TW9
UNIVERSITY OF BOLTON
SCHOOL OF ENGINEERING
MENG/BENG (HONS) IN
MECHANICAL/AUTOMOBILE ENGINEERING
SEMESTER 2 EXAMINATION 2015/2016
ENGINEERING PRINCIPLES 2
MODULE NO: AME4053
Date: Monday 16th May 2016
Time: 10:00 – 12:00 noon
INSTRUCTIONS TO CANDIDATES:
There are SIX questions on this
paper.
Answer ANY FOUR questions.
All questions carry equal marks.
Marks for parts of questions are
shown in brackets.
CANDIDATES REQUIRE:
Formula sheet (attached)
Page 2 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Q1
a)
A diver jumps off a 5m high diving board. The diver enters the water 1.5
seconds later. Taking g= 10 m/s2 determine:
i) The speed the diver generates in his jumps
(4 marks)
ii) His maximum height measured from the top of the diving board (3 marks)
iii) The speed with which he enters the water
b)
(3marks)
A flywheel is accelerated uniformly from rest for 3 seconds until it is rotating at
210 rev/min. It revolves at this speed for 4 minutes and is then decelerated
uniformly at the rate of 5 rad/s2 until it comes to rest. For the flywheel
determine:
i) The uniform angular acceleration
(5 marks)
ii) The deceleration time
(5 marks)
iii) The total number of complete revolutions made
(5 marks)
Total 25 marks
Q2 a) using the parallel axis theorem Determine the second moment of area and
radius of gyration about axis QQ of the triangle BCD shown in Figure Q2a.
Figure Q2a
(15 marks)
b) A solid circular section bar of diameter 20 mm is subjected to a pure bending
moment of 0.3 kN·m. If E = 2x1011 N/m2. Determine the resulting radius of curvature
of the neutral layer of this beam and the maximum bending stress.
(10 marks)
Total 25 marks
Please turn the page
Page 3 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Q3
a) An internal combustion engine of 44.74 kW transmits power to the car wheels
of an automobile at 300 rev/min. Determine :
i)
The minimum permissible diameter of the solid circular section steel
shaft, if the maximum shear stress in the shaft is limited to 50 MPa.
ii)
The resulting angle of twist of the shaft due to the applied torque over a
length of 2 m ,giving that the rigidity modulus G = 70 GPa.
(15 marks)
b) A solid disc flywheel has a mass of 120kg and radius of gyration of
100mm.from rest a constant torque of 10N·m is applied for a periods of 2
seconds. Ignoring any frictional effect determine:
i)
The moment of inertia of the flywheel
(4 marks)
ii)
The angular acceleration
(3 marks)
iii)
The angular velocity after 2 seconds
(3 marks)
Total 25 marks
Q4
a)
A thin cylinder 75 mm internal diameter ,250 mm long with walls 2.5 mm thick
is subjected to an internal pressure of 75MN/m2. Determine:
i) the change in internal diameter
ii) the change in length
iii) the Hoop and longitudinal stresses
Take E =200GN/m2 and Poisson’s ratio = 0.3
(5 marks)
(5 marks)
(5 marks)
b) A tensile load of 800 kN is applied axially to a solid circular cross –section bar.
If the unstrained bar has a length of 500 mm and a diameter of 90 mm,
calculate the change in volume of the bar when the load is applied.
Take E = 200 GPa and Poisson’s ratio = 0.3
(10 marks)
Total 25 marks
Please turn the page
Page 4 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Q5 integrate
23
i) ∫1
d𝑥
(3 marks)
ii) ∫0 𝑒 𝑥 𝑑
(4 marks)
𝑥
2
iii) ∫(1 + 2𝑥)4 𝑑𝑥
2
iv) ∫1 ln 𝑥 𝑑𝑥
1
v) ∫ √9− 2 𝑑𝑥
𝑥
using substitution method
(4 marks)
using the integration by parts method (4 marks)
using the substitution method when 𝑥 = 3 sin 𝑢
(10 marks)
Total 25 marks
Q6
a) Find the derivative of
i) sin (-2 𝑥)
(3 marks)
2
ii) Cos 3 𝑥
(3 marks)
b) Differentiate using the product rule i) y= 𝑒 𝑥 Sin 2 𝑥
ii) y= 𝑥 4 Ln (2 𝑥)
c) Differentiate using the quotient rule i) y=
𝑆𝑖𝑛𝑥
(3 marks)
(4 marks)
√𝑥
ii) y =
(3 marks)
(2𝑥+1)(3𝑥−1)
(𝑥+5)
(4 marks)
d) Differentiate by the function of function method
y= (4𝑥 − 5)6
(5 marks)
Total 25marks
Page 5 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Formula Sheet.
Stress and Strain:
F
 
A
u

L
E


F
A
  shear strain



G

K
x 
y 
z 

V
y
E
z
E
E
V

V
V
x
E
 long
 long
 lat  
 volumetric
 lat


y
E
x
E

z
E
.......etc
.................etc
E
31  2 
E
G
21   
K 
Static Equilibrium:
∑ 𝐹𝑥 = 0 ; ∑ 𝐹𝑦 = 0 ; ∑ 𝐹𝑧 = 0 ; ∑ 𝑀𝑥 = 0 ; ∑ 𝑀𝑦 = 0 ; ∑ 𝑀𝑧 = 0
Page 6 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Thin Pressure Vessels:
 hoop 
pd
2t
pd
4t
pd
1  2 l
 longitudinal 
4tE
p
 diametral 
1   d 2
4tE
 longitudinal 
For Cylindrical Shells:
V 
pd
5  4 V
4tE
For Spherical Shells:
V 
3 pd
1   V
4 tE
2nd Moments of Area
Rectangle
I=
bd3
12
Circle
I=
πd 4
64
Polar J =
d 4
32
Parallel Axis Theorem
Ixx = IGG + Ah2
Bending
M  E
I

y

R
Torsion
T  G
J

r


Motion
v = u + at
v2 = u2 + 2as
u v 
t
 2 
2 = 1 + t
22  12  2
 12 
t
 2 
s= 
 
s = ut + ½ at2
  1t  t 2
1
2
Page 7 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Speed
= Distance
Time
Acceleration =
s = r
V = r
a = r
Torque and Angular
T  I
I  mk 2
P  T
Energy and Momentum
Potential Energy = mgh
Kinetic Energy
Linear
Angular
Momentum
Linear
Angular
= ½ mv2
=½ I2
= mv
= I
Vibrations
Linear Stiffness k 
F

Circular frequency n 
Frequency f n 
x  r cos t
k
m
n
1

2 Tn
v   r 2  x 2  r sin t
a   2 x
f 
T
1
T
2

F  ma
Velocity
Time
Page 8 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Integration
∫ 𝑥 𝑛 ⋅ 𝑑𝑥 =
𝑋 𝑛+1
+𝐶
𝑛+1
∫ (𝑎𝑥 + 𝑏)𝑛 ⋅ 𝑑𝑥 =
(𝑛 ≠ −1)
(𝑎𝑥 + 𝑏)𝑛+1
+𝑐
𝑎(𝑛 + 1)
∫
1
⋅ 𝑑𝑥 = 𝐿𝑛|𝑥| + 𝐶
𝑥
∫
1
1
⋅ 𝑑𝑥 = ln|𝑎𝑥 + 𝑏| + 𝑐
𝑎𝑥 + 𝑏
𝑎
∫ 𝑒 𝑥 ⋅ 𝑑𝑥 = 𝑒 𝑥 + 𝑐
∫ 𝑒 𝑚𝑥 ⋅ 𝑑𝑥 =
1 𝑚𝑥
𝑒 +𝑐
𝑚
∫ cos 𝑥. 𝑑𝑥 = sin 𝑥 + 𝑐
∫ cos 𝑛𝑥. 𝑑𝑥 =
1
sin 𝑛𝑥 + 𝑐
𝑛
∫ sin 𝑥. 𝑑𝑥 = −cos 𝑥 + 𝑐
1
∫ sin 𝑛𝑥. 𝑑𝑥 = − cos 𝑛𝑥 + 𝑐
𝑛
∫ sec 2 𝑥. 𝑑𝑥 = tan 𝑥 + 𝑐
∫ sec 2 𝑛𝑥. 𝑑𝑥 =
∫
1
√1 −
𝑥2
1
tan 𝑛𝑥 + 𝑐
𝑛
⋅ 𝑑𝑥 = sin−1 𝑥 + 𝑐
∫
𝑥
⋅ 𝑑𝑥 = sin−1 ( ) + 𝑐
𝑎
√𝑎2 − 𝑥 2
∫
1
⋅ 𝑑𝑥 = tan−1 𝑥 + 𝑐
1 + 𝑥2
∫
1
𝑎2
1
1
𝑥
⋅ 𝑑𝑥 = 𝑡𝑎𝑛−1 ( ) + 𝑐
2
+𝑥
𝑎
𝑎
(𝑛 ≠ −1)
Page 9 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
𝑦 =𝑢⋅𝑣
𝑡ℎ𝑒𝑛
𝑢
𝑦=𝑣
𝑑𝑦
𝑑𝑢
𝑑𝑣
=𝑣
+𝑢
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑡ℎ𝑒𝑛
𝑑𝑦
𝑑𝑥
=
𝑣
𝑑𝑢
𝑑𝑣
+𝑢
𝑑𝑥
𝑑𝑥
𝑣2
Differential Equations
Auxiliary equations for differential equations of the form
𝑎
𝑑2𝑦
𝑑𝑦
+𝑏
+ 𝑐𝑦 = 0
2
𝑑𝑥
𝑑𝑥
Real and Different routes
𝑦 = 𝐴𝑒 𝛼𝑥 + 𝐵𝑒 𝛽𝑥
Repeated (Real and Equal)
𝑦 = 𝑒 𝛼𝑥 (𝐴 + 𝐵𝑥)
Complex (𝑝 +
𝑖𝑞)
−
𝑦 = 𝑒 𝑝𝑥 (𝐴 cos 𝑞𝑥 + 𝐵 sin 𝑞𝑥)
Numerical Methods
𝛥𝑥 =
(𝑏 − 𝑎)
𝑛
Simpson Rule
𝐴𝑟𝑒𝑎 ≈
𝛥𝑥
(𝑦 + 4𝑦1 + 2𝑦2 + 4𝑦3 + 2𝑦4 … + 4𝑦𝑛−1 + 𝑦𝑛 )
3 0
Trapezoidal Rule
𝐴𝑟𝑒𝑎 ≈ 𝛥𝑥 (
𝑦0
𝑦𝑛
+ 𝑦1 + 𝑦2 + 𝑦3 + 𝑦4 … + )
2
2
Mid-Point Rule
𝐴𝑟𝑒𝑎 ≈ 𝛥𝑥 (𝑦1 + 𝑦2 + 𝑦3 + 𝑦4 … + 𝑦𝑛 )
Page 10 of 10
School of Engineering
MEng/BEng (Hons) in Mechanical Engineering
Semester 2 Examination 2015/2016
Engineering Principles 2
Module No. AME4053
Statistics
Frequency Tables
∑ 𝑥2𝑓
√
𝜎=
− (𝑥̅ )2
∑𝑓
𝑥̅ =
∑ 𝑥𝑓
∑𝑓
Binomial Distribution
Mean =𝜇 𝑜𝑟 𝑋̅ = 𝑛𝑝
Standard deviation =√𝑛𝑝(1 − 𝑝)
𝑃𝑟 (𝑋) = 𝑛𝐶𝑥 𝑝 𝑥 (1 − 𝑃)𝑛−𝑥 Where 𝑛𝐶𝑥 =
𝑥
𝑛
∑ ( ) (𝑝)𝑥 (1 − 𝑃)(𝑛−𝑥)
𝑥
𝑥=0
Product Moment
𝑥̅ =
𝛴𝑥
𝑛
𝑠𝑥𝑥 = 𝛴𝑥 2 − 𝑛𝑥̅ 2
𝑠𝑦𝑦 = 𝛴𝑦 2 − 𝑛y̅ 2
𝑠𝑥𝑦 = 𝛴𝑥𝑦 − 𝑛𝑥̅ 𝑦̅ =
𝑠𝑥𝑦
√𝑠𝑥𝑥 . 𝑠𝑦𝑦
𝑛!
𝑥!(𝑛−𝑥)!