Task 1 – Use of mathematical methods for a variety of engineering examples.

 Use the method of dimensional analysis to verify whether (or not) the following formulae are dimensionally valid.

Note that u and v are velocities, a is an acceleration, s is a distance and t is a time.

1. s = (2)(u+v)t²
2. 2v² = 2u² + 4as

• F = ma

1. An oil company bores a hole 100m deep. Estimate the cost of boring if the cost is £150 for drilling the first metre with an increase in cost of £10 per metre for each succeeding metre.
2. The value of a CNC milling machine, originally valued at £240000, depreciates 15% per annum.

(i)Calculate its value after 3 years.

(ii)The machine is sold when its value is less than £125000. After how many years is the machine sold?

1. The instantaneous voltage v in a capacitive circuit is related to time t by the exponential function:

v = Ve^−t/CR       where V, C and R are constants.

Determine v, correct to 4 significant figures, when t = 50 ms, C = 7 µF, R = 40 kΩ and V = 350 volts.

1. Two voltage phasors V₁ =100V and V₂ =190V are shown in the diagram below represented by arrows. Use trigonometry to determine the value of their resultant (i.e. length OA) and the angle the resultant makes with the horizontal, V₁.
2. The length, l, of a heavy cable hanging under gravity, is given by the hyperbolic function l = 2c sh(L/2c). Find the value of l, to 2 decimal places, when c = 50 and L = 70

(Note: show your working out)

1. Using dimensional analysis, determine the values for a, b and c in F = M^a V^b R^c which result with the derivation of the formula relating the relationship between the centrifugal force, F, of a particle dependent on its mass, M, its velocity, V, and the radius, R, of the curvature of its path.
2. Using dimensional analysis, determine the values for a, b and c in V = P^a d^b g^c    which result with the derivation of the formula relating the velocity of a gas particle, V, in terms of the pressure, P, experienced by it, its density d and its gravity g.

Task 2 – Applications of statistical techniques           

 

1. The length in millimetres of a batch of 36 electrical components are as shown below:

2.10 2.29 2.32 2.21 2.14 2.22

2.26 2.15 2.21 2.17 2.28 2.15

2.15 2.25 2.23 2.11 2.27 2.35

2.24 2.05 2.29 2.18 2.24 2.16

2.15 2.22 2.14 2.27 2.19 2.21

2.23 2.05 2.13 2.26 2.16 2.12

 

(i) Form a frequency distribution table of the lengths having four classes.

(ii) Form a cumulative frequency distribution table.

(iii) Determine the mean and the standard deviation for your grouped data.

1. A machine is producing a large number of bolts automatically. In a box of these bolts, 85% are within the allowable tolerance values with respect to their diameter, the remainder being outside of the diameter tolerance values.

Five bolts are drawn at random from the box. Use binomial distribution to determine the probabilities that (i) one of the bolts and (ii) more than three of the five bolts are outside of the diameter tolerance values.

NOTE: Round your answers at your final answer only to obtain a much more accurate answer here.

1. A component is classed as defective if it has a diameter of less than 50mm. In a batch of 400 components, the mean diameter is 52mm and the standard deviation is 1.8mm. Assuming the diameters are normally distributed, determine how many are likely to be classed as defective.
2. An automatic CNC machine produces bolts which are stamped out of a length of metal. Various faults may generally occur during the production process, such as, the heads or the threads may be incorrectly formed, the length might be incorrect, and so on.

Assuming that 4 bolts out of every 100 produced are defective in some way. If a sample of 50 bolts is drawn at random, then the manufacturer might be satisfied that his defect rate is still 4% provided that there are no more than 2 defective bolts in the sample. The manufacturer’s quality manager has formulated the hypotheses: H₀ : p=0.04 The manufacturer has also made a decision that should the defect rate rise to 6% or more, he will take some action to protect the companies good name.

Using the binomial distribution determine the probabilities of getting 0,1,2 or 3 defective bolts in the sample and interpret your statistical hypothesis test results (refer to the error types) in the provision of justified advice for the manufacturer.

 

1. The quality control department within an aluminium foundry are carrying out some analysis of production measurement data. They wish to display the data graphically to enable non-technical staff to better understand the measures which they are calculating. They utilise CMM technology to measure the diameters, in millimetres, of 60 small holes bored in their ‘Panther’ brand engine castings and the results are as shown.

 

2.011–2.014                 7,                2.016–2.019                 14,

2.021–2.024                 25,              2.026–2.029                 8,

2.031–2.034                 6

 

 

• Determine analytically (by calculation) the mean and median diameters for the grouped data.
• Use an appropriate computer software package to generate a histogram depicting the results shown in the data table above.
• Using the histogram that you have created determine the mean, median and modal values of the distribution commenting on any differences between the values determined in part (i).