constant. The constant E is 5.3 cm.

Measurements were made (see Figure 1) of the length l in cm, for applied weights in kilograms as given in the following table:

Using an acceleration due to gravity of 9.8 kgms-2 , use linear least squares to estimate the spring constant k by regressing F(l) against (l-E).

Figure 1: Diagram of the mass-spring system for testing Hooke’s Law.

Question Two [7.5 marks]

A trough of length L has a cross section in the shape of a semicircle with radius r (see Figure 2). When filled with water to within a distance h of the top, the volume V of water is

Find the depth of water within the trough to within 0.01 feet (using the method of bisection) with L = 10 feet, r = one foot and V = 12.4 foot³.

Figure 2: Diagram of the semi-circular trough partially filled with water.

Question Three [7.5 marks]

The following data describe the position versus time of a car travelling down a road. Compare the performance of the forward difference and central difference techniques for calculating the velocity of the car at each point in time.

Question Four [7.5 marks]

The Fresnel integrals are used in the study of the diffraction of light and are given by:

Construct a table of values for c(t) and s(t) for t = 0.1, 0.2, …,1.0 using the composite trapezoidal method.