THE UNIVERSITY OF HONG KONG
BACHELOR OF ENGINEERING: LEVEL 1 EXAMINATION
FOUNDATIONS OF ENGINEERING MECHANICS (ENGG1010)
Date: May 17, 2011 Time: 2:30p.m.- 5:30p.m.
This paper contains SIX questions (THREE in Section A and THREE in Section B). Full
marks can be obtained by the correct solutions to FOUR questions (of which TWO must be
from each Section). All questions carry the same marks.
Electronic Calculators:
Only approved calculators as announced by the Examinations Secretary can be used in this
examination. It is candidates' responsibility to ensure that their calculator operates
satisfactorily, and candidates must record the name and type of the calculator used on the
front page of the examination script.
Foundations of Engineering Mechanics (ENGG1010) Page2
SECTION A
Al. (a) When a body is in equilibrium and there are only three coplanar forces (no couple or
moment) acting on it, show that the three coplanar forces must be concurrent or
parallel. (3 marks)
(b) Figure Al shows a jack for lifting up one side of a car at its front or rear for the
purpose of servicing one of its wheels (e.g. due to a punctured tyre). The jack is
made of aluminum and therefore it is very light. The jack is operated by a winding
handle (not shown in Figure Al) which is inserted into the hole at G for turning
member GH. Member GH passes through two big pins at C and F. It is only
allowed to turn inside the hole of the big pin at F and prevented from axial
movement relative to this big pin. There are screw threads on more than half of
member GH such that the big pin at C will move along member GH and thereby lift
(or unlift) member BCE when member GH is being turned.
When a car is being lifted using this jack, the weight of the car is gradually taken up
by component AB. It is given that the vertical loading taken up by component AB
is 300 kgfwhen the jack is in a position as shown in Figure Al.
(i) Draw a free body diagram of the jack with exclusion of component AB and
therefore determine the vertical and horizontal reaction forces of the ground at
point D (Hint : You may assume that the vertical force acting at point B is also
300 kgf). (6 marks)
(ii) Draw a free body diagram of member DEF and therefore determine the forces
acting on it at points E and F. (8 marks)
(iii) If the diameter of the pin at E, which is under double shear, is 10 mm,
determine the shear stress that this pin is subjected to. (4 marks)
(iv) Explain why a curved surface is designed for the base of this jack. (4 marks)
[Qn. Al is cont'd on Page 3]
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Foundations of Engineering Mechanics (ENGG 101 0) Page 3
[Qn. A1 is cont'd]
E
E
~
\
\
\
\
e\
\()~*,-+----------~
.... _ ...
E
E
8
.....
30mm 25mm 20mm
Figure A1
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Foundations ofEngineering Mechanics (ENGG1010) Page4
A2. (a) Figure A2(a) shows a bus wheelarch panel. Determine the second moments of area
of this panel with respect to
(i) the x- and y-axes as shown in Figure A2(a); and (4 marks)
(ii) the x- and y-axes through the centroid of the panel. (6 marks)
y
I. O.Bm .. 1.. 0.9m ..I .. 0.5m .. I
Figure A2(a)
(b) The two faces of the clamp shown in Figure A2(b) are 260 mm apart when the two
stainless-steel bolts connecting them are unstretched. A force P is applied to
separate the faces of the clamp so that an aluminum alloy bar with a length of
260.60 mm can be inserted as shown in Figure A2(b). The cross-sectional areas of
each of the bolts and the bar are 120 mm2 and 625 mm2 , respectively. After the load
is removed, the temperature is raised 100°C. Determine
(i) the axial stresses in the bolts and in the bar; and (12 marks)
(ii) the distance between the faces of the clamp. (3 marks)
Assume that the moduli of elasticity of stainless-steel and aluminum alloy are
190 GPa and 73 GPa, respectively, and the corresponding coefficients of
thermal expansion are 17.3 X 1o-6 I oc and 22.5 X 1o-6 I °C.
[Qn. A2 is cont'd on Page 5]
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Foundations of Engineering Mechanics (ENGG1010) Page 5
[Qn. A2 is cont'd]
40mm I 260mm l40mm
Figure A2(b)
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Foundations ofEngineering Mechanics (ENGGIOIO) Page 6
A3. (a) By considering the geometry of a deformed element in pure shear, show that the
Young's modulus of elasticity E, the shear modulus of elasticity G and the
Poisson's ratio u of a linearly elastic material are governed by the following
relationship :
G= E (10 marks)
2(1 + u)
(b) A shaft AD consisting of three segments is subjected to torques as shown in Figure
A3. The segment AB is made of copper and has a shear modulus of elasticity 40
GPa, segment BC is made of aluminum alloy and has a shear modulus of elasticity
30 GPa, and segment CD is made of steel and has a shear modulus of elasticity 80
GP a. The materials of the three segments are linearly elastic.
Determine the following :
(i) Angle of twist (in degrees) between points A and D. (9 marks)
(ii) Rate of twist (degrees per meter) in each of the 3 segments. (3 marks)
(iii) Maximum shear stress in the shaft. (3 marks)
Copper Alumlnum Steel
cross section cross section cross section
B c
3500Nm
1----- 2000Nm
\. 400
.\. mm .\. 600
.\
Figure A3
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Foundations of Engineering Mechanics (EN GG 101 0) Page 7
SECTIONB
B1. (a) A light rail vehicle (LRV) takes 1 minute to go from one stop to the next. The three
phases of the motion are described by the following velocity functions :
0,; t,; Ts: v(t) ~ 20(1-e-~ JmS 1
where v(l) ~ 15 ms- 1
T 5, t 5, 50s : v(t) = 15 ms- 1
1
50 5, t 5, 60s: v(t) = 15 cos(;o (t- so)) ms-
Determine the following :
(i) The time Tin seconds at which the LRV stops accelerating. (2 marks)
(ii) The maximum acceleration in ms-2 during the first T seconds. (2 marks)
(iii) The distance travelled in metres during the first T seconds. (3 marks)
(iv) The distance travelled in metres during the final 10 seconds. (3 marks)
(v) The total distance in metres between the two stops. (2 marks)
(b) As shown in Figure B1(a), a ball of mass m=10 kg rotates around the vertical pole
in a horizontal circular path of radium R=1 m. The strings are attached to rings that
rotate around the pole with negligible friction. The rings are also prevented from
moving vertically along the pole.
/
/
--- .................. ,
/
I
/ '\ drag
I I
I
I
\
'''
--..,...r:: ----
- ............ _,....
.......
/
I
I
/
'I
I
I I
'' ///
/
~-
(a) (b)
Figure B1
[Qn. B1 is cont'd on Page 8]
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Foundations ofEngineering Mechanics (ENGG1010) Page 8
[Qn. Bl is cont'd]
(i) Determine the angular rotational speed ro for the ball when the two strings
have the same tension, T. Determine T. (4 marks)
(ii) If string Bin Figure Bl(a) is removed, the rig becomes Figure Bl(b). The ball
starts with a high rotating speed by an impulsive push and is gradually slowed
down by air drag, -kv2 , where v is the instantaneous tangential speed of the
ball, and k=O.Ol kg/m. The angle between the string and the pole, 8, changes
from nearly 90deg down to Odeg as the ball decelerates by friction in a helical
path. Determine the speeds of the ball, v, when 8 reaches 55deg and 35deg,
respectively. (4 marks)
(iii) Determine the time taken for the ball to rotate from 8 = 55deg to 35deg in the
helical path by considering the tangential motion. (5 marks)
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Foundations ofEngineering Mechanics (ENGG1010) Page 9
B2. Take density ofwater = 1,000 kgm·3, acceleration due to gravity= 9.81 ms· 2.
(a) The inclined differential manometer of Figure B2(a) contains carbon tetrachloride
(relative density = 1.59). Initially the pressure differential between pipes A and B,
which contain a brine (relative density = 1.1 ), is zero as illustrated in the figure. It
is desired that the manometer give a differential reading of 0.3 m (measured along
the inclined tube) for a pressure differential of 0.5 kPa. Determine the required
angle of inclination, e. (1 0 marks)
Brine
Carbon
Tetrachloride
Figure B2(a)
(b) As in Figure B2(b), a 3 m high, 6 m wide rectangular gate is hinged at the top edge
at A, and is restrained by a fixed ridge at B.
(i) Determine the hydrostatic force exerted on the gate by the 5 m high water, and
the depth ofthe centre of pressure. (10 marks)
(ii) Assuming that the hinge A is frictionless, calculate the restraining force to be
applied at B. (5 marks)
E
N
- ••}-----'
A
Water
.--Gate
..,E
8
--------------~--~~~-------
Figure B2(b)
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Foundations ofEngineering Mechanics (ENGG1010) Page 10
B3. (a) For the flow shown in Figure B3(a), estimate the pressure p 1 and velocity V1, if the
velocity of the jet issuing from the nozzle V2 = 20 m/s, and the mercury (relative
density= 13.6) manometer reading H= 10 cm. (9 marks)
-------------., - - - - -
_ _ _ _ _ _ _____j - - -
Hg
... ·:·.•
Figure B3(a)
(b) A 90° conical plug is used to regulate the flow of air (density = 1.20 kg/m3) from
the pipe shown in Figure B3(b). The air leaves the edges of the cone with a uniform
thickness of 0.02 m. If viscous effects are negligible and the flow rate is 0.5 m3/s,
determine
(i) the pressure within the pipe, (8 marks)
(ii) the impact force by the flow on the plug. (8 marks)
Q=o.sm'n
Figure B3(b)
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