Part IB Paper 1: Mechanics Examples Paper Kinematics

Transcription

1 Engneerng Trpos Part B SECOND YEAR Straghtforward questons are marked t Trpos standard questons are marked *. Part B Paper 1: Mechancs Examples Paper Knematcs 1 ---~..-~---~---." 'SSUED ON t 7 JAN 20t4 t 1. A lamna s n the fonn of a trangle ABC, Fg. 1. Corner A has velocty and acceleraton as shown, parallel to BC and BA respectvely. The angular velocty of the lamna at the gven nstant s 2 rad/s antclockwse and s ncreasng at the rate of 1 rad/s2. () Where s the nstantaneous centre for the lamna? () What are the magntudes ofthe veloctes ofb and C? () Where s the centre of curvature of the path ofa? (v) What s the magntude of the acceleraton ofb? (v) Where s the centre of curvature ofthe path ofb? (v) What s the magntude of the acceleraton ofc? (v) What s the magntude of the acceleraton of the pont on the lamna whch s nstantaneously concdent wth? 4m B~ ~C 3m A!4.5 mls 2.. \ o 2rad/s 1 rad/s 2 Fg An engne runs at 6000 rpm and has a crank: AB 40 mm long and a connectng rod BD 120 mm long. Sketch the velocty dagram for an arbtrary crank angle DAB. What s the acceleraton ofthe pston: when angle DAB s (bottom dead centre - see lecture notes); when angle DBA s 90?

2 t3. The rod AB s 2 m long and has the acceleratons shown n Fg. 3 at ts ends. Sketch the acceleraton dagram for the rod and fnd ts angular velocty and acceleraton. /lmls 2 A~==========2=m===========BJ,:~ ~3~QOO '~2m1S2 Fg n the mechansm shown n Fg. 2, AB rotates wth constant angular velocty 0). Draw a velocty dagram and show that the angular veloctes of BC and CD are both -0). Calculate these angular veloctes usng vectors. (Attach unt vectors to each lnk and dfferentate an expresson for vector AD.) By drawng an acceleraton dagram or by dfferentatng your vector expresson from part, determne the components of the acceleraton of G (the md-pont of BC) parallel and perpendcular to AD? (c) How are the components affected f 0) s doubled, reversed n drecton? C A a B Fg The rgd member ABC rotates n ts own plane wth constant angular velocty 0) (Fg. 4). At the gven nstant block D s sldng wth constant velocty v relatve to ABC, and the dstance BD s a. Wrte an equaton for the poston ofpont D. What are the components of the acceleraton of D along and perpendcular to BC? Show that the acceleraton of a pont on the rodbe adjacent to D s a:t 01 towards A. a ~ D C Fg. 4 2

3 6. A wre hoop of radus R rotates n ts own plane wth unfonn angular velocty aj{ about an axs OZ through a pont 0 n ts crcumference (Fg. 5). Q s the centre of the hoop. A bead P moves round the hoop n the same sense wth a unfonn speed v relatve to the hoop. (c) Show that the absolute angular velocty of vector QP s (0) + vr)k. Wrte an expresson for the poston vector of P n tenns of unt vectors el and e2 and dfferentate t twce to detennne the absolute acceleraton of P. Express the absolute acceleraton of P n cartesan coordnates (, j) n tenns of the angle 8 (= OQP). Take OQ as the j-axs nstantaneously. o Fg. 5 7*. Fg. 6 shows one ofthe vanes (or blades) of a centrfugal pump mpeller whch turns wth a constant clockwse angular velocty of 300 rev/mn. Flud enters the mpeller through the central hole, s 'flung' out radally by the vanes, and s then collected n a volute (not shown). The vane surface has a constant radus of curvature R (wth centre at C), and the flud s observed to have an absolute velocty whose radal component s 3 mls at dscharge from the vane. The speed of the elements of the flud measured relatve to the vane ncreases at the rate of24 mls 2 just before they leave the vane. Show that the velocty of the flud relatve to the mpeller at dscharge from the vane s 3.../2 mls. By wrtng a poston vector to the partcle va pont C and usng the same approach as n Queston 6, or otherwse, fnd the magntude and drecton of the absolute acceleraton of an element of the flud just before t leaves the mpeller. 3

4 Fg Lnk AC passes through a fxed swvel B (Fg. 7). AB s 200 mm, BC s 100 mm and the velocty and acceleraton ofa are as shown. Draw the velocty dagram for the rod and fnd the angular velocty of the rod and the velocty ofpont C. Draw the acceleraton dagram for the rod and fnd the acceleraton of pont C. ~60m/s2 '#' 3~ C='=~~==~745~O A OJ. Xonrod Fg. 7 /4.8m1s 4

5 9. At a certan nstant the ends A and B of the hydraulc ram shown n Fg. 8 have zero acceleraton. The ram s extendng wth a speed of 1 mls and the ol flow s ncreasng at a rate correspondng to an acceleraton of 1 mls 2. What are the horzontal and vertcal components of the acceleraton of the pont G on the cylnder, mdway between A and B, when AB s 1 m? s t possble to specfy whether the acceleraton of G s up or down? Hnt: Look at the acceleraton of a pont n polar co-ordnates, n the data book. What equatons can you wrte down, gven that B has zero acceleraton relatve to A? Ol out +. A B W ~.==={~~O~.~G::~~~~ tol n Fg. 8 *10. Fg. 9 shows, to scale, a plane mechansm n whch the jack AB operates between a fxed hnge A and the centre pont B of the rgd lnk CD. f the slder C s movng at the gven nstant wth a constant velocty of 960 mmls, left to rght, determne: the velocty of sldng of the jack pston relatve to ts cylnder, and the angular veloctes of AB and CD. the sldng acceleraton ofthe pston relatve to the cylnder. 170mm _l_._.~._._._._._._._._._._~._ ~ Fg. 9! 5

6 *11. The wheel, radus r, of a car when turnng a comer traces out a path of radus R wth a unfonn velocty v. (c) (d) What s the acceleraton ofpont Q, the centre ofthe wheel? What s the velocty and acceleraton, relatve to Q, of pont P on the tyre whch s nstantaneously n contact wth the ground? Show that the absolute acceleraton of P has components: v 2 1r towards the centre of the wheel and v 2 1 R radally outwards from the centre ofthe curve. Sketch the paths traced-out by Q and by P as the wheel rolls along ts curved path. Account for the outward drecton of the radal acceleraton of P on geometrc grounds. For further practce try the followng B Mechancs Trpos questons: 2000 Qs 3 & 6; 2001 Qs 1 & Q2; 2003 Q6; 2004 Q2a,b; 2005 Q6; 2006 Q5, 6; 2007 Q3b; 2008 Q2a,b (gnore any parts of the above questons whch ask about forces, torques, &c., untl these have been covered later n the course.) ANSWERS 1. () mdway between A and B () 3 mls; 8.54 mls () 2 m from A on BA extended (v) 16.8 mls 2 (v) m from B on BA (v) 22.7 mls 2 (v) 10.6 mls ms- 2 upwards 616 ms 2 downwards radls clockwse or antclockwse; 0.75 radls 2 antclockwse aro2 ; 0.5 aro 2 they are quadrupled unaffected 5. - aro 2 ; 2vro - aro Rael-R(m+v/R)2e2 (c) - (ro 2 R + v 2 /R + 2rov) sne + { (ro 2 R + v 2 /R + 2rov) cose - ro 2 R} j ms- 2, 86.7 antclockwse from nward radus radls clockwse, 3.79 mls at 26.6 to AB; 143 ms- 2 at 68.6 to AB ms 2, 1.0 ms 2, no ms, mas = 1.93 radls ant-clockwse; a.t:d = 4.90 radls antclockwse 1.94 ms- 2. DCebon 6