Part 1:
Data has been gathered from a lifting system used to transport structural components. The system consists of a drum and cable. The following data was obtained when the drum lowers the load (assume constant acceleration).
Drum diameter = 0.6m
Mass of load = 2kg
Initial velocity = 0 m/s
Time to descend = A secs
Distance travelled = 0.4m
You have been asked to use this information to determine:
a) The final linear velocity of the load
b) The linear acceleration of the load
c) The final angular velocity of the drum
d) The angular acceleration of the drum
e) The tension force in the cable
f) The torque applied to the drum
In determining the above quantities, you should clearly state the formulae used and apply dimensional analysis techniques to show that all values/units used are homogeneous.
From the formulae used (above) you should apply dimensional analysis techniques to develop two equations for power (in terms of linear velocity and angular velocity).
Part 2:
a) An excavator is purchased from new for £20,000. It is thought that the machine depreciates according to the following law; . Here represents the time in years, is the initial cost and is the value after t years. k can be found in the table on page 5. Determine the time it takes for the machine to reach a value of £10,000.
b) A building site is shown below. Determine the perimeter of the site and calculate the area.
c) An electrical cable is to be suspended across two supports. The supports are a distance L metres apart. The cable forms the shape of a catenary of the form y = c cosh (x/c) where x = L/2. You have been asked to determine the fixing point height (y) if the minimum clearance (c) is to be 3m at the centre of the catenary.
Part 3:
a) The annual cost of hiring a machine is £8,000. The contract with the hiring company states the hiring cost is due to increase by £B each year. Determine the cost of hiring the machine during in year 12. Also calculate how much the machine will have cost in total during the first 15 years.
b) A drill is to have seven speeds ranging from 25 rev/min to C rev/min. If the speeds form a geometric progression, determine their value, each correct to the nearest integer.
Part 4:
Your company currently sources hightensile steel bolts from ParkerFasteners. A random sample of 42 bolts is selected at random and their nominal diameter is measured to the nearest hundredth of a millimetre. Your company specifies that the diameter of the bolt must lie between 10.20mm and 10.45mm inclusive. The results are as follows:
PARKERFASTENERS (Diameters of bolts in mm)
10.39 10.36 10.38 10.36 10.37 10.40 10.28
10.40 10.36 10.28 10.42 10.34 10.46 10.35
10.36 10.49 10.35 10.45 10.29 10.39 10.38
10.38 10.35 10.42 10.30 10.26 10.37 10.33
10.37 10.34 10.34 10.32 10.33 10.30 10.38
10.48 10.35 10.38 10.27 10.37 Y Z
a) Summarise the data using computer software for the following techniques:
i) A frequency distribution table.
ii) The arithmetic mean.
iii) The standard deviation.
iv) A histogram.
b) The bolts are delivered in batches of 500. Assuming that the diameter of the components follow a normal distribution, use a normal distribution curve to calculate the number of plugs from each supplier that can be expected to be outside of the tolerance specified (between 10.20mm and 10.45mm inclusive). You should assume the mean and standard deviation you have already calculated in a) is the same for the batch of 500.
c) One of the machines producing the bolts is causing quality problems as P% of the bolts produced on this machine has been found to be defective. Find the probability of finding 0, 1, 2, 3, and 4 defective parts in a sample of 30 parts (assuming a binomial distribution). You should also present a graphical illustration of the probabilities using appropriate computer software.
d) The machine discussed in part c) is repaired. Your manager claims that after the repair the machine is still producing at least P% defective bolts. A sample of 50 bolts is taken and one is found to be defective. Use a hypothesis test to interpret the results and hence indicate whether to accept or reject your manager’s claim at a 5% significance level.
e) Your manager has asked you to present and summarise, using appropriate software, the statistical data you have been investigating in Part 4 in a method that can be understood by nontechnical colleagues.
Student

A

k

D

L

B

C

Y

Z

P

1

1.96

0.10

25

3.45

150

450

10.20

10.45

8.1

2

1.97

0.11

26

3.50

155

455

10.21

10.44

8.2

3

1.98

0.12

27

3.55

160

460

10.22

10.43

8.3

4

1.99

0.13

28

3.60

165

465

10.23

10.42

8.4

5

2.00

0.14

29

3.65

170

470

10.24

10.41

8.5

6

2.01

0.15

30

3.70

175

480

10.25

10.40

8.6

7

2.02

0.16

31

3.75

180

490

10.26

10.39

8.7

8

2.03

0.17

32

3.80

185

495

10.27

10.38

8.8

9

2.04

0.18

33

3.85

190

500

10.28

10.37

8.9

10

2.05

0.19

34

3.90

195

505

10.29

10.36

9.0

11

2.06

0.20

35

3.95

200

510

10.30

10.35

9.1

12

2.07

0.21

36

4.00

205

515

10.31

10.34

9.2

13

2.08

0.22

37

4.05

210

520

10.32

10.33

9.3

14

2.09

0.23

38

4.10

215

525

10.33

10.32

9.4

15

2.10

0.24

39

4.15

220

530

10.34

10.31

9.5

16

2.11

0.25

40

4.20

225

535

10.35

10.3

9.6

17

2.12

0.26

41

4.25

230

540

10.36

10.29

9.7

18

2.13

0.27

42

4.30

235

545

10.37

10.28

9.8

19

2.14

0.28

43

4.35

240

550

10.38

10.27

9.9

20

2.15

0.29

44

4.40

245

555

10.39

10.26

10.0

21

2.16

0.30

45

4.45

250

560

10.40

10.25

10.1

22

2.17

0.31

46

4.50

255

565

10.41

10.24

10.2

23

2.18

0.32

47

4.55

260

570

10.42

10.23

10.3

24

2.19

0.33

48

4.60

265

575

10.43

10.22

10.4

25

2.20

0.34

49

4.65

270

580

10.44

10.21

10.5

26

2.21

0.35

50

4.70

275

585

10.45

10.20

10.6

