EN40: Dynamics and Vibrations. Final Examination Friday May : 2pm-5pm

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EN4: Dyaics ad Vibratios Fial Exaiatio Friday May 8 15: p-5p School of Egieerig Brow Uiversity NAME: Geeral Istructios No collaboratio of ay kid is peritted o this exaiatio.

1 EN4: Dyaics ad Vibratios Fial Exaiatio Friday May 8 15: p-5p School of Egieerig Brow Uiversity NAME: Geeral Istructios No collaboratio of ay kid is peritted o this exaiatio. You ay brig double sided pages of referece otes. No other aterial ay be cosulted Write all your solutios i the space provided. No sheets should be added to the exa. Make diagras ad sketches as clear as possible, ad show all your derivatios clearly. Icoplete solutios will receive oly partial credit, eve if the aswer is correct. If you fid you are uable to coplete part of a questio, proceed to the ext part. Please iitial the stateet below to show that you have read it `By affixig y ae to this paper, I affir that I have executed the exaiatio i accordace with the Acadeic Hoor ode of Brow Uiversity. PLEASE WRITE YOUR NAME ABOVE ALSO! 1- [4 poits] 1 [1 POINTS] [1 POINTS] TOTAL [6 POINTS]

2 FOR PROBLEMS 1- WRITE YOUR ANSWER IN THE SPAE PROVIDED. ONLY THE ANSWER APPEARING IN THE SPAE PROVIDED WILL BE GRADED. ILLEGIBLE ANSWERS WILL NOT REEIVE REDIT. 1. The platfor show i the figure vibrates horizotally with a displaceet xt ( ) = X(1 cos ωt). Its horizotal acceleratio is x(t) (a) at ( ) = Xω (1 cos ωt) (b) at ( ) = Xω siωt (c) at ( ) = Xω cosωt (d) at ( ) = Xωsiωt Straight lie otio forulas give ω a( t) = d x / dt = X cosωt ANSWER ( POINTS). A 1 kg ass rests o a horizotal surface with frictio coefficiet.5. At tie t= the surface begis to vibrate horizotally with a displaceet xt ( ) = X(1 cos ωt). The frequecy of vibratio is ω = 1 rad/sec. For each value of X listed below, state whether or ot slip will occur at the cotact betwee the ass ad the surface. (a) X = 1c SLIP NO SLIP (b) X = c SLIP NO SLIP (c) X = 4c SLIP NO SLIP (d) X = 8c SLIP NO SLIP x(t) =1kg If o slip occurs the block has the sae acceleratio as the surface. F=a gives the oral force at the cotact as N=g ad the tagetial force as T = X ω cosωt. Slip occurs if the axiu value of T exceeds µ N X > µ g / ω = 5 /1 = 5c ( POINTS)

3 3. A particle starts at rest at poit A ad travels with costat tagetial acceleratio aroud a circular path with radius R=4. After secods it has a speed of 4 /s. The acceleratio at tie t= sec is B j e e r (a) (b) (c) (d) a= e + 4 e / s r a= 4e + e / s r a= e + 4 e / s r a= 4e + e / s r R i A The tagetial acceleratio is = / = 4/= /. The oral acceleratio is a V t s V / R= 16 / 4 = 4 / s. Note that the oral acceleratio is towards the ceter of the circle so i the egative e r directio. ANSWER B ( POINTS) 4. The ed of the dashpot at A oves with a prescribed tie depedet displaceet yt (). With the defiitios ω = k /, ζ = c / k K = 1 the equatio of otio for st () is 1 d s ζ ds (a) + + s = L + Ky() t ω dt ω dt 1 d s ζ ds ζ dy (b) + + s= L + K ω dt ω dt ω dt 1 d s ζ ds ζ dy (c) + + s= L + K yt () + ω dt ω dt ω dt (d) 1 d s ζ ds K d y + + s= L + ω dt ω dt ω dt s(t) B k,l A c y(t)=y siωt Draw a FBD, use F=a ad the sprig/daper force equatios to see that d s ds dy + c + ks = kl + c. Divide through by k to see that the equatio looks like (B) dt dt dt ANSWER B ( POINTS) 3

4 5. A electric vehicle with ass kg is powered by a battery that is capable of developig a axiu power of kw. Neglectig air resistace, the shortest possible tie for the vehicle to accelerate fro rest to /s while travelig o level groud is approxiately (a).5 sec (b) 1 sec (c) sec (d) 4 sec Use the work-ke relatio: Power*tie=(chage i KE) so tie = (chage i KE)/Power = ( / ) / = s ANSWER ( POINTS) 6. The bod i a diatoic olecule has a potetial eergy that is approxiated usig the fuctiov( r) = E ( r / d)exp( r / d), where r is the distace betwee atos ad E ad d are costats. Whe the atos are separated by a distace r=d, the force of attractio betwee the is j 1 i r F (a) F = E exp( ) / ( d ) (b) F = E exp( ) (c) F = E exp( ) / d (d) F = E exp( ) / d The force-potetial eergy forula gives F = dv / dr = E (exp( r / d) / d r exp( r / d) / d ) = E exp( ) / d ANSWER ( POINTS) 4

5 7. A satellite with ass 5kg i a circular low earth orbit has velocity 8 k/s. A rocket is fired that exerts a thrust o the satellite (directed parallel to the velocity of the satellite), which varies with tie as show i the figure. Neglectig the ipulse exerted by gravity o the satellite, the velocity of the satellite just after the rocket is fired (at t=1 sec) is (a) 6 k/s (b) 9 k/s (c) 1 k/s (d) 1 k/s The ipulse of the force is the area uder the thrust-tie curve, i.e. 1Ns. The ipulseoetu equatio gives the chage i velocity as v= I / = 1 / 5 = / s= k/ s ANSWER ( POINTS) 8. A spherical rock saple is dropped fro rest fro height h oto a 45 degree slope. The collisio has a restitutio coefficiet e ad is frictioless. The oral ad tagetial velocities of the sphere after ipact are (a) v = e gh vt = gh (b) v = e gh vt = gh (c) v = e gh vt = e gh (d) v = e gh v = e gh t h j i 45 t Eergy coservatio gives the velocity of the saple just before the ipact as v= ghj= gh( t )/. Sice the collisio is frictioless the tagetial velocity does ot chage. The restitutio forula gives the oral velocity after collisio as v 1 = ev. Therefore v = gh v = e gh t ANSWER B ( POINTS) 5

9. The figure shows the path of a charged particle with ass i a scatterig experiet. The ucleus ca be assued to be statioary, ad exerts a repulsive radial force o the charged particle.

6 9. The figure shows the path of a charged particle with ass i a scatterig experiet. The ucleus ca be assued to be statioary, ad exerts a repulsive radial force o the charged particle. Idetify whether the stateets below are true or false r Path j i Nucleus (a) Liear oetu of the charged particle is coserved T F (b) The total eergy of the syste is coserved T F (c) Agular oetu of the charged particle is coserved about the origi T F (d) The forces actig o the charged particle do o work T F ( POINTS) 1. I the ethyl alcohol olecule show i the figure, the atos ca be idealized as particles ad the bods betwee atos as sprigs. The olecule has (a) 18 degrees of freedo ad 1 vibratio odes (b) 18 degrees of freedo ad 18 vibratio odes (c) 7 degrees of freedo ad 1 vibratio odes (d) 7 degrees of freedo ad 7 vibratio odes 9 particles, so 7 DOF. There are 6 rigid body odes, so 1 vibratio odes ANSWER ( POINTS) 11. The syste show i figure (a) is critically daped. The syste i figure (b) ust therefore have a dapig factor (a) ζ = 1/ (b) ζ = 1/ (c) ζ = (d) ζ = 1 (e) Noe of the above. k k c c k c (a) (b) Sice the first syste is critically daped c / k = 1. The effective sprig stiffess ad dashpot coefficiet for the secod syste are k/ ad c/. Therefore ζ = ( c /)/ k / = ( c / k)/ = 1/ ANSWER A ( POINTS) 6

7 1. A vibratio isolatio platfor ca be idealized as a sprig-ass syste. It has a atural frequecy of Hz. The base is excited at a frequecy 4Hz. Idetify whether each of the chages to the syste listed below will icrease or decrease the steady-state vibratio aplitude of the syste: (a) Double the dashpot coefficiet c INREASE DEREASE c k y(t) (b) Double the sprig stiffess k INREASE DEREASE (c) Double the ass INREASE DEREASE (d) Double the frequecy of the base excitatio to 8Hz. INREASE DEREASE Sice the excitatio frequecy is twice the atural frequecy the origial syste is i the vibratio isolatio regie. I this regie icreasig c will icrease the vibratio aplitude. Doublig the sprig stiffess will icrease the atural frequecy by a factor of ad so icreases the aplitude. Doublig the ass decreases the atural frequecy by the sae factor ad so reduces the aplitude. Doublig the frequecy reduces the vibratio aplitude (check the graphs of M for the base excited syste to see these) ( POINTS) 13. The disk show i the figure rolls without slip, with a clockwise agular velocity. Fro the list below, pick the vector that best describes the acceleratio of poit ω (a) Zero (b) (c) (d) (e) Poit has a zero tagetial acceleratio ad a vertical acceleratio. If you do t reeber this you ca use the fact that the acceleratio of the ceter is α Ri ad the use the rigid body acceleratio forula to see that a = αri+ αk ( Rj) + ωk ωk ( Rj ) ANSWER D ( POINTS) 7

8 14. The ass oet of iertia of the thi disk about a axis parallel to the k directio passig through O is (a) (b) I = R / I = R R j O i (c) (d) I = 3 R / I = R The ass oet of iertia about the ceter is R /+ R = 3 R / R / The parallel axis theore gives ANSWER ( POINTS) 15. The disk show i the figure swigs freely about O. At tie t= the ceter of the disk is level with O, ad the disk is statioary. Whe the ceter of the disk is iediately below O, its agular speed is g (a) ω = 3 R g (b) ω = 3R g (c) ω = R R O ω (d) ω = g R (e) Noe of the above Eergy coservatio gives PE + KE = cost gr + I ω /= gr + 3 R ω /4= ω = g /3R ANSWER A ( POINTS) 8

9 16. The figure shows a propeller with total ass oet of iertia 15 kg. The propeller is drive by a otor (ot show) that exerts a oet M o the shaft. At the istat show, the propeller has zero agular velocity ad has a agular acceleratio of 1 rad s -. At this istat, the rate of work doe by the oet o the propeller is M (a) Zero (b) 15 Watts (c) 75 Watts (d) 75 Watts Power = torque x agular speed so zero ANSWER A ( POINTS) 17. The figure shows a proposed desig for a suspesio. The liks AB ad D are rigid, ad the sprig has stiffess k. The static equilibriu cofiguratio is =, ad the equatio of otio for the syste is d L + ( g + kl) cos cos si + = dt The atural (agular) frequecy for sall aplitude oscillatios of is L A L B D (a) (b) ω ω = = g + kl L g + kl L (c) 1 ( g + kl) ω = L (d) ω = ( g + kl) L Liearize ( cos 1, si ) ad rearrage the equatio ito stadard for L d + = ( g + kl) dt The coefficiet of the first ter is 1/ ω ANSWER A ( POINTS) 9

10 18. I the figure show the lik AB rotates couter-clockwise with costat agular speed 4 rad/s. The velocity of ad the agular velocity of lik B are (a) v =, ωb = 4 k rad / s (b) v =, ωb = 4 k rad / s B A j i (c) v = 8 i / s, ω = B (d) v = 8 i / s, ω = B vb = 4k j= 8i v = 8 i+ ω Bk (i j) = ( ωb 8) i+ ωb j. Sice ust ove i the i directio, ω =, v = 8i B ANSWER D ( POINTS) 19. I the figure show the lik AB rotates couter-clockwise with costat agular speed 4 rad/s. The acceleratio of ad the agular acceleratio of lik B are (a) (b) (c) a = 3 j / s, α = B α B a = 3 i / s, = 16 k rad / s α B a = 3 i / s, = 16 k rad / s B A j i (d) a = 3 i / s, α = B B αb ωb αb αb a = 3 i / s, α B = 16 k rad / s a = 3j a = 3 j+ k (i j) (i j) = i+ ( 3) j. The acceleratio of is zero i the j directio ad therefore ANSWER B ( POINTS) 1

11 . I the figure show gear A has 3 teeth ad gear B has 4 teeth. Gear A rotates clockwise at 4 rad/s. Gear B rotates (a) clockwise at 16/3 rad/s (b) clockwise at 3/16 rad/s (c) couterclockwise at 16/3 rad/s (d) couterclockwise at 3 rad/s A B Fro HW6 we kow that the gear radius is proportioal to the ube of teeth. ω r = ω r ω = 4 3 / 4 = 16 / 3 rad/s. (The positive sig eas couterclockwise) B B A A B ANSWER ( POINTS) 11

12 1 The figure shows a siple idealizatio of a force sesor. Its purpose is to easure the force F, by providig a electrical sigal that is proportioal to the legth s of the sprig. At tie t= the syste is at rest, ad F=. At tie t=1s a costat force of F=1N is applied to the ass. The figure below shows the variatio of s with tie for <t<5s. s k,l c F 1.1 Usig the graph provided, calculate values for the followig quatities. (a) The period of vibratio (1 POINT) cycles takes 1 sec so T=.5s. (b) The daped atural frequecy ω d (1 POINT) ωd = π / T = 4 πrad / s 1

13 (c) The log decreet of the vibratio δ (be careful to use the correct origi) (1 POINT) The first peak has aplitude.7; the third has aplitude. so the forula for log 1 decreet gives δ = log(.7 /.) =.66 (d) The dapig factor of the syste ζ (1 POINT) δ Fro the forula ζ = =.1 4π + δ (e) The udaped atural frequecy of the syste ω (1 POINT) 4π + δ Fro the forula ω = = 1.6 rad / s T (f) The u-stretched legth of the sprig L (1 POINT) The legth of the sprig ust be equal to its ustretched legth before the force is applied, so L = 1c (g) The sprig stiffess k (1 POINT) After the oscillatios die out, the sprig has stretched by 1c after the 1N force is applied. Therefore k = 1 /.1 = 1 N / (h) The ass (1 POINT) We kow that ω = k / = k / ω = 1 / (1.6) = 6.8kg (this ust be the world s heaviest force sesor do t desig oe like this!) (i) The dashpot coefficiet c. (1 POINT) We have ζ = c / k c = ζ k = 158 Ns / 13

14 1. The sesor is ow used to easure a force that vibrates haroically Ft ( ) = F siωt. The figure below shows the steady-state variatio of the sprig legth s with tie. alculate the aplitude of the force F. Note that the frequecy of the force is equal to the atural frequecy (the period of vibratio is equal to the period i the first figure). This eas the syste is at resoace, ad we ca use the forula for the aplitude at resoace 1 1 X = KF M ax F F = ζ kx = = 1N k ζ (3 POINTS) 14

15 A crate of ass is pulled up a slope with agle by a iextesible cable that is wrapped aroud a pulley. The cotact betwee the crate ad slope has frictio coefficiet µ. The pulley has ass ad radius R ad ass oet of iertia R /. It is rotated couterclockwise by a otor attached to a axle at its ceter, which exerts a oet (torque) with agitude Q o the pulley. The bearigs supportig the axle of the pulley are frictioless. The goal of this proble is to fid a forula for the agular acceleratio α of the pulley. ω,α Q R Frictio coefficiet µ t.1 O the figures provided below, draw free body diagras for the pulley ad crate. R y T t R x Q g T N µn (3 POINTS). Write dow F= a for the crate, expressig both forces ad acceleratios as copoets i the { t,} basis show i the figure. ( N g cos ) + ( g si + µ N T ) t = att ( POINTS).3 Write dow r F+ Q= rg ag + IGα for the pulley, statig what poit you take oets about Moets about the ceter of the pulley gives 1 ( Q TR) k = R αk (1 POINT) 15

16 .4 Write dow a kieatics equatio relatig the agular acceleratio of the pulley α to the acceleratio of the crate. at = Rα (1 POINT).5 Hece, show that α Q g (si cos ) 3 R R µ = + The equatio syste fro the precedig parts ca be solved to see that N = g cos Q 1 T = Rα R 1 Q at = Rα = g siµ N T = Rα + g si + µ g cos R 3 Q Rα = g ( si + µ cos) R which reduces to the stated aswer. (3 POINTS) 16

EN40: Dynamics and Vibrations. Final Examination Friday May : 2pm-5pm

EN40: Dynamics and Vibrations. Final Examination Friday May : 2pm-5pm EN4: Dyaics ad Vibratios Fial Exaiatio Friday May 8 15: p-5p School of Egieerig Brow Uiversity NAME: Geeral Istructios No collaboratio of ay kid is peritted o this exaiatio. You ay brig double sided pages