Learning Outcome 1.1
Draw graphs involving algebraic, trigonometric and logarithmic data from a variety of scientific and engineering sources, and determine realistic estimates for variables using graphical estimation techniques
1) The voltage v across an inductor L, in LR electrical circuit drops exponentially over time t (s). The relationship is:
v = Ee-t/τ
Where the emf, E, and t are constants. Use the table of values for t and v and logarithms to plot an appropriate straight line graph.
|
t
|
0.00
|
0.010
|
0.020
|
0.025
|
0.030
|
0.040
|
0.050
|
|
v
|
5.50
|
2.950
|
1.600
|
1.150
|
0.850
|
0.450
|
0.250
|
Use your graph to estimate the values of E and t. How long did it takes to half the initial voltage?
Learning Outcome 1.2
Make estimates and determine engineering parameters from graphs, diagrams, charts and data tables
2) The following data are for cold-worked carbon steel tested in a tensile test:
Original diameter : 12.78 mm
Final diameter : 8.43 mm
Original gauge length : 49.8 mm
Final gauge length : 61.51 mm
|
Load
(kN)
|
5.5
|
11.7
|
17.0
|
22.1
|
27.4
|
40.5
|
53.1
|
60.5
|
64.7
|
66.9
|
68.7
|
70.5
|
71.5
|
72.1
|
72.8
|
|
Extension
(mm)
|
0.010
|
0.020
|
0.030
|
0.040
|
0.050
|
0.076
|
0.102
|
0.127
|
0.152
|
0.178
|
0.203
|
0.254
|
0.305
|
0.356
|
0.508
|
The maximum load was 75.1 kN. Plot the stress - strain graph and determine:
a. realistic estimation of the modulus of elasticity
b. relationship between stress and strain
Learning Outcome 1.3
Determine the numerical integral of scientific and engineering functions
3) a. Use the trapezium rule with eight strips to estimate the area of an ellipse with a circumference defined by:
2 0∫Π/2√(5 + sin2t)dt
b. Repeat (a.) using Simpson's rules
Learning Outcome 1.4
Estimate values for scientific and engineering functions using iterative techniques.
4) Use the Newton-Raphson method correct to 4 d.p's. to find the roots of the following equation.
t2e3t = 25
Learning Outcome 2.1
Represent force systems, motion parameters and wave forms as vectors and determine required engineering parameters using analytical and graphical methods
5) a. A ship is heading in a direction of N 50° E at a speed which in still water would be 20km.h. It is carried off course by a current of 8km/h in a direction of E 60° S.
(N 40° E = 50 degrees from the north axis in a east direction)
(E 60° S = 60 degrees from the east axis in a south direction)
i. Calculate the ships actual speed.
ii. Calculate the ships actual direction.
b. Determine the resultant force acting on the eyebolt shown in the diagram below as result of the four forces shown in the figure below, analytically and graphically. Compare and comment on the two results
Learning Outcome 2.2
Represent linear vector equations in matrix form and solve the system of linear equations using Gaussian elimination.
6) Use Gaussian elimination; calculate the tensions, T1, T2,T3 in a simple framework are given by the simultaneous equations:
5T1, 5T2, 5T3 = 7.0
T1, 2T2, 4T3 = 2.4
4T1, 2T2, = 4.0
Learning Outcome 2.3
Use vector geometry to model and solve appropriate engineering problems
7) A force of (2i -2j + k) newtons acts on a line through point P having coordinates (0, 6, and -4) metres. Determine the moment vector and its magnitude about point Q having co-ordinates (4, -5, 3) metres.