**Unit Learning Outcomes**

LO1: Identify the relevance of mathematical methods to a variety of conceptualised engineering examples.

LO2: Investigate applications of statistical techniques to interpret, organise and present data

**Assignment Brief and Guidance**

**Scenario:**

You work as a Test Engineer for a global manufacturer of electrical and mechanical components and systems. Your Line Manager is responsible for delegating to you and your colleagues the testing of theory, principles, and hypotheses from several worldwide company divisions. They have asked you to undertake a series of such evaluations.

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**Activity:**

**Part 1:**

a) Hooke’s Law is given by the simple expression 𝐹 = 𝑘𝑥, which defines the force required to extend a spring by a distance 𝑥, where 𝑘 is the spring’s stiffness constant.

Use dimensional analysis to determine the dimensions of 𝑘.

b) A mass 𝑚, suspended by a spring of stiffness 𝑘, is set into simple harmonic motion. Theperiod of oscillation is given by:

Using dimensional analysis, demonstrate that the period of oscillation does, indeed, have units of time

c) A sphere of radius 𝑟, is moving through a fluid of density 𝜌, at velocity 𝑣. The velocity is sufficiently high such that the effect of the fluid’s viscosity may be ignored. It is postulated that the resistive force acting on the sphere:

**𝐹 ∝ 𝑟 𝑣 p**

Use dimensional analysis to confirm the relationship and develop a final formula for the resistive force, F.

d) A digital frequency synthesiser, manufactured by your Signals Division, generates voltage samples of a linear waveform, measured in mV. The samples are generated as values 1, 5, 9, 13, 17, 21………etc.

Your Test Department colleague has asked you to assist by determining a formula that describes the sequence in terms of the nth term of that sequence, and then to use that formula to determine the value of the 273rd sample, which has failed to be recorded correctly.

The final stage of the process requires the summation of all samples up to and including the 273rd sample. Determine a formula to produce the result of this summation.

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e) A digital chip, made by your Microelectronics Division, produces the start of a count sequence as follows: 10, 14, 20, 28, …

It is known that the sequence is based on a quadratic formula of the form 𝑎𝑛2 + 𝑏n + 𝑐, where 𝑎, 𝑏, & 𝑐 are unknown coefficients, and 𝑛𝑛 is the sample number. Determine the unknown coefficients a, b, & c, and therefore, the equation of the quadratic. Now apply your formula to work out the value of the 10th count.

f) A series electrical circuit for a computer game which you are testing, features a capacitor (C) discharging via a 270 kΩ resistor (R). The capacitor is initially charged to a voltage of 15 V. The voltage across the capacitor (Vc) may be described by the following equation, where t represents time.

g) One of your commonly used laboratory instantaneous test signal voltages (vs) is described

by the equation…

where f = 500 kHz and t, represents time.

Make the time t, the subject of this formula, and hence determine the first point in time when the instantaneous signal voltage has a magnitude of +5V.

Use this software or this software to draw at least two cycles of this signal and annotate the drawing so that your non-technical colleagues may understand the relevant information it contains.

h) An electric cable strung between supporting posts, takes the form of the following curve, which is called a catenary, and which has the following relationship:

a = minimum height above the ground of the cable

d = distance between the supporting posts

(i) If the height of the posts is 10 m and the minimum height of the cable above the ground is

8 m, determine the distance between the posts.

(ii) Determine the length of the cable s, between the posts in (i) above, given by the following

equation:

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**Part 2:**

a) Your Manufacturing Division produces capacitors with a nominal value of 47 μF, with a tolerance of +/- 10%. A random sample of ten capacitors was taken, and their value was measured by a colleague, to check that they remain within tolerance. The results are as follows:

(i) Calculate the mean capacitance for these samples.

(ii) Determine the standard deviation for the samples.

(iii) Produce a Tally Chart showing the frequency of measured capacitances.

b) Further analysis of a larger sample of the capacitors shows that 85% are within the allowable tolerance value.

The remainder exceed the tolerance. You select eight capacitors at random from the sample. Determine the probabilities that:

(i) Two of the eight capacitors exceed the tolerance.

(ii) More than two of the eight capacitors exceed the tolerance.

c) Your analysis shows that the mean capacitance of a batch of 500 of the capacitors you have selected is 46 μF, with a standard deviation of 4 μF. Assuming the capacitors are normally distributed, determine the number of capacitors likely to have values between 42 μF and 50 μF.

d) The Quality Assurance Department is anxious to improve the nominal value of the capacitors to ensure more of them fall within the rated tolerance band. The manufacturing process is adjusted in an attempt to bring about this improvement. Following the adjustment, a further sample of 100 capacitors is taken in order to determine if the adjustment has resulted in an improvement to the nominal value of the capacitors. Analysing this second sample, it is noted that the mean value is now 46.5 μF, with a standard deviation of 2 μF.

By comparing these new results of the capacitor values following the manufacturing adjustment, with the values obtained in part (c) recorded before the adjustment, show whether or not you agree with the hypothesis that the adjustment to the manufacturing process has had a beneficial effect by making the capacitor values closer to the desired manufactured nominal value of 47 μF.