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SINGAPORE POLYTECHNIC MM2105 / MM8311

2008/2009 SEMESTER ONE EXAMINATION

Diploma in Mechanical Engineering

2nd Year Full-time

Diploma in Aeronautical Engineering

2nd Year Full-time

Diploma in Bioengineering

2nd Year Full-time

Diploma in Mechatronics

2nd Year Full-time

Diploma in Mechanical Engineering3rd Year Evenings-Only

MECHANICS II Time Allowed: 2 Hours

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Instructions to Candidates:

1. The examination rules set out at the back page of the answer booklet are to be complied

with.

2. This paper consists of 5 questions.

3. Answer any FOURquestions.

4. Marks for questions are shown and candidates should allocate their time in proportion to

the marks.

5. A List of Formulae is provided on page 7. Take g = 9.81 m/s.

6. This examination paper consists of 8 pages.

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ANSWER ANY FOUR QUESTIONS

Q1. A cantilever beam 10 m long having an I cross section as shown in Fig. Q1a

carries an uniformly distributed load of 250 N/m over its whole length.

(a) Calculate the second moment of area about its horizontal centroidal axis.

(5 marks)

(b) Calculate the maximum bending moment exerted on the beam and hence the

maximum bending stress and state whether it is tensile, compressive or both.

(10 marks)

A second cantilever beam 10 m long is formed by joining TWO I cross sectionstogether as shown in Fig. Q1b. They have the same cross section as given in Fig. Q1a.

(c) Show that the second moment of area about its horizontal centroidal axis is

82.983 106 mm4. (5 marks)

(d) If the maximum bending stress is limited to 100 MN/m2, what is the maximum

uniformly distributed load that it can carry over its whole length?

(5 marks)

2105sem12008-09 see page 3

Fig Q1a Fig Q1b

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Q2.

(a) A hollow circular shaft, with external diameterD = 400 mm, and internal diameterdmm, is subjected to torsional loading. The maximum allowable shear stress is 50

MN/m2 and polar second moment of area, J= 2.13 10 3 m4.

(i) Show from the polar second moment of area J, that the internal diameter,

d = 250 mm. (3 marks)

(ii) Calculate the twisting moment (torque) and power transmitted at 200 rev/min.

(7 marks)

(iii) If external diameterD and maximum allowable shear stress remain unchanged;

state the effects (i.e. increase or decrease) on twisting moment and power

transmitted, when internal diameterddecreases. (2 marks)

(b) A 10 kg mass attached to the end of a 1.5 metre long rope is whirled in a vertical

plane at 80 rev/min. With relevant free body diagrams, calculate:

(i) the maximum tension in the rope (4 marks)

(ii) the minimum tension in the rope (4 marks)

(iii) the minimum speed in revolution per minute at which the

mass must be whirled to keep the mass in circular motion. (5 marks)

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Q3. A hoist drum raises a cage of mass 950 kgwith an initial acceleration of 0.7 m/s2.

The drum has a mass of 200 kgand an effective diameter of 0.8 m and a radius of

gyration of 0.32 m. There is a frictional torque of 260Nm at the drum bearings.

(a) Sketch the Free Body Diagrams of:

(i) the cage (2 marks)

(ii) the hoist drum (3 marks)

(b) Determine the:

(i) tension in the cord (3 marks)

(ii) torque required at the drum. (7 marks)

After the initial acceleration, the velocity of the cage is maintained constant at 6 m/s.

(c) Determine the torque required for the cage to be raised at this velocity.

(5 marks)

Before reaching the top, the cage moves up with deceleration for another 25 m,

(d) Determine the angular deceleration of the hoist drum. (3 marks)

(e) Sketch the velocity time graph for the whole upward motion of the cage.

(2 marks)

2105sem12008-09 see page 5

Cage

HoistDrum

Fig. Q3

Motion

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Q4. A 4-wheeled vehicle of total mass 1400 kg is accelerated from rest to 60 km/h in a

distance of 480 m up an incline of 1 in 8. It has two axles, each of which together withthe wheels has a mass of 100 kg and radius of gyration of 300 mm about its axis of

rotation. The diameter of each wheel is 660 mm. The tractive resistance is 400 N.

With the aid of a sketch , (2 marks)

calculate, for the speed of 60 km/h,

(a) the moment of inertia of each axle about its axis of rotation. (3

marks)

(b) total potential energy of the vehicle (3 marks)

(c) total kinetic energy of the vehicle (6 marks)

(d) the required tractive effort by using the Principle of Conservation of Energy

(6 marks)

When the vehicle reaches 60 km/h, the tractive effort is removed. If the tractive

resistance remains the same, calculate;

(e) the speed (in km/h) of the vehicle when it travels a further 80 m up

the inclined plane. (5 marks)

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(5 marks)Q5. A crab winch consists of a compound gear train and two pulleys as shown in Fig.Q5.

The diameters of the effort and load pulleys are 400mm and 250mm respectively.

When using the machine, it was found that 300N would lift a load of 180 kg.

(a) Show that the velocity ratio is 6.4. (3 marks)

(b) Determine the mechanical advantage, efficiency and the friction effort in

lifting a load of 180kg. (10 marks)

(c) Determine the power required if the load of 180kg is lifted at a velocity of

0.05m/s. (5 marks)

(d) When the efficiency is 72%, determine the load that can be raised by an effort

of 160N. (5 marks)

(e) Name two ways in which the velocity ratio could be increased. (2 marks)

**********

WE

1 2 3 4

Fig. Q5

N1 = 35 teeth

N2 = 60 teeth

N3 = 30 teeth

N4 = 70 teeth

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LIST OF FORMULAE

Kinematics

v = u + at s = ut + at2 s = (u + v)t v2 = u2 + 2as

Second Moment of Area

I = bd3 I = (d14 d2

4 ) J = (d14 d2

4 )

12 64 32

Torsion Equation Bending Equation

T = G = M = E =

J L r I R y

Simple Lifting Machines Centripetal Force

E = Ei + Ef ; Ei = W C.F = mr2 or mv2

VR r

VR = D (Wheel and Axle)

d

VR = 2D (Differential Wheel and Axle)

d1 - d2

VR = DN (Worm and Wheel Lifting Hoist)

dn

VR = 2R (Screw Jack) np

N2 N4 D

VR = -------- x --- (Crab Winch)

N1 N3 d

Principle of Conservation of Energy

PE0 + KE0 + WD = PE1 + KE1

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Reference Table for Simple Loadings

Simply supported

beam with central

point load

Simply supported

beam with

uniformly

distributed load

(UDL)

Cantilever beam

with central point

load

Cantilever beam

with

uniformly

distributed load

(UDL)

F

B

D

S

F

D

B

M

D

Note: W denotes a concentrated (point) load and may be defined inNewton or kilo-Newton.

w denotes a UDL and may be defined inNewton per metre or kilo-Newton per metre.

W

L

w(N/m )

L

W

L

w(N/m )

L

00

WL00

+W/2

W/2 00+ wL/2

wL/200

W

00

wL2/2

00

WL00

+wL2/8

00

WL/4

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