ENGR20004 S2 Exam 2014 PDF

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The University of Melbourne Department of Mechanical Engineering

Semester 2 2014 ENGR20004 Engineering Mechanics

Reading time: 15 minutes

Writing time: 3 hours

This paper has five pages of questions and one page of formulas (not including this page). Authorised materials School-approved calculators may be used. No other materials are authorised.

Instructions to invigilators Script books should be provided. This paper may be taken by students at the end of the exam. Paper to be held at Ballieu Library.

Instructions to students Answer all five questions. The total mark is 100. Start your answer to each question on a new page. A formula sheet is provided at the beginning of this paper.

Formulas δ=

PL , AE

δT = α∆T L,

g = 9.81 m s−2 , X

G=

U1-2 = T2 − T1 ,

X

I=

Z

mi v i = mv,

1 T = mv2 , 2

v B = v A + (v B|A )t ,

My M d2 u = , , dx2 EI I X Z t2 F dt = G(t2 ), G(t1 ) +

y2 dA,

U 1-2 =

σ=−

Z

t1

F · ds,

P = F · v = T ω,

1-2

(vB|A )t = rω,

v B|A = ω × r B|A ,

(aB|A )n = rω 2 ,

aB = aA + (aB|A )n + (aB|A )t ,

aB|A = ω × (ω × r B|A ) + α × r B|A , X X F = ma, MP = Iα + mad.

1

(aB|A )t = rα,

Question 1 The frame ABCD, supported by pinned joint A and roller joint (slider) D, is supporting a 1-kN point force at C. D 1 kN 30 mm A

B

40 mm

C

40 mm

(a) Determine the reactions at A and D (5 marks). (b) Draw the free-body diagram of ABC and determine all forces and moments acting on it (5 marks). (c) Draw and label the normal-force, shear-force and bending-moment diagrams of ABC (10 marks).

2

Question 2 A

10 kN

B

500 mm 500 mm

(a) The steel bar shown (Young’s modulus, E = 200 GPa; coefficient of thermal expansion, α = 12 × 10−6 K−1 ; cross-sectional area, A = 100 mm2 ) is subjected to the 10-kN axial load. Determine the displacement at B (4 marks). A

B P

500 mm 500 mm

(b) The steel bar of part (a) is now subjected to the axial load P . Determine the displacement at B in terms of P (4 marks). A

∆T = 20 K

B

500 mm 500 mm

(c) The steel bar of part (a) is now subjected to the 20-K temperature rise. Determine the displacement at B (4 marks).

A

∆T = 20 K 10 kN

B

500 mm 500 mm 0.1 mm

(d) The steel bar of part (a) is now subjected to both the 10-kN axial load and the 20-K temperature rise, and bounded by a 0.1-mm gap to a fixed wall. Determine the reactions at A and B (8 marks).

3

Question 3 b

Section A-A y

1 3h

Neutral axis

2 3h

(a) Show from first principles that the second moment of area I about the neutral axis of the triangular cross section A-A shown is given by I = bh3 /36. The triangle has two equal sides (8 marks). L/2

L/2 F

A A M F L/4

+ L/2

x L

(b) The simply supported beam shown (L = 1 m) with uniform cross section A-A (b = 100 mm, h = 150 mm) from part (a) is supporting the point load F = 1 kN. The bending-moment diagram is given below it. Determine the maximum tensile bending stress through the section at x = L/2 (4 marks). (c) Under the loading conditions shown in part (b), determine the bending deflection at x = L/2. The beam is made from steel with Young’s modulus given by E = 200 GPa (8 marks).

4

Question 4 vA vB vC mA mB mC y x

A fireworks display, is made from three masses (mA = mB = 100 g, mC = 200 g) as shown. Gravity points down. (a) The fireworks display is held together and launched vertically upwards from the ground with initial velocity v A = v B = v C = 100 m s−1 . Determine the maximum height relative to ground of the three masses if joined for the whole duration (6 marks). vB vA mB

mA mC vC

(b) Now, the fireworks display is launched as described in part (a), but the masses then split. Just before the split, the velocity is given by v A = v B = v C = 2 m s −1 j. Just after the split, the velocity of A is given by v A = −5 m s −1 i + 5 m s−1 j and the velocity of C is given by v C = −2 m s−1 j. Determine the velocity of B just after the split (6 marks). (c) Determine the maximum height relative to ground of the centre of mass after the split described in part (b) (2 marks). (d) Determine the total energy required for the split described in part (b) to occur (6 marks).

5

Question 5 300 mm

θ

y

G 400 mm x

A

A motorcycle rider is doing a wheelie as shown without the rear wheel slipping. At the instant shown, the speed of point A is 15 m s−1 towards the left, the acceleration of point A is 0 m s−2 , θ˙ = 3 s−1 and ¨θ = 0 s−2 . Gravity points down. (a) Determine the velocity of point G, v G = vGxi + vGy j (4 marks). (b) Show that the acceleration of point G is given by aG = aGxi + aGy j = 2.7 m s−2 i − 3.6 m s−2 j (4 marks).

G A

(c) Complete the free-body diagram of the combined 200-kg mass without the rear wheel as shown to determine the magnitude of the torque applied to the rear wheel at A. Treat the combined mass as a rigid body with centre of gravity at G (8 marks). (d) Given that the radius of the wheel is 250 mm, determine the power applied to the rear wheel at A (4 marks).

End of exam

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Library Course Work Collections

Author/s: Mechanical Engineering Title: Engineering Mechanics, 2014 Semester 2, Engr20004 Date: 2014 Persistent Link: http://hdl.handle.net/11343/52306...