MECH ENG 1007/1007UAC Page 1 of 13 PLEASE SEE NEXT PAGE Examination for the Degree of Bachelor of Engineering Semester 2, 2019 MECH ENG 1007/1007UAC, ENGINEERING MECHANICS - DYNAMICS - UAC COURSE ID: 103972 Official Reading Time: 10 mins Writing Time: 180 mins Total Duration: 190 mins Questions Time Marks Answer ALL thirteen questions 180 mins 100 total Instructions for Candidates Answer Section A multiple choice questions (1-10) on the answer sheet provided. Answer Section B long answer questions (11-13) in the blue book provided. This is a closed book examination (Refer to Permitted Materials). Begin each answer on a new page. Examination materials must not be removed from the examination room. The allocated marks for each question are indicated. It is recommended that you tear off the back two sheets of this booklet (containing relevant formulae) for easier reference. Permitted Materials One blue book. One multiple-choice answer sheet. Formula sheets are provided at the back of the question booklet. English language dictionary is permitted. Drawing Instruments and Rules are permitted. Calculator without remote communications capability is permitted. The calculator memory should be cleared. DO NOT COMMENCE WRITING UNTIL INSTRUCTED TO DO SO MECH ENG 1007/1007UAC Page 2 of 13 PLEASE SEE NEXT PAGE Note: You may consider g = 9.8 m.s-2 throughout this examination. Section A Questions 1-10 are MULTIPLE-CHOICE and should be answered in the table provided on the answer sheet. Ensure the answer sheet has your name and student number clearly indicated. Correct multiple-choice answers score 4 marks. No answer scores zero marks. An incorrect answer scores -1 mark. Q1. A ball falls from rest in oil, with an acceleration function given by = − , where the drag coefficient is = 0.65 s-1, and the positive coordinate direction is downward. Which of the following values is closest to the terminal velocity of the ball? [4 marks] a) = 12.0 m/s b) = 13.6 m/s c) = 15.1 m/s d) = 24.3 m/s Q2. A particle begins from rest with an acceleration function = 45 (m/s2), where t is in units of seconds. Which of the following values is closest to the displacement of the particle after an elapsed time of = 0.2 s? [4 marks] a) ∆ = 0.17 cm b) ∆ = 0.42 cm c) ∆ = 1.53 cm d) ∆ = 5.09 cm MECH ENG 1007/1007UAC Page 3 of 13 PLEASE SEE NEXT PAGE Q3. An automobile travels at a constant speed of = 47 km/h around a flat track described in the horizontal plane by = 0.016, where x and y are both in metres. Which of the following values is closest to the normal acceleration of the automobile when = 40 m? [4 marks] a) = 0.87 m/s2 b) = 1.03 m/s2 c) = 1.27 m/s2 d) = 9.80 m/s2 Question 4 pertains to the apparatus shown in Figure 1. A block slides over a frictionless curved surface. At the instant shown, the block is at location A where it has a speed of = 28 m/s. The apparatus is aligned vertically in the Earth’s gravitational field. The instantaneous radius of curvature of the surface at A is = 400m. Q4. Which of the following values is closest to the normal force per unit mass, acting from the surface onto the block at A? [4 marks] a) /! = 12.23 N/kg b) /! = 8.49 N/kg c) /! = 5.55 N/kg d) /! = 5.07 N/kg MECH ENG 1007/1007UAC Page 4 of 13 PLEASE SEE NEXT PAGE Questions 5 and 6 pertain to the mass-pulley system shown in Figure 2. You may assume that the pulley is massless and frictionless. The system, which is in the vertical plane, is released from rest. !" = 5 kg; !# = 20 kg Q5. Which of the following values is closest to the tension in the cable at the moment of release? [4 marks] a) $ = 196.0 N b) $ = 147.0 N c) $ = 98.0 N d) $ = 78.4 N Q6. Which of the following values is closest to the speed of the masses after B has fallen a distance of 12.0 cm? [4 marks] a) = 0.12 m/s b) = 0.77 m/s c) = 1.03 m/s d) = 1.19 m/s Question 7 refers to the direct central impact shown in Figure 3. Data: !" = 2 kg; |"| = 4.0 m/s !# = 3 kg; |#| = 2.4 m/s ' = 0.85 MECH ENG 1007/1007UAC Page 5 of 13 PLEASE SEE NEXT PAGE Q7. Which of the following values is closest to the speed of block A immediately after the impact? [4 marks] a) ()"/ ( = 2.11 m/s b) ()"/ ( = 2.48 m/s c) ()"/ ( = 2.96 m/s d) ()"/ ( = 3.10 m/s Q8. A mass/pulley system, shown in Figure 4, has a light, inextensible cord that remains taut. The pulley may be modelled as a uniform disc. If the total system (A+B+P) has a kinetic energy $ = 20.0 J at an instant, which of the following values is closest to the angular velocity of the pulley [4 marks] a) *+ = 3.75 rad/s b) *+ = 4.92 rad/s c) *+ = 6.28 rad/s d) *+ = 7.95 rad/s MECH ENG 1007/1007UAC Page 6 of 13 PLEASE SEE NEXT PAGE Q9. External forces (,- and ,) are applied to an unhinged object, as shown in Figure 5 below. You may neglect gravity. Data: Force: |,-| = 50 N; |,| = 40 N; . = 30° Object: 0 ̅ = 12.8 kg.m2 Which of the following values is closest to the angular acceleration of the object (with anti-clockwise defined as positive)? [4 marks] a) 2 = −7.27 rad/s2 b) 2 = −0.78 rad/s2 c) 2 = +7.08 rad/s2 d) 2 = +17.17 rad/s2 Q10. Two masses of negligible size are welded to the ends of a massless, rigid beam, as shown in Figure 6. Which of the following values is closest to the mass moment of inertia of the composite object, with respect to a rotation axis that passes through the centre of mass, G, and is perpendicular to the beam axis? Hint: Find the location of the centre of mass first. [4 marks] a) 0 ̅ = 0.63 kg.m2 b) 0 ̅ = 2.19 kg.m2 c) 0 ̅ = 3.00 kg.m2 d) 0 ̅ = 3.14 kg.m2 MECH ENG 1007/1007UAC Page 7 of 13 PLEASE SEE NEXT PAGE Section B Questions 11 to13 are long answer questions and should be answered in the blue booklet provided. Show all relevant working. Ensure the blue booklet has your name and student number clearly indicated. Q11. The mass-spring pulley system shown in Figure 7 is released from rest, with the spring initially uncompressed. You may assume: The spring is linear and massless, with stiffness 4 = 800 N/m Axle friction of 56 = 2.65 N.m in the pulley opposes the pulley rotation There is no friction between B and the inclined surface. The cord remains taut and does not slip over the pulley. The pulley may NOT be modelled as a uniform disk. Other physical data: !" = 12 kg; !# = 36 kg; !+ = 16 kg 7+ = 0.2 m (pulley radius); 4+ = 0.175 m (radius of gyration) a) Draw complete and accurate free-body diagrams for masses A, B and the pulley P. [6 marks] Q11 continued on next page… MECH ENG 1007/1007UAC Page 8 of 13 PLEASE SEE NEXT PAGE …Q11 continued b) Determine an expression for the total work of all forces on the system (A+B+P) in terms of a displacement of mass B a distance “s” down the inclined plane. [5 marks] c) Hence or otherwise determine the maximum displacement (89:) of the system before it comes to rest. [3 marks] d) Show that the distance B travels for the system to achieve maximum kinetic energy (- satisfies the relationship - = ;<=> [6 marks] MECH ENG 1007/1007UAC Page 9 of 13 PLEASE SEE NEXT PAGE Q12. Two balls collide. Their configuration at the moment of impact is shown in Figure 8. The masses of the disks are: !" = 10 kg and !# = 8 kg. Their initial velocities are: )? = 8 m.s-1, )@ = 4 m.s-1 in the directions indicated in Figure 8. ' = 0.7 a) Determine the speed (#/) and direction from the vertical () of B immediately after the impact. [16 Marks] b) What is the percentage mechanical energy loss in the impact? [4 Marks] MECH ENG 1007/1007UAC Page 10 of 13 PLEASE SEE NEXT PAGE Q13. An unconstrained thin beam has a small attached mass at P, as shown in Figure 9. The composite body is at rest, lying on a frictionless horizontal surface. Data: A = 0.4 m; |,| = 20.0 N; !BCDE = 12.0 kg; !F = 4.0 kg. G denotes the centre of mass of the composite body, NOT that of the beam. An external force applied at the location Q, as shown in Figure 9. a) Determine the location of the centre of mass (G) of the composite body. [3 Marks] b) Determine the acceleration of the centre of mass, HG , at this instant [3 Marks] c) Determine the angular acceleration () of the beam at this instant? [9 Marks] Hint: To determine the mass moment of inertia of the composite body use the Parallel Axes Theorem. d) Determine the acceleration of point R (HI) at this instant. [5 Marks] END OF EXAMINATION PAPER FORMULA SHEET TO FOLLOW MECH ENG 1007/1007UAC Page 11 of 13 PLEASE SEE NEXT PAGE MECH ENG 1007, Dynamics Formula Sheet Rectangular Co-ordinates J = K̂ + M̂ ) = N K̂ + N M ̂ H = O K̂ + O M ̂ (Dots indicate time derivatives) Polar Co-ordinates J = PQRS ) = PNQRS + P.NQRT H = UPO − P.N VQRS + UP.O + 2PN.NVQRT Tangential and Normal Co-ordinates ) = QRW H = NQRW + XYZ QR Radius of Curvature = U-[\]YV ^ Y_ |\]]| , where ` = a\ a: and `` = aY\ a:Y Relative Motion (in non-rotating co-ordinates) J# = J" + J#/", )# = )" + )#/", H# = H" + H#/" Transformation of a vector between 2 right-handed co-ordinate systems rotated with respect to each other. If b = : K̂ + \M̂ = :` K̂` + \`M̂` where K̂, M̂ are a pair of orthogonal unit vectors, and K̂`, M̂` is another pair rotated degrees anti-clockwise with respect to the un-primed pair, then the vector components are related by: d:`\`e = f gh . sin . − sin . cos .n d :\e Newton’s Second Law for a system of particles o ,p p = !qHG where !q is the total mass of the system, HG = JO G is the acceleration of the centre of mass, JG, which is defined by: JG = ∑ !pJpp !q_ Newton’s Universal Law of Gravitation s = G8t8YSY where G = 6.673x10-11 m3kg-1s-2 Central Force Motion P.N = = gh Kinetic Energy of a point particle of mass m and velocity v $ = 12 ! MECH ENG 1007/1007UAC Page 12 of 13 PLEASE SEE NEXT PAGE Principle of Work/Energy u-→ = ∆$, where u-→ is the work of forces on a system between states 1 and 2. Generally, for a force, u-→ = w s d cos y; . For a pure couple, u-→ = w 5 d.. For a conservative force, u-→ = −∆z, where V is the potential energy of the force. Work of Selected Conservative Forces Uniform gravity u-→ = −!∆ (y is positive in upward vertical direction) Universal gravity u-→ = {5! | -SY − - St} Linear Spring u-→ = − - 4~ − ~- Potential Energy Functions For a linear spring: z = - 4~ For a mass in a uniform gravitational field: z = ! For a mass in a general gravitational field: z = − G8S Power and Efficiency Power = aaW = , ∙ ) = s ghy, Efficiency = = WW S W S Principle of Impulse and Momentum 0!' = = , = Δ W W- Coefficient of Restitution ' = # ` − "` " − # Relative Motion of 2 points on a Rigid Body For a polar co-ordinate system attached at A, )# = )" + U)#/"VT = )" + P#"*QRT H# = H" + H#/" = H" + UH#/"VS + UH#/"VT = H" + −P#"*QRS + P#"2QRT (Note: * is the angular velocity of the link, and 2 its angular acceleration) Some old maths formulae you may have forgotten For the quadratic equation, ax2 + bx + c = 0, the roots are given by = − ± √ − 4g2 The sine and cosine rules " 9 = # = ¡ ¢ = + g − 2g gh£ MECH ENG 1007/1007UAC Page 13 of 13 PLEASE SEE NEXT PAGE Rigid Body Dynamics mrI d2 ; 0 = 0 ̅ + !; ¤G = 0*̅; Σ5 = ¤NG = 02̅; $ = 12 !G + 1 2 0*̅ 2 2 1 MRIcylinder ; = 0pW 89;; = 0W̅¦p 9§; = 57; 2 5 2 MRI sphere ; 0S̅¨ = -- 5A; 0 = 54I (4I is the radius of gyration) Principle of Angular Impulse and Momentum ¤ ≡ 0* Impulse = dtMH
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