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SINGAPORE POLYTECHNIC

2010/2011 SEMESTER ONE EXAMINATION

Diploma in Mechanical Engineering2nd Year (B) Full-time

Diploma in Aeronautical Engineering2nd Year (B) Full-time

Diploma in Bioengineering2nd Year (B) Full-time

Diploma in Mechatronics2nd Year (A)Full-time

Diploma in Resort Facilities Services and Management2nd Year (B) Full-time

Diploma in Mechanical Engineering3rd Year (B) 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 compliedwith.

2. This paper consists of 5 questions.

3. Answer any FOUR questions.

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

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

6. The Reference Table for Simple Loadings is provided on page 8.

7. This examination paper consists of 8 pages.

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ANSWER ANY FOUR QUESTIONS

Q1. A 4 m long cantilever beam carries a uniformly distributed load of w N/m over itswhole length. Its cross-section is as shown in Fig Q1. The allowable tensile andcompressive stresses are 40 MN/m and 65 MN/m respectively.

(a) Show that the position of the centroid of the cross section from datum A-A is202.5 mm. (3 marks)

(b) Show the second moment of area about its horizontal neutral axis is 89.22 10 -6 m4

(6 marks)

(c) Determine the maximum UDL, w N/m it can carry. (10 marks)

(d) Determine the factor of safety if the uniformly distributed load is replaced by aconcentrated load of 5 kN at the free end. (6 marks)

Fig Q1.

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G

202.5mm

300mm

200mm

N A

Flange & Web thickness20mm

A A

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Q2. (a) A hollow shaft is to transmit 2 MW at 2400 rev/min . The external diameter, D is to be 2 times the internal diameter, d . The maximum torque is 30 per cent greater thanthe mean torque. The maximum shear stress is limited to 150MN/ m .Taking G=85 GN/m 2 , determine

(i) the mean torque and the maximum torque that can be transmitted. (3 marks)(ii) the internal diameter, d , and the external diameter, D, of the shaft. (7 marks)(iii) the maximum angle of twist in a length of 0.5 m. (3 marks)

(b) A pilot candidate, with a mass of 60 kg , is tested in a centrifuge with a radius of 12m. [see Fig Q2(b)]. The centrifuge rotates in the horizontal plane with a constantangular velocity of 1 revolution every second.

(i) Sketch the free body diagram of the centrifuge, indicatingthe direction of the force. (4 marks)

(ii) Calculate the force exerted on the pilot, indicating the direction of theforce. (4marks)

If the centrifuge starts from rest and is subjected to a constant angular acceleration, of 0. 2rad/s 2,

(iii) what is the magnitude of the total force exerted on the pilot after 20 s?(4 marks)

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12 m

pilot

Pivot, centre

of rotation

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Fig Q2(b).

Q3. A hoist drum lowers a concrete block of mass 350 kg vertically from a building on tothe ground (see Fig Q3). The drum has a mass of 220 kg and an effective diameter of 0.9 m with a radius of gyration of 0.32 m. There is a frictional torque of 160 Nm at thedrum bearings.

The concrete block is lowered under the action of a braking torque with accelerationof 1.2 m/s2 until a velocity of 9.5 m/s is reached. Thereafter, it moves with uniformvelocity for 20 m. Finally, it moves down with deceleration of 1.85 m/s2 to come to

rest on the ground.

(a) Sketch the Free Body Diagrams of:(i) the concrete block (2 marks)(ii) the hoist drum (3 marks)

(b) Determine the tension in the cable and braking torque required at the drumduring the acceleration stage. (10 marks)

(c) Determine the braking torque required at the drum when the velocity ismaintained constant at 9.5 m/s. (5 marks)

(d) Determine the total distance descended by the concrete block. (5 marks)

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Bearings

HoistDrumConcrete

Block

Fig. Q3

Motion

Smooth pulley

Cable

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Q4. In a mine conveying system (see Fig Q4), a wagon is pulled up by a cable-drumsystem. The wagon, at maximum load weighs 2 tonnes in total. The wagon has four wheels, of diameter of 0.5 m. Each axle (together with two wheels) has a radius of gyration of 0.4 m and weighs 150 kg .

The wagon is being pulled up and down by a cable (of negligible weight) wrappedround a drum of diameter 1.0 m, with a mass moment of inertia of 250 kg m2. Thefrictional resistance faced by the wagon is 500 N, while the frictional torque at thedrum is 200 Nm. The wagon is designed to move up from rest, a 20 m inclinedistance in 5 s. The incline is 1 in 10.

Fig Q4.

(a) Show that the two axles with the four wheels have a total mass moment of inertiaof 48.0 kg m2.

(2 mark)(b) Determine the linear velocity and acceleration of the trolley at position 2.

(5 marks)

(c) Determine the total kinetic energy of the trolley at position 2. (6 marks)

Using the Principle of Conservation of Energy,

(d) Find the torque required at the drum to pull the trolley up to position 2 and hencefind the maximum power required. (12 marks)

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1

2Trolley

Drum

Cable

20 m

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N 1 N 3 d

Principle of Conservation of Energy

PE 0 + KE 0 + WD = PE 1 + KE 1

Reference Table for Simple Loadings

Simply supported

beam with centralpoint load

Simply supportedbeam with

uniformlydistributed load

(UDL)

Cantilever beam

with point load

Cantilever beamwith

uniformlydistributed load

(UDL)

FBD

SFD

BMD

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W

L

w (N/m )

L

W

L

w (N/m )

L

00

WL

00

+W/2

W/2

00

+ wL/2

wL/2

00W

00

wL 2/2

00

WL00

+wL 2/8

00

WL/4

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Note : W denotes a concentrated ( point ) load and may be defined in Newton or kilo-Newton .w denotes a UDL and may be defined in Newton per metre or kilo-Newton per metre.

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