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1 EN40: Dynacs and Vbatons Fnal Exanaton Wednesday May School of Engneeng own Unvesty NAME: Geneal Instuctons No collaboaton of any knd s petted on ths exanaton. You ay use double sded pages of efeence notes. No othe ateal ay be consulted Wte all you solutons n the space povded. No sheets should be added to the exa. Make dagas and sketches as clea as possble, and show all you devatons clealy. Incoplete solutons wll eceve only patal cedt, even f the answe s coect. If you fnd you ae unable to coplete pat of a queston, poceed to the next pat. Please ntal the stateent below to show that you have ead t `y affxng y nae to ths pape, I aff that I have executed the exanaton n accodance wth the Acadec Hono Code of own Unvesty. PLEASE WITE YOU NAME AOVE ALSO! 1-10: (0 PTS) 11: (15 PTS) 1: (5 PTS) 13: (10 PTS) 14: (10 PTS) TOTAL (60 PTS)

2 FO POLEMS 1-10 WITE YOU ANSWE IN THE SPACE POVIDED. ONLY THE ANSWE APPEAING IN THE SPACE POVIDED WILL E GADED. ILLEGILE ANSWES WILL NOT ECEIVE CEDIT. 1. The fgue on the ght shows a dsc wth adus ollng on the gound wthout slp. A and ae two ponts on the edge of the dsc. The cente of the dsc oves wth velocty v. 1.1 What s the speed of pont A? A v j (a) Need oe nfo (b) v A v C (c) v A v (d) v A 3v (e) v A v ecall v, and v A k A/ C so v A k A/ C ( v / ) k j v 1. What s the speed of pont? (a) Need oe nfo (b) v v (c) v v v v (d) 3v (e) v ANSWE E (1 POINT) v k / C so v ( v / ) k ( j) vj v and the speed s the agntude of v. A vetcal downwad foce F s appled at the edge of a dsk of adus as shown. The dsk then otates clockwse by 90 degees, whle ollng wthout slp. The foce acts vetcally thoughout and acts on a fxed ateal pont n the dsk. What s the wok done by F? (a) Need oe nfo (b) F (c) F / (d) F (e) F ANSWE C (1 POINT) F F The wok done s F d Fj ( dx dyj ) Fdy F. 0 0 ANSWE ( POINTS) 1

3 3. In the fgue shown, the sall gea otates counteclockwse wth angula speed. The centes of both geas ae fxed. The lage ng gea has angula speed (a) clockwse (b) counteclockwse (c) clockwse (d) counteclockwse (e) None of the above ng gea The ponts on the two bodes whee they touch ust have the sae veloctes. So / and snce ths s postve the otaton decton s counteclockwse ANSWE D ( POINTS) 4. Mak each of the foces below as consevatve (C) o non-consevatve (NC) (a) Gavty (b) The foce actng between two agnets (c) uoyancy (d) A dag C C C NC ( POINTS) 5. Two objects of asses 1 and ae placed on a table and ae connected by a spng as shown n the fgue below. Assue thee s no fcton between the table suface and the objects. If 1 s held fxed, the natual fequency of vbaton of s found to be. When s held fxed, ass 1 has natual fequency of vbaton (a) Need oe nfo (b) (c) 1 (d) 1 (e) 1 1 oth ae just sple spng ass systes - 1 k / 1 k / 1 / 1 ANSWE D ( POINTS)

4 6. A oto wth total ass M=50kg has otatng ntenal ass of =1kg that otates on a shaft wth eccentcty e=1 at angula ate 100 ad/s. The engne s ounted on vbaton solaton pads wth stffness k= N/ and a dashpot coeffcent c=50 Ns/. The syste s found to have a sevee vbaton poble. Wll the followng changes educe the vbaton apltude? (a) Incease the stffness of the pads k (b) Decease the stffness of the pads k (c) Incease the speed of the oto (d) Decease the dashpot coeffcent fo the pads c YES YES YES NO c M e k The esonant fequency s n / ad/s. The syste s theefoe at esonance. The dapng facto c / km Wth ths the esonant peak s vey shap, so changng k, o changng the oto speed wll take the syste away fo esonance and so decease the vbaton apltude. Deceasng c wll educe and so ake the vbaton wose. ( POINTS) 7. A ass-spng syste shown n the fgue s subjected to a haonc foce F( t) F0 snt. Let x(t) denote the deflecton of the ass fo ts statc equlbu poston. What s the equaton of the oton fo ths syste? k A d x k 0 (a) A k F x sn t dt (b) d x ka k F0 x sn t dt k (c) d x k ka F0 x sn t dt (d) d x kak F0 x sn t dt ( ka k ) The spngs ae n paallel and so have effectve stffness ka k. Ths s a standad undaped foced syste, so has the standad EOM wth ths effectve stffness ANSWE A ( POINTS) F(t) 8. The fgue shows a block wth ass / on a wedge wth dentcal ass. The ente assebly acceleates to the ght. All contacts ae fctonless. What foce P s necessay to ensue that the block eans a fxed dstance d fo the base of the wedge? (a) P = g / 45 0 (b) P = g/ (c) P = g (d) P g / (e) None of the above A FD fo the sall ass s shown. Newton fo the sall ass gves ( N / ) ( N / g / ) j ( / ) a N g / a g. F=a fo the whole syste gves P=g ANSWE A ( POINTS) P N d / N g/ 3

5 9. A satellte ccles the eath n a ccula low-eath obt wth adus. In ths obt the total enegy of the satellte (.e. knetc plus potental enegy) s E. Appopate ocket buns ae then used to tansfe the satellte nto a ccula geosynchonous obt wth adus 5. The total enegy (KE+PE) of the satellte n the new obt s (a) E/5 (b) 5E j (c) E (d) 4E (e) 4E/5 Ccula oton wth gavtatonal foce Fg GM / e so F=a n the adal decton gves v / GM /. The total enegy s GM / v / GM /. So E GM / changes to GM / 10 E / 5 when s nceased fo to 5. ANSWE A ( POINTS) 10. The echancal behavo of soe polyec ateals can be dealzed as a spng-dashpot cobnaton as shown n the fgue. The followng tests ae conducted on a polye specen to detene ts effectve stffness k and dashpot c k coeffcent c. () A ass of 100kg s suspended fo the ba, and ts statc deflecton s easued to be 1. () The ass s then stuck to set t n oton, and ts vbaton esponse s easued. It s found that the apltude of the sxth oscllaton s M M educed to about 30% of that of the fst oscllaton. The stffness and dapng coeffcent ae appoxately (a) k=10 5 N -1, c=770 Ns -1 (b) k=10 6 N -1, c=770 Ns -1 (c) k=10 5 N -1, c=40 Ns -1 (d) k=10 6 N -1, c=640 Ns -1 (e) None of the above Fo the statc test 3 6 F g kx k g / x / N /. The log deceent s (1 / 5) log(1 / 0.3) and so Fnally c / k c k 766 / ( ) ANSWE ( POINTS) 4

6 11. A ba of ass s suppoted by two olles, whch spn apdly n opposte dectons as shown n the fgue. At the nstant shown, the cente of ass of the ba s a dstance x fo olle. Hozontal oton of the ba s essted by a spng wth stffness k, whch s unstetched when x=0. The coeffcent of knetc fcton between the olles and the ba s. k A d x 11.1 Daw a fee body daga showng the foces actng on the ba T A N A x F S T g N d [ POINTS] 11. Wte down Newton s law of oton and the equaton fo otatonal oton fo the ba. (The thckness of the ba can be neglected) d x Newton s law gves ( TA T Fs ) ( N NA g) j dt Fo otatonal oton we have N ( d x) N x 0 A 11.3 Hence, show that x(t) satsfes the equaton of oton d x g k x g dt d The two contacts slp, so TA N A, T N and the spng foce law gves Fs kx. Also fo the otatonal equaton and the j coponent of F=a we have that N N A g d N N A 1 0 x Add these equatons to see that N gx / d then substtute back to see N g(1 x / d). Theefoe A TA T Fs g(1 x / d) kx and the coponent of the EOM gves d x g(1 x / d) kx. Ths can be eaanged nto the equaton gven. dt 5

7 11.4 Hence, fnd an expesson fo the natual fequency of vbaton of the syste. eaange nto standad fo d x k g g x 0 dt d k / g / d The natual fequency s theefoe n k g d [ POINTS] 11.5 If the syste s eleased fo est wth x=0, what s the esultng apltude of vbaton? g If we set y x n the EOM and note that the ntal condtons fo y ae k / g / d g y and dy/dt=0 at t=0, we can ead off the soluton fo the standad esults. k / g / d g y cosnt. k / g / d g The vbaton apltude s theefoe Y0 k / g / d 11.6 Descbe befly the oton of the syste f k g / d exta cedt f you answe s n vese. (no calculatons ae equed). One pont It appeas to e If k g / d The eo s case On the handout we gave you Fo ths we can see Fo lage values of t, x Aexp( t) Wth A abtay, and k / g / d Instead of vbaton Ths s just tanslaton. So, the ba oves to the ght And soon vanshes fo sght [ POINTS] 6

8 1. The fgue shows a pston-cank echans. The cank A otates wth constant angula speed of ad/sec. At the nstant shown, calculate the followng quanttes, expessng you answe n the {,j,k} bass shown. j 3 C 4 A 1.1 The velocty of pont v k / O k 3j 6 / s [ POINTS] 1. The velocty of pont C and the angula velocty of ebe C Note that C ust ove n the decton. Theefoe vc v Ck C / 6 Ck ( 4 3 j) vc The and j coponents of ths equaton gve two equatons fo, v. Clealy 0 v 6. C c C c 7

9 13. Consde a ng of adus ass and adus, ntally spnnng at angula speed 0 n the clockwse decton. The thckness of the ng s neglgble copaed to ts adus. The spnnng ng s placed on a hozontal suface wth coeffcent of fcton. The ng ntally slps on the suface, then begns to oll wthout slp. j 13.1 Daw the fee body daga fo the ng just afte t coes nto contact wth the suface. g j T N [ POINTS] 13. Show that, when the ng s slppng, the acceleaton of the cente of the ng s a g and ts angula acceleaton s α g / k Newtons law gves T ( N g) j ag The otatonal equaton of oton gves T IGk ecall also that snce the contact slps T Solvng these gves N=g, T g, a G N and fo a ng g g / IG [ POINTS] 8

10 13.3 If t = 0 when the ng coes nto contact wth the suface, at what te does ollng wthout slp coence? Integatng the angula acceleaton gves 0 t 0 gt / Integatng the lnea acceleaton gves v gt. Fo ollng wthout slp v and theefoe gt 0 gt t 0 / ( g) 13.4 What ae the velocty of the cente of the ng and the angula velocty of the ng when ollng wthout slp coences? Substtutng fo the tes n the expessons fo the velocty and angula velocty gves 0 / v 0 / so the angula velocty of the ng s always halved, egadless of the coeffcent of fcton 9

11 14. The fgue shows the scsso-lft echans on an acaft cateng tuck. Mebes A and CD both have length L. Mebe A otates about A and oves though a slde at ; slaly, ebe C otates about D and oves along a slde at C. The payload has ass, and s ased by a oent (o toque) M appled to ebe A at pont A. D M A C h 14.1 Wte down the heght h of the ass n tes of. Hence, detene a foula fo the vetcal speed dh/dt of the payload n tes of d / dt elevant vaables. and any othe Sple geoety gves sn dh d h L L cos dt dt [ POINTS] 14. Assue that the vetcal speed of the ass s constant. Usng enegy ethods, detene the oent M as a functon of and any othe elevant vaables. The tuck s a consevatve syste so the ate of wok done by the oent M has to be equal to ts ate of change of potental enegy (snce the speed s constant) d dh d Theefoe M g gl cos M gl cos dt dt dt 10

12 14.3 Now, assue that the oent M s constant. Use enegy ethods to fnd a foula fo the vetcal acceleaton of the ass. In ths case the ate of wok done by M has to be the ate of change of PE and KE. Theefoe d d 1 dh dh dh d h M gh g dt dt dt dt dt dt 1 d h M g L cos dt [5 POINTS] 11