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SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS

SEMESTER 2 EXAMINATIONS 2011/2012

DYNAMICS

ME247

DR. N.D.D. MICHÉ

Time allowed:THREE hours

Answer:AnyFOUR from SIX questions

Each question carries 25 marks

This is a CLOSED-BOOK examination

Items permitted:Any approved calculator

Items supplied:Formulae sheet (attached - page 8)

Marks for whole and part questions are indicated in brackets ( )

Question 1

A force of 500 N acts on point A in the z direction where the cables AB, AC, and AD are joined as shown in Figure Q1 below:

Figure Q1

The system of forces and cable tensions is in equilibrium.

(a)What are the necessary conditions for static equilibrium?

(5 marks)

(b)Draw the free-body diagram isolating the system at point A.

(4 marks)

(c)Express the unit vectors of AB, AC and AD.

(6 marks)

(d)Determine the expression for the force vectors of the tensions in cablesTAB, TAC, TAD in terms of unit vectors and magnitudes TAB, TAC, TAD.

(3 marks)

(e)Calculate the magnitudes of the tension in cables AB, AC and AD for this system in equilibrium.

(7 marks)

Question 2

(a)What is meant by the centroid of a body?

(5 marks)

(b)Determine the location of the centroid of the composite area with a cut-out shown in Figure Q2(a) below:

Figure Q2(a)

(10 marks)

(c)Determine the area moment of inertia about Ox for the area shown under the curve in Figure Q2(b) below:

Figure Q2(b)

(10 marks)

Question 3

A jet aircraft pulls up into a vertical curve about point C of constant radius ρ=1500m as shown in Figure Q3 below. As it passes the position at point B, where θ = 30°, its speed is 1000 km/hr and was decreasing at a constant rate of 15km/hr per second between points A and B.

Figure Q3

(a)Describe the normal-tangential coordinate system and what curvilinear motion means.

(6 marks)

(b)Calculate the velocity of the aircraft at point A.

(6 marks)

(c)Determine the tangential and normal components of the acceleration of the aircraft at point B.

(6 marks)

(d)Deduce the and components of the acceleration of the aircraft at point B in the (x,y) coordinate system shown in Figure Q3.

(5 marks)

(e)Calculate the magnitude of the overall acceleration of the aircraft at point B.

(2 marks)

Question 4

Figure Q4 shows a slider-crank mechanism typical of a reciprocating engine. Crank OB of length r = 125 mm has a constant clockwise rotational speed of 1500 rev/min and connects to sliding piston A via a connecting rod AB whose centre of gravity G is located as shown in Figure Q4. Consider the instant where the position of crank OB is at an angle θ = 60° for your calculations and the angle β between connecting rod AB and the horizontal to be β = 18°.

Figure Q4

(a)What is a rigid body? Name the different types of rigid-body plane motion.

(5 marks)

(b)According to the mechanism shown in Figure Q4, determine the expressions of velocity vectors vB, vA, ωAB and ωOB in terms of components (i,j,k) and velocity magnitudes vB, vA, ωAB and ωOB respectively.

(4 marks)

(c)Calculate the magnitudes of velocity vB of point B, velocity vA of point A and the angular velocity ωABof link AB at the instant shown in Figure Q4.

(10 marks)

(d)Calculate the velocity vG of point G at the instant shown in Figure Q4.

(6 marks)

Question 5

Figure Q5 shows a pulley mechanism hoisting a 200 kg wood log up a 30° ramp by releasing from rest a 125 kg concrete block A. The coefficient of kinetic friction between the log and the ramp is µk = 0.5.

Assume that g = 9.81 m/s2.

Figure Q5

(a)Draw the free-body diagrams of the mass A, pulley system at C and log D.

(5 marks)

(b)Calculate the friction force on the log being pulled up the ramp.

(4 marks)

(c)Determine the dependent motion relationship between the displacement sA of mass A and displacement sC of pulley C. Derive the relationship between the accelerations of A and C.

(4 marks)

(d)Calculate the accelerations of pulley C, block A and the tension T in the cable attached to A when A is released.

(8 marks)

(e)Determine the velocity of block A as it hits the ground at B.

(4 marks)

Question 6

Figure Q6 shows a 5.5 kg lever OA of mass moment of inertia about O, IO=0.35kg.m2, connected to a spring of spring coefficient, k = 52 N/m. The lever OA is initially at rest when θ = π/2 rad, and the spring is un-stretched in this position. It then drops in a clockwise direction as shown in Figure Q6.

Assume that g=9.81m/s2.

Figure Q6

(a)Explain briefly the different types of potential energy.

(6 marks)

(b)Calculate the stretch lengths s1 and s2 of the spring when θ = π/2 rad and θ=0rad respectively.

(4 marks)

Question 6 continues on the next page

Question 6 (continued)

(c)Determine the expressions of kinetic and potential energy of the system when θ=π/2rad and θ = 0 rad respectively.

(8 marks)

(d)Calculate the angular velocity of the rod when θ = 0 rad.

(7 marks)

Formula Sheet

Vector Notation:

; ;

Unit Vector : uA= A/A= (Ax/A)i + (Ay/A)j+(Az/A)k

where, ,

Centroid :

Area Moment of Inertia :

Polar Coordinates:

  • Velocity : ; Acceleration :

where,

Equation of motion

  • ; ; where, or , ,
  • Constant acceleration: , ,

Relative general plane motion:

Energy:

  • Elastic potential Energy :
  • Gravitational potential Energy :
  • Linear motion Kinetic Energy :
  • Rotation Kinetic Energy :

ME247 (2011/2012)Page 1 of 9