SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ

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1 s SCHOOL OF COMPUTING, ENGINEERING ND MTHEMTICS SEMESTER EXMINTIONS 011/01 DYNMICS ME47 DR. N.D.D. MICHÉ Tme allowed: THREE hours nswer: ny FOUR from SIX questons Each queston carres 5 marks Ths s a CLOSED-BOOK eamnaton Items permtted: ny approved calculator Items suppled: Formulae sheet (attached - page 8) Marks for whole and part questons are ndcated n brackets ( ) May/June 01 Page 1 of 8

2 Queston 1 force of 500 N acts on pont n the z drecton where the cables B, C, and D are joned as shown n Fgure Q1 below: Fgure Q1 The system of forces and cable tensons s n equlbrum. What are the necessary condtons for statc equlbrum? (b) Draw the free-body dagram solatng the system at pont. Epress the unt vectors of B, C and D. (d) Determne the epresson for the force vectors of the tensons n cables TB, TC, TD n terms of unt vectors and magntudes TB, TC, TD. (3 marks) (e) Calculate the magntudes of the tenson n cables B, C and D for ths system n equlbrum. (7 marks) ME47 (011/01) Page of 8

3 Queston What s meant by the centrod of a body? (b) Determne the locaton (, y) of the centrod of the composte area wth a cutout shown n Fgure Q below: y m m 3m O 3 m 3m Fgure Q (10 marks) Determne the area moment of nerta about O for the area shown under the curve n Fgure Q(b) below: y y= 9 9 m O 1 m Fgure Q(b) (10 marks) ME47 (011/01) Page 3 of 8

4 Queston 3 jet arcraft pulls up nto a vertcal curve about pont C of constant radus ρ = 1500 m as shown n Fgure Q3 below. s t passes the poston at pont B, where θ = 30, ts speed s 1000 km/hr and was decreasng at a constant rate of 15 km/hr per second between ponts and B. θ y B Fgure Q3 Descrbe the normal-tangental coordnate system and what curvlnear moton means. (b) Calculate the velocty of the arcraft at pont. (d) Determne the tangental and normal components of the acceleraton of the arcraft at pont B. Deduce the and y components of the acceleraton of the arcraft at pont B n the (,y) coordnate system shown n Fgure Q3. (e) Calculate the magntude of the overall acceleraton of the arcraft at pont B. ( marks) ME47 (011/01) Page 4 of 8

5 Queston 4 Fgure Q4 shows a slder-crank mechansm typcal of a recprocatng engne. Crank OB of length r = 15 mm has a constant clockwse rotatonal speed of 1500 rev/mn and connects to sldng pston va a connectng rod B whose centre of gravty G s located as shown n Fgure Q4. Consder the nstant where the poston of crank OB s at an angle θ = 60 for your calculatons and the angle β between connectng rod B and the horzontal to be β = 18. Fgure Q4 What s a rgd body? Name the dfferent types of rgd-body plane moton. (b) ccordng to the mechansm shown n Fgure Q4, determne the epressons of velocty vectors vb, v, ωb and ωob n terms of components (, j, k) and velocty magntudes vb, v, ωb and ωob respectvely. Calculate the magntudes of velocty vb of pont B, velocty v of pont and the angular velocty ωb of lnk B at the nstant shown n Fgure Q4. (10 marks) (d) Calculate the velocty vg of pont G at the nstant shown n Fgure Q4. ME47 (011/01) Page 5 of 8

6 Queston 5 Fgure Q5 shows a pulley mechansm hostng a 00 kg wood log up a 30 ramp by releasng from rest a 15 kg concrete block. The coeffcent of knetc frcton between the log and the ramp s µk = 0.5. ssume that g = 9.81 m/s. D Fgure Q5 Draw the free-body dagrams of the mass, pulley system at C and log D. (b) Calculate the frcton force on the log beng pulled up the ramp. Determne the dependent moton relatonshp between the dsplacement s of mass and dsplacement sc of pulley C. Derve the relatonshp between the acceleratons of and C. (d) Calculate the acceleratons of pulley C, block and the tenson T n the cable attached to when s released. (8 marks) (e) Determne the velocty of block as t hts the ground at B. ME47 (011/01) Page 6 of 8

7 Queston 6 Fgure Q6 shows a 5.5 kg lever O of mass moment of nerta about O, IO = 0.35 kg.m, connected to a sprng of sprng coeffcent, k = 5 N/m. The lever O s ntally at rest when θ = π/ rad, and the sprng s un-stretched n ths poston. It then drops n a clockwse drecton as shown n Fgure Q6. ssume that g = 9.81 m/s. Fgure Q6 Eplan brefly the dfferent types of potental energy. (b) Calculate the stretch lengths s1 and s of the sprng when θ = π/ rad and θ = 0 rad respectvely. Determne the epressons of knetc and potental energy of the system when θ = π/ rad and θ = 0 rad respectvely. (8 marks) (d) Calculate the angular velocty of the rod when θ = π/ rad. (7 marks) ME47 (011/01) Page 7 of 8

8 Formula Sheet Vector Notaton: F R = F ; F Ry = F F ) ( y R FR + F ; Unt Vector : u = / = (/) + (y/)j+(z/)k Ry θ = tan 1 = ( Ry ) FR F where, = + + y z, Centrod : = rea Moment of Inerta : d d y = yd d I = y d = y = y n d d = n n 1 Polar Coordnates: Velocty : n+ 1 n d = n + 1 ( ) v= vr + vθ = r + ( r θ ) ; cceleraton : r r θ a = r θ + r θ where, a r = ( ) θ ( ) a ar + = a θ Equaton of moton dv dv F n = man ; F t = ma a t = = v t ; where, dt ds or a t = rα, v = rω, a n = rω = v / r = vω Constant acceleraton: v = v 0 + a t, s s + v t + 1 a t v = v + ac s s c =, ( ) 0 0 c 0 0 VB = V + ωb rb Relatve general plane moton: Energy: 1 Ve = ks Elastc potental Energy : V Gravtatonal potental Energy : g = mgh 1 T = mv Lnear moton Knetc Energy : 1 T G = I G ω Rotaton Knetc Energy : ME47 (011/01) Page 8 of 8