A B C. Solution: a The cart moves down the incline with acceleration a

Transcription

1 PHYS 154 Prctice Tet Solution Spring 18 Q1: [4] A cr trvel long the pth hown in the picture with uniforl increing pee. In which point i the ccelertion iu in gnitue? ) A b) B c) C ) Nowhere. The ccelertion i unifor. The net ccelertion in ever point cn be coniere hving tngentil coponent t, ecribing how the gnitue of the velocit chnge, n ril coponent r, ecribing how the irection of the velocit chnge:. t r The pee of the cr i increing uniforl, o t i contnt (the length of the repective vector rrow i the e in ever point). However, the gnitue of r i proportionl to the qure of the pee, o it i zero in point B (tright pth), n lrger in A thn in C (e curvture r, but lrger pee). Conequentl the gnitue of the net ccelertion i lrget in A. A B C Q: [4] A cnnon ounte on crt lie own on frictionle rp. The cnnon fire projectile verticll upwr. Where oe the projectile fll? ) Behin the crt. b) Right bck in the crt. c) In front of the crt. ) It epen on the ngle of inclintion. e) It epen on the uzzle pee of the projectile. The crt ove own the incline with ccelertion gin, where the projectile ccelertion i g. So, ince the crt ccelertion h horizontl coponent, it will pee up in -irection leving the projectile behin. The eiet w to unertn thi i to conier the pecil ce when the crt i initill t ret on the top of the rp. It lunche the projectile tright up n then it ccelerte own the lope. Obvioul, the projectile won t hve n reon to follow the crt own the incline. Since n rguent houl be verifible b thi prticulr ce, we conclue tht thi houl hol even if the crt h initill certin velocit. Q3: [4] A bll roll off the ege of tble. The horizontl coponent of the bll velocit rein contnt uring it entire trjector becue ) The bll i not cte upon b n force. b) The net force cting on the bll i zero. c) The verticl coponent of the force vrie proportionll with grvit. ) The bll i not cte b force long the horizontl irection. e) None of the bove. Solution: The net force cting on the bll i it weight, which i verticl. Therefore, onl the verticl coponent of the velocit will vr while the horizontl one will t contnt. Q4: [4] An rrow i lunche with n initil pee v t non-zero ngle >45 with repect to the horizontl. Then, t it iu height ) the pee of the rrow i v = v in. b) the pee of the rrow i v = v. c) the pee of the rrow i v =. ) the verticl iplceent i the iu poible for the repective horizontl rnge. e) None of the bove. Solution: The projectile reche the e horizontl rnge for two initil ngle: one lower thn 45 n the other one lrger. Since the initil ngle i lrger thn 45, the iu height i lrger. 1

2 PHYS 154 Prctice Tet Solution Spring 18 Q5: [4] A bttlehip fire two projectile t two trget the figure. Which trget get hit firt? ) A b) B c) Both trget get hit t the e tie. ) Depen on which one i eviler. Solution: b The proble k which projectile pen horter tie in the ir. Uing the totl tie pent in the ir b projectile oving between point t the e verticl level, t v in g, tot we ee tht the totl tie i horter when the initil ngle i ller, uch tht the trget B will be hit firt. A B Q6: [4] A hunter i irectl t trget t the e verticl level. If the bullet leve the gun t pee v = /, n it ie the trget b h = 4.9, wht i the itnce to the trget? ) trget b) 49 c) 98 h ) inufficient infortion h The bullet ie the trget verticll n it i lunche horizontll, uch tht it initil velocit h no verticl coponent. Therefore the totl tie necer to rech the trget horizontl poition i given b h gt t h g 1 tot tot 1. In the e tie, the bullet trvel horizontl itnce vt tot Q7: [4] An object i oving with contnt velocit. Which of the following tteent i true? ) A contnt force i pplie in the irection of otion. b) There i no force cting on the object. c) The ccelertion ut be contnt. ) The net force on the object i zero. e) In Aritotle tie there w force cting on the object; ince Glileo, there i no force. Solution: Accoring to Newton 1 t Lw of Motion, oving object will ten to conerve it otion (contnt velocit) long the net force cting on it i zero. Q8: [4] A contnt net force F i pplie to object of increing. Which of the grph ot likel ecribe the epenence of the object ccelertion on the? ) A b) B c) C ) D e) None Solution: c B Newton n lw F F, tht i the ccelertion vrie with 1. Thi epenenc i bet oele b curve C. Notice tht it i not liner. A C B D

3 PHYS 154 Prctice Tet Solution Spring 18 Q9: [4] A prticle of trt fro ret n ove cte b contnt force F. If the prticle trvel itnce to rech pee v, which of the epreion below ot likel give the force F? ) v b) v c) v ) v v e) None of the bove. The force i contnt, o the prticle ove with uniforl ccelerte otion. Uing Newton n lw, we get v v F v F, tht i the ccelertion vrie with 1. Thi epenenc i bet oele b curve C. Notice tht it i not liner. Q1: [4] A book ret on horizontl tble. If we enote it weight w, n the norl t the urfce of contct n, which of the following tteent i true? ) Since the re equl n oppoite, w n n for pir ction-rection. b) n i inignificnt copre to the force eerte b the Erth on the book. c) Since n cncel out w, one nee no force to lift the book. ) The velocit of the te i iultneoul zero n non-zero. Solution: The force o not for pir ction-rection (the ct on the e object). Since the bo i t ret, n i equl in gnitue to weight. An, finll, if one trie to ove the bo fro ret, force nee to be pplie ince the book will be ccelerte. We t with () ince the velocit of the te i reltive: it i zero in it own fre but it i finite in other inertil fre oving with repect to it. Q11: [4] A book ret on n incline urfce. If the inclintion of the urfce i e lrger, wht hppen with the norl force on the book? n ) incree b) ecree c) t the e ) i lrger thn the ttic friction. e) None of the bove. gco Solution: b Since there i no ccelertion perpeniculr on the incline n g co, where i the inclintion ngle. Conequentl, when incree, n ecree ince co ecree. g Q1: [4] A og in n elevtor levitte, brel touching the floor. Which of the following tteent i true? ) The og h no weight. b) The floor puhe the og upwr with force equl to it weight o it levitte. c) The og ccelertion i equl in gnitue n oppoite to the elevtor ccelertion. ) The gnitue of elevtor ccelertion i equl to the grvittionl ccelertion. e) Thi i not poible. The oberver i everel inebrite or oething... Solution: ) The og oe hve weight long the elevtor i in the proiit of the Erth. b) Levittion en zero norl force o there i no puh fro the urfce. c) A gnitue cnnot be equte to vector. e) An, finll, the objective relit oen't cre bout runken tupor. 3

4 PHYS 154 Prctice Tet Solution Spring 18 Q13: [4] The bo in the figure i hel in equilibriu b rope ping over pulle n connecte to wll. There re four force contributing to thi equilibriu: weight, norl, tenion n friction. In thi orer, which of the following object re cte b the force foring ction-rection pir with thee force? ) erth, rp, pulle, rp b) erth, rp, wll, rp c) rp, rp, pulle, pulle ) rp, rp, wll, rp e) None of the bove, becue the rope i not perpeniculr on the wll, o the rection to the tenion i cte both on the wll n pulle. wll pulle rp erth Pir ction-rection pper between object tht ct on ech other (gent v. ubject). In our ce, the weight i cte b erth, the norl b the rp, the tenion b the pulle, n the friction b the rp. Q14: [4] A book i plce on rp with vrible ngle. The ngle i initill ll, o the book t t ret. A tuent incree the ngle ver lowl. She note tht the book uenl trt to lie own the rp when the ngle i bout 7º. Wht i the coefficient of ttic friction between the book n the rp? ).45 b).51 c).89 ) Inufficient infortion, becue the i not given. e) None of the bove. gco f g n gin Solution: b Initill, the book t t ret becue the ttic friction tche the force tht pull the book own the rp, tht i, the prllel coponent of the weight, f =gin. A the ngle incree, the prllel coponent incree o the ttic friction incree well. However, the ttic friction cnnot incree inefinitel, but up to vlue f _ =µ N correponing to n ngle. The llet perturbtion t tht ngle will brek the equilibriu n lunch the book own the lope. Therefore, ince t tht ngle the friction i till equl to the prllel coponent of weight: g co in g in tn tn co Q15: [4] A contnt force F = 9 N i pplie horizontll to bo of = 3 kg contributing to it otion on floor with coefficient of kinetic friction µ k =.4. Tking the poitive irection to the right, which of the ccelertion v. tie grph on the right ot likel repreent the ccelertion of the bo? ) A n F b) B f k c) C g ) D e) Inufficient infortion, becue µ k i unknown. A t C t B t D t Solution: b The ccelertion i contnt, becue the net force cting on the bo i contnt. Therefore, the correct nwer houl be either B or D. To figure out if the ccelertion i poitive or negtive, we hve to clculte the net force long the irection of otion. Be on the force igr, Newton n lw ictte tht F f. k Uing the efinition of kinetic friction, n the fct tht in thi ce the norl i equl to the weight, F F kn F kg kg.9, which inicte tht B i correct. 4

5 PHYS 154 Prctice Tet Solution Spring 18 P1: A cnnon i cpble of lunching projectile with pee v = 44.3 /. Now it trie to hit trget t itnce R = 1 t the e level. ) [5] Clculte the lunching ngle require to hit the trget. R v g in. Therefore The rnge i given b gr 1 1 gr in in v v 9.8 / in in 15 v v iniu pee b) [3] Wht other initil ngle woul give the e rnge? Uing the forul for the rnge n the propert of the ine function to hve the e vlue for ngle α n 18º α, it cn be hown tht the projectile reche the e horizontl rnge for two initil ngle ( n ): one lower thn 45 n the other one lrger. R v g in v g in So, in our ce the lterntive ngle woul be c) [5] Clculte n rw on the figure the n coponent of v. Lbel the v, v. (If ou in t olve prt (), ue = 15 ). Uing the coponent repreenttion on the figure we ee tht: v v co 44.3 co v v in 44.3 in ) [8] Clculte the n coponent of the velocit t tie t =. fter the projectile w lunche. Drw the on the figure (the point i locte t 85 ). Lbel the v, v. At oent t., the coponent of the velocit re v v 4.8 Since v v gt 11.5 / 9.81 /. 8.1 v i negtive, the vector coponent. v point ownwr, ketche in the figure. e) [5] Uing v n v, clculte the gnitue n irection of the velocit in tht point. At n oent long the trjector the velocit of the projectile i given b it n coponent: Mgnitue: Therefore n 1 Direction: tn v v v v v v tn or 1.7 below the horizontl.,. f) [4] Mrk on the figure the point where the pee of the projectile i iniu. Wht i the velocit in tht point? The iniu pee i t iu ltitue, where the velocit i horizontl, tht i v = v = 4.8 /. v v R 5

6 PHYS 154 Prctice Tet Solution Spring 18 P: Two boe re relee iultneoul fro the e height h = 4.9. Bo 1 i lunche with velocit v own ooth rp (of length = 8.5, incline t ngle = 3 ), with contnt ccelertion = 5. / oriente prllel with the incline. Bo i roppe fro ret, verticll bove the botto of the rp. h Bo 1 Bo ) [5] Clculte the n -coponent of the ccelertion = (, ). Sketch n lbel the vector coponent on the figure. Notice tht the ngle between the ccelertion vector i given b the inclintion of the rp. Therefore, co, in 4.5,.6 h b) [5] Write bolicl epreion for the poition r t, of Bo 1 in the given te of coorinte. The epreion houl contin the initil coponent of velocit (v,v ), the tie t n the known h n (, ). Bo 1 ove with contnt ccelertion, uch tht it poition i given b 1 1 r t v t t, v t t. 1 After pting the epreion to the conition of Bo 1, we get 1 1, r t v t t h v t t 1 c) [5] Write bolicl epreion for the poition r t, of Bo in the given te of coorinte. The epreion houl contin the tie t n the preter, h n g., g, uch tht it poition i given b Bo ove with ccelertion 1, r t h gt ) [5] Clculte the tie it tke until Bo reche the botto of the rp. We cn ue the eqution of otion for Bo : when the bo touche the groun 1 h gt t h g 1. e) [1] Clculte the coponent (v,v ) n then the pee v uch tht Bo 1 reche the botto of the rp t the e tie Bo. When the boe collie, their poition coincie, uch tht 1 1 v t t v t t 6. r1 t r t h v t t h v gt g t 3.6 Therefore, the gnitue of the initil velocit i v v v

7 PHYS 154 Prctice Tet Solution Spring 18 P3: Three og hzroul left lone in the houe kick off prt b etroing pillow. To iize the fun, the pull in three ifferent irection, hown in the jcent figure. Let u nlze the otion of the pillow in the light of the following etil: it i =. kg n it i initill t ret; the og pull with force of trength F 1 = 5 N, F = 9 N, n F 3 = 5 N; the irection of the force re given b ngle inicte on the figure below. The trteg of choice i to ue Newton n lw of otion. F 3 F F 1 ) [3] The cle-up force re repreente on the jcent force igr ebee in te of -e. Ue thi fre to ketch the vector coponent of the force tht hve both - n -coponent. Denote the F 1, F 1, etc. b) [6] For ech force, clculte the - n -coponent. Before plugging in the nuericl vlue, provie epreion in ter of force gnitue n irection ngle ( 1 n 3 ). F F co 5co3 45 N F F in 5in3 6 N F F F 9 N F F co 5co56 9 N F F in 5in N F 3 3 =56º F 1 F 3 F F 3 1 =3º F 1 F 1 c) [4] Ue the coponent ou foun bove to clculte the coponent of the net force F cting on the pillow (neglecting friction). F F1 F F3 16 N F F1 F F3 1 N ) [4] Ue the coponent to clculte the gnitue n irection of the net force F. F F F N F F 1 tn 37 e) [5] To viulize thi reultnt, ppl the til-to-tip etho to the three force in the jcent fre to obtin the net-force grphicll. Recll tht, when ppling the til-to-tip etho, the vector cn be chine up in n orer. So, for viibilit, I trte with F 3 n then I followe with F 1 n F. Then, the net force connect the til of F 3 to the rrow tip of F. Obvioul, ou cn o the opertion in n orer n the reultnt will be the e (tr it!). Note tht I rke the ngle on the figure o ou know wht w clculte bove. The gnitue F woul be the length of the re rrow. f) [3] Appl Newton n lw to clculte the gnitue of the ccelertion of the pillow. F F 1 g) [5] Auing tht the pillow t in one piece (for little while) n the og pull on it with contnt force, wht i the itnce it trvel in tie Δt = 1. econ, n how ft will it ove fter thi tie? Becue the ccelertion i contnt, the iplceent n the velocit of the pillow will be given b t t v t v v , t t F 3 F F 1 F 7

8 PHYS 154 Prctice Tet Solution Spring 18 P4: A bo of =. kg i initill t ret (but right on the verge of oving) in point in the ile of rp incline t n ngle = 3º, with coefficient of kinetic friction μ k =.3. Beie the uul force, the bo i lo cte b force F = 6 N prllel to the incline, hown on the figure. ) [5] On the figure, rw the vector igr with the force cting on the bo. However, even though the incline i rough, o not repreent the friction et, becue we firt nee to figure out it irection. Sketch the vector coponent of the weight in the provie coorinte te. b) [8] Write Newton n lw long the n irection, with the coponent of the weight written in ter of g n. Note tht the irection of the friction epen on the blnce between the -coponent W of the weight n force F: if F>W the bo i bout to ove up the incline; otherwie it lie own. So, when writing Newton n lw, inclue the friction ±f k, where + t for own the incline n for up the incline. I hve lre nubere the eqution for ou. -i: g in F fk (1) -i: n g co () c) [7] Clculte the ifference between the -coponent of the weight n force F. Be on the ifference, i the friction upwr or ownwr the incline? Ue the efinition of kinetic friction n cobine it with the eqution () bove to fin n epreion for friction in ter of μ k,, g, n. If the -coponent of F i lrger thn the -coponent of the weight, the bo ove up the incline. We ee tht g in F 5 N, tht i, inee, the bo ove up n the friction i own correponing to poitive ign in eqution (1) bove. Uing the efinition of kinetic friction n the eqution for the norl force, we obtin fk kn fk kg co n g co ) [5] Subtitute the friction epreion in eqution (1) bove n fin n epreion for ccelertion in ter of given quntitie. Then clculte it nuericll. Uing the poitive ign in eqution (1), we cn olve for ccelertion in ter of given quntitie: F g in F fk g in k g co 3. A epecte, the ccelertion i negtive, ening tht the bo ccelerte up the incline. e) [5] Depening on the irection of otion tht ou figure out, clculte the velocit of the bo either in point or point. The itnce = 1.5. The bo will en up in point in the origin of the coorinte te. The ccelertion i contnt, o we cn ue one of the eqution for uniforl ccelerte otion. Since we on t hve the tie, the eiet w to fin the pee i fro: v v v 8.3, tht i, the velocit i 8.3 / upwr the incline. 1 F gco g f k n gin 8

9 PHYS 154 Prctice Tet Solution Spring 18 P5: Two boe of e 1 = 1.5 kg n = 3.5 kg re connecte b cble ping over n iel pulle. The boe cn lie on two etric frictionle incline of ngle = 4, hown on the figure. ) [6] Sketch the free bo vector force igr for ech of the two boe. Then plit the weight long the te of coorinte provie on the figure. Write the repective coponent in ter of the repective, g n ngle. Ech of the two boe i cte b it weight, norl n tenion. The coponent re given on the figure. b) [7] Write out Newton n lw for ech of the two boe long the repective n -irection. M 1 : -i: T1 1 g in 1 -i n1 1g co M : -i: T g in -i ng co c) [7] Ue the eqution bove to clculte bolicl epreion for the ccelertion of the two e. Onl then clculte the ccelertion nuericll. Since the pulle i iel, the tenion i the e on it two ie, o let enote both of the with T: T1 T T. Thence we hve T 1g in 1 T 1g in Aing the eqution ter b ter (or ubtituting T fro one eqution into the other), we obtin 1 T 1 g in T g in 1 gin.5. ) [1] Ue the reult of prt (b) n (c) to clculte bolicl epreion for the tenion in the cble. S tht the iu tenion tht the cble cn utin without breking i T = 1 N. Wht i the iniu ngle tht the two incline cn be tilte before the cble np? We cn ubtitute the ccelertion into one of the eqution for the tenion. For intnce 1 T 1 g in 1 T 1 g in 1 g in g in 1 g in 1 g in 1 g in 1 T gin. So, we ee tht the ngle correponing to the iu tenion i given b T T gin in g gin 1 gco gco gin 1 9