Ch5 Problems

7
1 Problems 5-1 Frldiun •nd N ......... 1 l.awll 1. (I) If lhc cocflicicnt of kinetic frlcliou a ZZ-q erate and the tloor ill 0.30, what horir.antalforce ia required to mOYe the crate at a needy •peed llm'OU the floor? What horizmdal force ia required if 11-.. ia zero? Z. (I) A fon:e of 35.0 N il required to start a bDJ: moW!,g 1120111 I horizontal OODI2'Cte 11oor. (11) What U the c:oefliQcltt of Italic frictioll betwea lfle boz lllld the fioDI'/ (b) If the farce lhc boll aaleleratu at Cl.60 m/;.. What ilthe of kinetic frill1km? 3. (I) Suppoee you are ataDcliJ!c on a !laiD -=laalillg at D.20_1· What minimum coe:ftl.c:Wit ol Italic :friction muat emt between yoar feet and the Ooor if you not to llide? 4. (I) The coeflicient of atatic :friction hard rubber and normal strut pnem.ent is aboul 0.90. On how lteep a hill (muimmn angle) can you leave a car parbd? S. (I) What il the muimum IICilCk:ration a 1111: can wuiergo if the c:oefficiem of atalil: friction between the tires and the grOUDd II 0.90? 6. (fi) (a) A boz sits at rest on a roup 33" IDcll.ned plan.e. Draw the free-body diqnm.lhowiDg all the fon:ee aeting on the bOL (b) How would the diqram c:llaD&e if the box were a1idiDg down the plan.e. (c) How would it c:lwl&e if the box wme slidin& up the plan.e after an inil:i.al ahove7 7. (D) A 25.0-q boz is releMed on a 'IT" incline and accelerates downlhe indine at 0.30 m/r. Pl.ad the lric1iaa fmw impediDg ita motion. What il the coe:ftkient oftinelil: friction? L (fi) A car can decelerate at - 3.81.) m/r without atiddiD.& when coming to reat on a road. W1W would itl eralioa be if 1he road il iDcliued at 9.3" and the car IIIOVell uphill? Allume the ume atatic frlll1km coemaent. 9.. (fi) A d::ler movea down a 'IT" &lope at COJIItant What can you aay about the codllcieut of frlcliDD, 1J1i< 1 Aaaume the •peed is low enOU&h that air remw- CUI. be isnored. 10. (fi) A wet bar of IOifl alldea freely down a ramp 9.0 m long iDcliDcd 1118.00. How long does it 1:llb to Rid! the bo11om? Aaume 1'1: 0.000.. 1L (D) A boJ: ill pm a pulllao that it liidel the 11ooL How far will it,., that the of kineli<: fricti.on is 0.15 and the puah impu1a 1D initial of 3.5 m/17 1Z. (II) (•) Show that the minimum doppioa diltan.ce for an automobile tnvelln& at v a equal to tr/2 ,.._,,where,.._ il the coeflicieDt of datil: :friction between !be and the road, IIDd g II the aa:deration of grarity. (b) Wbat iJ thll diltance for a 1200-q car trave.Jm, 95 km/h if ,.._ = 0.65? (c) What wmW1 it be if the car wue on the Moon (the aa:el· eralion of gnvlty on the Moon ia about 1/6) but all dae atayed the aame? 13. (D) A 1280-kg car pullla 35().kg trailer. The car exerta a hori· zontal force of 3.6 X lo' N apilat 1he pound in order to accelerate. What force doe& the car exert on the trailer? Allume an effective friction codficiaat of 0.15 fm: the llailer. 1<&. (II) Police investigaton, eJ:IIIDlniD& the acene of an aa:i.dent invol'fing two c:an, meuurc 7:2.-m-long U:id IIWb of one of the cars. which nearly came to a stop before c:oiJ.id.in&. The of kinetic friaion between rubber and the pave- ment ia about 0.80. Ealimate the illitial speed of that car UIUlJiins a level road. 15. (II) Pilei of IIIOW on slippery room can become daageroul projecdlea II lhey me!L Caulder I dmnk of IDOW at the ridge of a roof with a lllopo of 34 •. (11) What il the minimum value of the c:oefficlem of atatk: frlctlon that will keep the llllOW tn.n slidiag down? (b) Aa the anow begin. to melt the c:oefficiad: of ltatic friction and the II10W filially lllpL Aauming that the dialllll£e from the chUDl to the edge of the roof il 6.0 m IIDd 1he coellidm!t of kinetic frictica iJ 0.1D calculate the lpeed of the- chank when It al1du off (e) If the eclce of the roof il lO.Om cround. eatimate the speed of the IIIDW wben it hitl the lfOIIIId. 16. (D) A 11111111 boz il bcld in place apind a roup ver1ica1 wall by aomeone p1llhina on it with a force direc:bld upward at 28" alxwe the horizontal The coeB!denta of &talk: aad ldnetk: friction between the box and wall life OM and D.30, f'DIPIIC- tively. The box lildea down unlea the applied fon:e baa magnitude 23 N. What ia the IIWI of the boll? 1'1. (D) 'lWo crates, of IDISI6S q and 125 ks ,life in contad IIIJd at reat on a borizon1a1 llllface (Fig. 5-32). A 650-N fm:a: il exerted on the 65-kg crate. If the coefficieat af kinetic fril:tion Is 0.18, calculate (a) the a.ccelention at the lyltem, IIDd (b) the forte that IIIICb crate eum on the otbllr. (c) Repeat wilh the crllles reveraed. FIGUU5-Jl Problem17. 1.L (fi) The crate abmm In Ff&. 5- 33. llel on a. plane tilted Ill an 111111e 8 = 25.0• to the horizomal. with 101: = 0.1!1. (•) DetermiDe the acceleration ol the Y crate u it l1idea down the plane. ( (b) If the cralll 11tart1 from 11111 &.tSm up the pia. from ita bile, what will 1he cralie'l speed m when il: reacbea the bottam of f 1he indine? FIGUR£5-JJ Crate on Inclined plan.e. Problema 18 and 19. D. (fi) A crate Ill gi- liD. Initial lpeed of 3.0 m/a up the 25.00 pl- ahown in fi&. 5-33. (a) How far up IIIII plaDe will it go? (b) How much time eJar-1 1lefln it to ill at.artlDg point? Alaume J£t - 0.17. 2:0. (D) '1\vo blocb IIUide of difl'mmt materWI COIIIleded toselher by a tbin down a plaDC ramp inclined at 8 to the holilontal shown in Pig. 5...,14 (blod; B il block A). 1be 1D811e1 at the blocb "'" and "'a, IIIJd the CXMIIJicieo1ll of friclion 10 and I-'D. If m,.. = mu = 5.0 kg, and i£A - 0.20 and I-'D - 030, deter- mille (11) the &aleieralion of 1he blocks and (b) the tenaion in the card, fDr an anp 8 = 32". FIGUR£5-M Problema 20 and 21. ,0 1 1 :52 QIAPTER 5 Using Newton's Laws: Fridion. Cira.dar Motion, Drag Forces

description

asdf

Transcript of Ch5 Problems

Page 1: Ch5 Problems

1 Problems 5-1 Frldiun •nd N ......... 1 l.awll 1. (I) If lhc cocflicicnt of kinetic frlcliou ~tween a ZZ-q erate

and the tloor ill 0.30, what horir.antalforce ia required to mOYe the crate at a needy •peed llm'OU the floor? What horizmdal force ia required if 11-.. ia zero?

Z. (I) A fon:e of 35.0 N il required to start a 6~q bDJ: moW!,g 1120111 I horizontal OODI2'Cte 11oor. (11) What U the c:oefliQcltt of Italic frictioll betwea lfle boz lllld the fioDI'/ (b) If the 3S~N farce CODtin~~et. lhc boll aaleleratu at Cl.60 m/;.. What ilthe ~t of kinetic frill1km?

3. (I) Suppoee you are ataDcliJ!c on a !laiD -=laalillg at D.20_1· What minimum coe:ftl.c:Wit ol Italic :friction muat emt between yoar feet and the Ooor if you ~ not to llide?

4. (I) The coeflicient of atatic :friction ~tween hard rubber and normal strut pnem.ent is aboul 0.90. On how lteep a hill (muimmn angle) can you leave a car parbd?

S. (I) What il the muimum IICilCk:ration a 1111: can wuiergo if the c:oefficiem of atalil: friction between the tires and the grOUDd II 0.90?

6. (fi) (a) A boz sits at rest on a roup 33" IDcll.ned plan.e. Draw the free-body diqnm.lhowiDg all the fon:ee aeting on the bOL (b) How would the diqram c:llaD&e if the box were a1idiDg down the plan.e. (c) How would it c:lwl&e if the box wme slidin& up the plan.e after an inil:i.al ahove7

7. (D) A 25.0-q boz is releMed on a 'IT" incline and accelerates downlhe indine at 0.30 m/r. Pl.ad the lric1iaa fmw impediDg ita motion. What il the coe:ftkient oftinelil: friction?

L (fi) A car can decelerate at - 3.81.) m/r without atiddiD.& when coming to reat on a ~ road. W1W would itl ~1-eralioa be if 1he road il iDcliued at 9.3" and the car IIIOVell

uphill? Allume the ume atatic frlll1km coemaent. 9.. (fi) A d::ler movea down a 'IT" &lope at COJIItant ~peed. What

can you aay about the codllcieut of frlcliDD, 1J1i< 1 Aaaume the •peed is low enOU&h that air remw- CUI. be isnored.

10. (fi) A wet bar of IOifl alldea freely down a ramp 9.0 m long iDcliDcd 1118.00. How long does it 1:llb to Rid! the bo11om? Aaume 1'1: ~ 0.000..

1L (D) A boJ: ill pm a pulllao that it liidel aero~~ the 11ooL How far will it,., p~ that the ~ld of kineli<: fricti.on is 0.15 and the puah impu1a 1D initial ~peed of 3.5 m/17

1Z. (II) (•) Show that the minimum doppioa diltan.ce for an automobile tnvelln& at ~peed v a equal to tr/2 ,.._,,where,.._ il the coeflicieDt of datil: :friction between !be tire~ and the road, IIDd g II the aa:deration of grarity. (b) Wbat iJ thll diltance for a 1200-q car trave.Jm, 95 km/h if ,.._ = 0.65? (c) What wmW1 it be if the car wue on the Moon (the aa:el· eralion of gnvlty on the Moon ia about 1/6) but all dae atayed the aame?

13. (D) A 1280-kg car pullla 35().kg trailer. The car exerta a hori· zontal force of 3.6 X lo' N apilat 1he pound in order to accelerate. What force doe& the car exert on the trailer? Allume an effective friction codficiaat of 0.15 fm: the llailer.

1<&. (II) Police investigaton, eJ:IIIDlniD& the acene of an aa:i.dent invol'fing two c:an, meuurc 7:2.-m-long U:id IIWb of one of the cars. which nearly came to a stop before c:oiJ.id.in&. The ~t of kinetic friaion between rubber and the pave­ment ia about 0.80. Ealimate the illitial speed of that car UIUlJiins a level road.

15. (II) Pilei of IIIOW on slippery room can become daageroul projecdlea II lhey me!L Caulder I dmnk of IDOW at the ridge of a roof with a lllopo of 34 •. (11) What il the minimum value of the c:oefficlem of atatk: frlctlon that will keep the llllOW tn.n slidiag down? (b) Aa the anow begin. to melt the c:oefficiad: of ltatic friction deer~ and the II10W filially lllpL Aauming that the dialllll£e from the chUDl to the edge of the roof il 6.0 m IIDd 1he coellidm!t of kinetic frictica iJ 0.1D calculate the lpeed of the- chank when It al1du off the~ (e) If the eclce of the roof il lO.Om a~ cround. eatimate the speed of the IIIDW wben it hitl the lfOIIIId.

16. (D) A 11111111 boz il bcld in place apind a roup ver1ica1 wall by aomeone p1llhina on it with a force direc:bld upward at 28" alxwe the horizontal The coeB!denta of &talk: aad ldnetk: friction between the box and wall life OM and D.30, f'DIPIIC­tively. The box lildea down unlea the applied fon:e baa magnitude 23 N. What ia the IIWI of the boll?

1'1. (D) 'lWo crates, of IDISI6S q and 125 ks ,life in contad IIIJd at reat on a borizon1a1 llllface (Fig. 5-32). A 650-N fm:a: il exerted on the 65-kg crate. If the coefficieat af kinetic fril:tion Is 0.18, calculate (a) the a.ccelention at the lyltem, IIDd (b) the forte that IIIICb crate eum on the otbllr. (c) Repeat wilh the crllles reveraed.

FIGUU5-Jl Problem17.

1.L (fi) The crate abmm In Ff&. 5- 33. llel on a. plane tilted Ill an 111111e 8 = 25.0• to the horizomal. with 101: = 0.1!1.

(•) DetermiDe the acceleration ol the Y crate u it l1idea down the plane.

(

(b) If the cralll 11tart1 from 11111 ~ &.tSm up the pia. from ita bile,

what will ~ 1he cralie'l speed ~ m when il: reacbea the bottam of

f 1he indine?

FIGUR£5-JJ Crate on Inclined plan.e. Problema 18 and 19.

D. (fi) A crate Ill gi- liD. Initial lpeed of 3.0 m/a up the 25.00 pl- ahown in fi&. 5-33. (a) How far up IIIII plaDe will it go? (b) How much time eJar-1 1lefln it return~ to ill at.artlDg point? Alaume J£t - 0.17.

2:0. (D) '1\vo blocb IIUide of difl'mmt materWI COIIIleded toselher by a tbin eon!,~ down a plaDC ramp inclined at m~DP 8 to the holilontal • shown in Pig. 5...,14 (blod; B il ~~~low block A). 1be 1D811e1 at the blocb ~ "'" and "'a, IIIJd the CXMIIJicieo1ll of friclion 10 ~A and I-'D. If m,.. = mu = 5.0 kg, and i£A - 0.20 and I-'D - 030, deter-mille (11) the &aleieralion of 1he blocks and (b) the tenaion in the card, fDr an anp 8 = 32".

FIGUR£5-M Problema 20 and 21.

,0 1

1 :52 QIAPTER 5 Using Newton's Laws: Fridion. Cira.dar Motion, Drag Forces

Page 2: Ch5 Problems

ZL (D) For two blocts. connected by a cord and lliclin,g down the focllne shown In Fla. 5-34 (see Problem 20), desc:rlbe the motioll. (11) i! P.A < p.11 , and (b) if II>A > II>B• (c) Doter­mine a formula. for tb.e acceleration of each block and the tenlllon Jlr in the cord in terms of "'A• m8 , end 6; interpret your rerultll in light of your anJWOII to (11) and (b).

22. (D) A flalbed tr111:k i& cany:ing a heavy crate. The coef&ient of static friction between the crate and the bed of the truck is 0.75. What ill the ma:dm.um rate at which the driver can decelerate and lltill. avoid ha~ the crate lllide agailut tho cab of the truck?

:D. (D) In Hg. S-3S tbe QOefliQent of atali.e friction between II1.UI mA and the table is 0.40, whereas the QOefficient of kineli.e friction it 0.30 (11) What miDi11mm value of mA will keep tb.o system from starting to move? (b) What value(a) ai mA will kup tb.e syatem mo-ring Ill constant speed?

mo= 2.0kg

FIGURE 5-35 Problems 23 and 24.

24. (D) Determine a formula. for the acoeleration of the syatem llhow:u in Hg. S-35 iD. term~ of mA,ma, and the man of the cord,lflc. Define any other variables needed.

2$, (D) A amall bhd. of m.aa m it given an initial speed q, up a ramp inclined at anglo 9 to tb.o horizontal. It travels a dllltance ti ap the ramp and comes to rest. (a) Determine a formula for the coe1licient of kinetic friction botwooD. bl<d and ramp. (b) What can you say about the value of the coe~nt of atalic fric1ion?

24i. (D) A 75·kt snowboarder h.u an Initial velocity of SJJm/s at the top of a :W iD.clilul (Hg. S-36). Aft.er diding dow:u the 110-m lont incline (on which tho coefficient of kinetic friction ill P.k ~ 0.18), the anowboerder h.u attained a velocity v. Tho mowboarder then alidea aloag a flat surtiu:o (on which P.k ~ 0.15) and comes to rest after a dl.stance x. Uee Newton._ eecond law to find tbe snowboardeo IUX)Oleration while on tho incline and while on tho flat IIW'fal:e. 'Then uae theee acoeleratiOD.A to determine ;x;.

l" ..;.;o mls 110m

~k = 0.18

28°{ :1...::. ~k ; 0.15

j .<

FIGURE 5-36 Problem 26.

7:1. (D) A package of IIWI m is dropped vonically onto a hori­zontal conveyor belt whose speed ill i1 = 1.5 m/r., and the coefli.c:iont of kinolic friction botwoon tho package and tho bell ill P.k - 0.70. (4) R>r bow much time does the pachge allde on the belt (anti! it Ia at rest relative to the belt)? (b) How far dnes the package move during this time?

28. (D) 'IWo manes "'A = 2.0 kg and ma = S.O kg ue on iD.clines and are connected together by a string aa show:u in Fla. 5-37. The QOeffic!ent of kinetic friction between ea.ch mau and iiB inclino iJ lilt = 0.30. If "'A movea up, and ma moves down, determine their acceleration.

RCURE 5-37 Problem 28.

111. (D) A child a1idee dnwu a Blide wilh a 34• incline, end at the bottom her !peed ill prec:iJely half what it would have boon If the allde bad been frlctionlea.. Calculate the coefficient of .tinetic friction between the ll1ido and the child.

31. (D) (a) Suppose the QOefficient of kinetic friction between "'A and the plane in 1'13. S-38 iJ /HJr. = 0.15, and that mA - m11 - 2.7 ks· AI mB mnves down, determine the J11&811itude of the acce!J::ralion of mA and ma, given 8 ~ 34•, (b) What anallest value of llok will keep tb.e syatem from acceleraling?

FIGURE s-:sa Problem 30.

3L (In) A 3.0-kg block lila on top of a S~tg bloct: which ill on a horizontal surface. The 5.~k,8 bloct is palled to the right with a force i as Ahawn ln Fig. S-39. The coef1!.cient of static friction between all surfaces is 0.60 and the kinetic coeffi­c!e:m ill 0 . .4a (a) What ill the miolmum value ofF n.eedt-rJ to move tho two bloc:D? (b) If tho force ill 10% greater than your .answer for (a), wbal iJ the aa::eleration of each bloclr.?

L_~ ~5.0 kg F -

FIGURE s-:st Problem 31.

Problems Ill

Page 3: Ch5 Problems

3Z. (DI) A 4.o.tg block is stacted on top of a 12.0-kg block, which Is aoceleralhlg along a horizonW table 111 a = 5.2 mjr ~5--W).Let P.k = p., = p.. (a)Whatmillimumcoellicient of fridion p. between the two blocks win p:revent the 4.().tg blcx:l l'rom s1l.dlng ofJ? (b) If p. Is oDI.y half thill mlnlmum value, what is tho -=l.eration of tho 4.0-q block wilh respect to the table, end (c) with rapect to the 12.0-kg blcx:l? (d) What ill tbe for:co that lll.'lat be applied to tbe 12.0-kg block in (a) and in (b), a.umio,g that tbe table Is frictlonka&?

FIGUREJ-40 Problem32.

I 4.0 .. I 12.0q -a=S.2mlal

33. (DI) A wma1l bW of mue m -'-on the rou&fl, alopmg side of a lrillll,gular block ofm.asa M which ille!f rests on a hori· zontal l'rie1ion1eM table • &hown in F!g. S-41. If tho coeffi. cient of static friction is p., dotermine tho minimum horizontal forre F applied toM

tb.al will - the anal! block m to start moving up the incline.

FIGURES-41 Problom33.

S-2 to S-4 Unifonn Cfrcua.r Motion 34. (I) What is tho lll.IIXimum spoed with which a 1200-k,g car

can rolllld e. turn of radlus80.0 m on e. flat road If the ooeffi· c:ieD.t of fri.cti.on between tirel and road ia 0.65?15 this Iellllt independent of the m.asa of the car?

35. (I) A child lilting 1.20m from the center of a merry-go· around mavea with a speed of 1.3Dm/&. Calculate (a) the centripoW acceleration of the child and (b) tho net bori· zonW force exerted on the child (ma&!ll = 22.5 kg).

36. (I) A jet plene traveling 1890 k:m/h (S2S m/a) pulls out of a dive by mo'ring in an arc of radius 4.80km. What iJ the plane's acceleration in (s?

37. (ll) II it pouible to 'IYbirl a buclcet of water fast enou,gh in a vertical cln:l.e 10 that the water won't fall out? If so, what ill the minimum speed? Define all quantities needed.

38. (ll) How fan (in rpm) must a centrifuge rotate if a particle 8.00cm from tbe uis of rotation ia to esperienoe an acceler· ation of t2S,(l()() t(s'l

39. (ll) Hi8hway curve~ a:te marked with a augge~~led apeed. If this speed is based on what would be aafe in wet wealhcr, estimate ru radha of curvature for a curve marll:ed SO hn/h. U10 Table 5-1.

40. (ll) At what minimum speed must a roller couter be traveling when upalde down at the top of a ~ (F"tg. S-42.) 10 that the p.uaengen do not fAll oat? Alsume a radius of curva· ture of7.6m.

FIGURE .5-42 Probkm40.

4L (ll) A sporll car crones the bottom of a valley with a radius of CUfVllture equal to 9S m. At the very bottom, the normal force 011 the driver ill twio& hiJ wei,ghL At what speed was the car traveling?

G. (ll) How large must the coc1ficient of atatic friclioD. be between the tirea and the road if a car is to round a level c:um: of radius8S m at a apeed of 9S k:m/h?

43. (ll) Suppoae the space ahuttle ia in orbit 400 km from the Barth'a -race, and llirdea ru Barth about ODeC every 90 min. Find the centripetal aoceloration of the space ahuttle In Ita arbiL Expreu your 11116wer In tetm.8 of g, the gra.vita· tioa.al aoc:elerati011at the Earth'e aurW:e.

44. (ll) A bucket of IIWII 2.00 kg ia whirled In a vertbl cln:le of radiu& 1.10 m. At the loweet point of ill moti011. the tclllion in the rope supportin,g the bucket is 25.0 N. (a) F'md the speed of tho bucket. (b) How fast mWlt the bucket move Ill the top of the circle 10 that the rope dooanot go alack?

45. (ll) How meny revolutiQilll per minlltc wOIIld a 22-m­diameter FenU wheel need to make for !he p1118eDge111 to feel "welghtle811" at tho topmost point?

46. (ll) Use dimensional analysis (Section 1-7) to obtain the form for the oentripetal acceleration, aa = tl-/r.

47. (ll) A jet pilot tokea hia airaaft in a ver1ic:al. loop (Fig. S-43). (.a) If the jet is moving at e. speed of 1200km/h at the lowest point of the loop, detezmiD.c the minimum radius of the circle so that tho centripetal aoc:oleration at the lowest polot does not exceed 6.0 g'&. (b) Calculate the 78-kg pilot's effective weight (the force with whk:h the aeat pushea up 011 him) at the bottom of the circle, and (c) at tbe top of the circle (usume the aame apeed).

f'IGUR£5-45 Ptoblem47.

48. (ll) A propoaed rpa.oe atatlon conslsta of a drallar tube that will rotate about ita center {like a tubular bieycle tire), Pi,g. S-44. The cin:le formed by the tnbe has a diameter of about 1.1 km. What muat be the rotation rpeed (revolutions per day) if an effect equal to gravity at the surface of the Barth (1.0 g) ia to be felt?

FIGURES-44 Problem48.

49. (ll) On an ice rin1r. two ekatcra of equal man grab hands and spin in a mutnal circle once every 2.S a. If we usume their armJ are each 0.80 m long and their Individual ma&!llea are 60.0 kg. how hard are they pulliDg on one another?

50. (ll) Redo Example S-11, predllely thil time, by not f&no:rlng dae wei&ht of the baD. whichrevol.vca 011 a lltri!Jg0.600m lottg. In particular, find the ll1.llj!!litude or lT, and the angle it makea with dae horizontal. [Hii#: Set the borizoa.tal component of iT equal to mt~a: allo, Iince there is no vertical motion, what can you say about the vertical component of iir 1]

1M OiAPIER 5 Using Newton's Laws: Friction, Cira.llar Motion, Drag Forces

Page 4: Ch5 Problems

51. (D) A coin is pLued 12.0 an from the uiJ of a rotating turDtabJ& of variable apeed. Whe:D tho ~peed of tho tumtable II dowly inacaed, \he coin remain~ med 011 the turntable until a rate at 35.0 rpm (revolutiolll per minute) ia readied, at whio:h poillt the coin slldel oft What is the coefficient of static friction between the coin md the turntable?

n. (D) 'lbe delign of a new road include• a atrai,Pt atretdl that i.a horizollllli. and flat but that JUdd=ly dipa down a atcep bill at 22". The tnmsition llhould be rounded with what minimum radiDa 10 that ~:an traveliDg 9S .km/h will DOt leue the road (Fig. S-45)?

FIGUR£5-45 Problem 52.

51. (D) A 975-q aportl car (incl.udiDa driver) CI'OIIIel the rounded top of a hill (radiua = 88.0 m) at 12.0 m/1. Determine (11) the normal furce eurted by the road on the ear, (b) the DllllDIIi force exerted by the car on the 72,0-kg driver, and (~) the ~ 11peed at which the DOI'IIllli. furce on the driver equab zero.

54. (D) two blocts, with muse~ mA 1.114 IIIJI, are ooonected to each other and to a centrlll poat by cord.t u &hown In PiS- 5-46. They rotate about the polt at frequeru::y I (revolution& per aecond) on a frictioulea borir.ontalll1ll'.fa.:e at diltancel r A and 1'JI from the polL Derive an aiaebraic c::aprcuion for the tenJian iD c:ad& ..:pam of the cord (UIIU!Ied muale.a).

-~ ----, ...... _____ .... " ,,. ' ' /( ,.,, ', \ .... ..."' ', ........... ____ ... .,~

........ ______ .,..,.,,. FIGURE 5-46 Problem S4.

55. (D) 'Jarun plml to crou a gorge by rwiDging in an an: from a hangiDg 'rille (Fig. 5-47). If hi.& um1 ue capable of eserting a force at 13Sll N on the rope, wbat is the muimum apccd be can 1olctate at the lowest point of Ilia awing? IDs 111111 il 78 kg and the viDe il S.2mlong.

fiGUIE~7 ProblemSS.

Sfi.. (D) A pilot performs an e...allve tna.n£UVet by diWII verti­cally at 310m/L If he can withltmd 1111. aa:eloraliml of 9.0 (' wilhcnat blacking out, at what altitude m•t he begin to pull out at tbe dive to awid c:ruhiD& Into 111e sea?

57. (ID) 'lbe poailion of a par1icle m~g In !be :cy plane i' pvea by r = 2.0coa(3.0rad/at}1 + 2.01in(3.0rad/ at)j, where r II Ill meten and t II Ill aeconda. (a) Show that thll Rprwents cin:ulai moliDn of l"'llliia 2.0 m centetod at the origin. (b) Determine the ..elocity and acceleraliDn ..ecton u 1'ullaiona of 1ima. (c) Detelllline the llpeed and magnitude of the acceleration.(~ Show that a= tl-/r. (1!) Show that the acceleration vector alwa}'l points towmd tbe ceo!er of the circle.

511. (Ill) If a CIIIV~ with 1. radiua of 8S m ia properly baDkcd £c. a car travelloa ~ km/h, wbat lllllllt be the coefBclent of ltatic fric1ion for a c:u not to llkid wbc:n traveliDg at 9S km/h1

59. (ID) A cane of radius 68 m ia banked for a design apecd of 85 .km/.b. If the coctficimt of Ita tic frlmiml i.a 6.30 (wet pave· ment), at what ranp of apeeds can a car lllfeJy lll.lke the ~? [Hiitt: Ccmlida the dim:tioo of~ rru:cion force when the~ &Gel too llow or too fast.)

•s-s Nonuniform drallu Molion *60.. (D) A part:ide atarting from rest revolvea with uniformly

lnc:reuing ~peed In a cloc:kwi..: c:i:rd& in the :cy plaDc. The center of the circle is at the origin of an z:y mordinate ayatem. At t = 0, tbe particle is at z = 0.0, y = 2.0 m. At t - 2.0 ., it bu made one-quarter at a revolution and ia at z = 2.0 m, y = 0.0. Determine (a} ita 11peed at t = 2.0 a, (b) the average velocity vector, and <~>the averase IICICeler· ation wc:tor during this intervaL

• 6L (ll) In Problem 60 asaume !be tangential 8CCieleratiolll. is llOIJ&tmt and determillc the compaiK'oDI:I of the iaaiJadalleOua IIICIZI.eratioa at (a) t - 0.0, (b) t - 1.0 a, and (c) t - 2.0 a.

• 62, (ll)An objectlllOvetln a ~ofradiua22m with itltpccd given by t1 - 3.6 + 1.St2, with t1 in meten per aecond and t In 8eiXlDdt. At t • 3.0 1, find (11) the tangential ~OD lllld (b) the radial~~Ceeleration.

• 63. (III) A partk.le rotates Ill a circle of radius 3.80 m. At a particular illltant itl acc:eleration is 1.15 m/s2 in a direc:tion that makel an angle at 38.0" to ita ditectlon at motion. Determine ita lpeeci (a) at thi& 1110111e11t and (b) 2.00 I later, -mg 0011.1tant tangential acceleration.

• 64. (Ill) An ob;ect oi.IIIUI m is <XIIIItnl.ined to move in a c:irc1e ol. ndiul t. Ita tanp!tial accelemtion as a flmction crt time il &iYm by "-= b + cr, whln b md care oonalanla.lt v = \II It , = o, determine the tansemw lllld radial c:ompaaenta of the {Orce, F~a and Fa, acliDg on the objed at any time t > 0.

• s-& velodty-Dependent Forces • 65. (I) Uae dimen.lion.al anai.)'Bis (Section 1-7) In BDmple 5-17

to detcnnine If the time Wllltlnt de 'I = m/b ar 'I - b/m. • 66. (ll) The terminal wlocity of a 3 x 10""' kg raindrop ia about

9 m/a. Allumfn& a drq fome 1D = -btl, determine («) the value of the oonatalll b and (b) the time required for IUr.h a drop, •tutiaa from rest, to reach 63% crt temlinal velocity.

• (11. (D) An obj.ect III<IYiDs vertically haa i = fo al t = 0. Determine a formula tor ill velocity u a function of time ..umiDg a rcaialiv~ fotec F = -IN u wdl u gravitr for ttro cuea: (•) f 0 iJ dowuward and (b) f 0 ia upwani.

Problems 1 SJ

Page 5: Ch5 Problems

• 68. (Dl) 1b.~ dra,g ftm:o on largo objects rucb as can, planes, and si:y divers movlns tbrough air Is more nearly Jb = -bll-. (11) For thi5 quadratic dope!ldo1100 on v, dotm:mino a formula for the tllnninal velocity Vf, of a vertically 1a1Jin.8 object. (b) A 7S·kg ai:y diver hall 11. tel'lllhW. velocity of about 60 m/s; determine the value of the constant b. (c) Sketdl a curve like that of Fig. S-27b for this case of PO ex r. For the sam~ te:nnin.al velocity, would this curve lie above or below that in Fig. S-27'? Bxpt.ln why.

*69. (Dl) A bicydist can c:out down a 7 !1' hill at a steady 9.5 i:m/h. If tb.e draa force is proportional to the square of the l!peed 11,110 that PO= -ctf, ~ (11) the value of the COOII!Uil c and (b) the average force !hat m111t be applied. in order to ~d the hill at 2$ km/h. The lll1IN of the cyclist plus bicycle is 80.0 kg. Ignore other types of friction.

• 70. (Dl) 'IWo drag f<m:e3 act on a bic:yde lll.d rider: lb2 duo to rolliD,g resistance, which is enentially velocity independent; and Fo 2 due to air re&lnan<:e, which Is proportional to r. For a apecific bike plus rider of total ll1.llll 78 kg. P01 "'4.0N; and for a rpeed of 2.2m/a, P02 F:; l.ON. (11) Show tbat the total drag force is

lb = 4.0 + o.zt.,Z, where v is in m/a, and PO is in N and oppote& the motion. (b) Determine at what llope angle 9 the bib and rider can c:oaat dawnhlll at a constant rpeed of 8.0m/a.

I General Problems '16.. A coffee cup on tb.e horizontal. dashboard. of a car slides

forward when the driver deceleratu from 4S kmfh to J:Qt ill 3.5 • or los.'lo but not if she docelerale! ill a longer tim«~. What is the coeffld.ent of static friction between the cup and !he daah? Anume the road lll.d the dalhboard are level (horizontal}.

'77. A 2.0-kg silverware drawer doe~~ not lllide readily. The owner gradually pulls with more and more force, and when the applied force mu:hes 9.0 N, the drawer l!llddellly opem, throwing all the uten:lils to th~ floor. What is the c:oefficlenl of static friction between the drawer and the cabinet?

711.. A roller coaster reaches the top of the steepest hl11 with a •peed of 6.0 km/h. It then de~nds the hill, which is at an average angle of 45• and is 45.0 m long. What will its apeed be when it n:acbe8 the bottom? Asaume 1-'k ~ O.U.

79. An 18.D-tg bor. il released on a 37.0" incline and aa:eleratu down the Incline at o.2:20 mfr. And the friction force impeding its motion. How large is the coe1ficient of friction?

110. A flat puck (m.&.!IS M) ill revolved ln a clrd.e on a friC1ionleas air hockey tab!~ top, and is h~ld ill thil orbit by a light cord wbfdl is conn.ected to a dangling mass (mass m) tbrough a c:enlral hole as shown in Fig. S-48. Show that the l!peed of the puck is given by f/ ~ YmgR/ M.

FIGURE .5-48 Problem 80.

• 7L (Dl) Determin~ a formula forth~ position and acceleration of a flllliD& object as a fUnction of time lf the obJect starts from :reat at t = 0 and unclergoee a Ieai!tive for<:e F- -btl, uinBDmpleS-17.

•n.. (Dl) A block of m.8llll m all.dea along a horizontal mrface lubricated with a thick oil which provides a dra,g force proportional to the square root of velocity:

1 Fo = -bv2.

If v = 1:1) at t = 0, determine v IID.d ~ aa funclioll!l of time.

• '73.. (ID) Sbaw that the maxlnwm dlatam:e 1he blod: In Problem 72 can travel is 2m WJ'l/3b.

• 74. (Ill) You. dive a!Jai&ht down into a pool of warer. You bit the Wlllerwith a rpeed of5.0m/a,and yourmua is 75kg.A&&umi:ag a drag force of th~ form Jb = -( 1.00 x 104 tg/s) v, how loog does it take you to reach 2% of your original speed? (Jp.ore any eiiecla of buoyancy.)

• 75.. (Dl) A motorboat traYeling at allpUd of 2.4 m/s llhulll off il! e:ogines at t ~ 0. How far does it trawl before coming to rert if it il noted that after 3.0a i:ls spe.ld. bas dropped. to half ita ~ ~e? Assume that 1fu: dreg for<:e of the water is proportional to i!.

111. A motorqd!st Is coasting with the eoglne off at e. steady rpeed of 20.0 m/a but ealen a 8andy stretdl whc:re the coef· fi.cient of kinetic friction is 0.70. Will th~ cyclist emerge from the sandy metch without haYing to awt the engine lf the sand !uta for 15m? If to, what will be tha apeod upon emerging?

liZ. In a "Rotor-ride• at a cami.val, people rotale ill a vc:rtical. cylindrically we.lled "room." (See Fig. S-49). If the room radiu& was 5.5 m, and the rotalion freqiiC!Iey 0.50 revo· lutiom per second wh~n tb.~ floor drops out, what minimum codflclem of l!tatic friC1ion keept the people from &lipping down? People on thil ride &aid they were "p:reued a,glin.st tb.e waiL" Ia there r:eally an outward force pressing them .apiwlt the wall? If 110, what lA its aour<:e? If not, what il the prop or deecription of their situation (besides nausea)? [Hint: Draw a free. body dla,gram for e. person.]

FJGURE .5-41 Proble:m. 82.

83. A device for traiDiDg utronauta IID.d jet fi8hter pilota is designed to rotate the trainee in a horizontal cin:le of radius 11.0 m.lf the f=:e Wt by the trainee is 7.45 times hc:r own weight, how fast iJ sh~ rotating? Express your answer in botbm/und rev/&

1M OiAPIER 5 Using Newton's Laws: Friction, Cira.llar Motion, Drag Forces

Page 6: Ch5 Problems

84. A 1250-kg car rounda a wrve of radius nm bllllked at an angl~ of 14• .If the car is travoling at 8S km/h, will a friction force be required? If so, how much and In what direction?

as. Dete:rmin~ the tangential and centripetal COIIIpOIIeDtl of th~ net force netted on a car (by the ground) when its speed is Z1 mls. aDd il bu a~leralcd to thiJ speed from reet in 9.0 s on a c:une ofiadius 4SOm. The car's mau is llSO kg.

86. The 70.o.tg c:limber in :fi3. 5-SO is supported in ~ •drimney" by tbe friction forces e:e:rted on his ~ and bade. The alali4: ~Is of frio­lion between his shoes and the wall, end between Ilk back and tbe wan. 111e 0.80 and 0.60, r~vely. What is the minimum normal force he DWSt exert? Aaume the wall! are vertical and that th~ static friction forces are both at tbelt maximum. Jguore his grip on the :rope.

FIGURE S-SO Problem86.

87. A amalliiiUII m ia set on ~ IJIIrface of a sphere, Fig. S-51. If the coefficient of static friction ia p., = 0.70, at what angle ~ would the mass !tart alldlng?

FIGUREs-51 Problem87.

i I I I '

~,/ I ,

! ' ()

Ill

88. A 2.8.0-kg blodl: is ClODDCelcd 1o an empty 2.00.kg blleket l1y a ClOtd IUIIlliD.g over a frictionless pull~y (H,g. S-52). 1h~ coefficient of llllltlc frll:tlon between tbe table and the block ia 0.45 aDd ~ ClOefli.c:ieD.t of tinetic friction between ~ table and the block is 0.32. Sand ia gradually added 28.0 kg

1o th~ bucket until the SJBtem just begins to move. (a) Olli:U!.ate the III8B8 of sand added 1o the bucket. (b) Calculate the ~lere.­tion of the syrtem.

FIGURES-52 Probl~m88.

19. A car is headlng down a a!lppety road at a llpeed of 95 km,lh. The millimum. diltanc:e within which il can stop without skiddlllg Is 66m. What Is the sharpest carve the car can u.egoliate on the icy su:rface at the same speed without skidcling?

Sit. What iJ the ai:Qekration e1perienl:ed by the tip of the 1.5-cm-long sweep second hand on your wrist watch?

'L An airpllllle traveliDg &1480 km/h need! 1o reveno itl c:ourao. The pilot decides 1o a.ooomplisb tbis by banking the wing! at anllll8le of 3s•. (a) FlDd the lime needed to reveoe c:ourse. (b) Desaibe any additional force the passenger~ es:perience during tbe tum. [Kw: Asallme an aerodynamic "lift" force that ac111 perpendicularly 1o the flat winga; see Fig. S-53.J

FIGUREJ-55 Ptoblem91.

92. A banked ~:UtYe ofrlldlu.s R in a new hlghway Ia dealgned 10

that a car travelillg at ·~ 'II) c:an negotiate the tnm aafoly on glare U:e (zero friction). If a car travels too &lowly then It win slip toward the ceD.ter of the circle. If it trave!J too fail, it will slip away from the center of the citde. If the coefficient of static friction in~:reuea. it ~possible for a car 1o ltay on th~ road wbile tmveling a1 a speed within a range from tt,m, to t.Omu:· Derive formu1u for 11mm and 11mu as fim.c1ionJ of p., .'17Q, aDd R.

93. A small bead of mass m Is C:OIISil'alned to sllde without friction inside a circular vertkal. hoop of radius r which rotatu about a vertical u:is (Fig. 5-54) at e. frequency f. (a) Determine the e.ng1o 6 where the bee.d will be in equilibrium-that ia, whoro it will have no tendency 1o move ap or down along the hoop. (b) If f = 2.00rev/• and r = 22.0cm, whatls6? (c) Can the bead ride 81

hiP as the uruer of the circle {6 = 90"}? Bxpl.aln..

FIGUil£ 5-54 Ptobkm93.

94. &:rth i8 'IWt ~ 111'1 inerti4l frame. We often make meume­menl:s in a reference frame fiud on the Barth, assuming Earth i& an iner1ia1 reference frame. But tbe Btmh rotates, 10

thiJ 8!11umptioll. iJ not quite valid. Show ~t thiJ anumption is off by 3 pam in 1000 by calculating the acceleration o1 an objeu Ill B!uth'a equator due to Earth'• dail.y rotalion, and compare to 8 = 9.80 mfal. the acceleration due to gravity.

"· While fi.shi:ag. you g:et bored and start to swing a ainker weight e.round in a circle below you on a 0.45-m pi~ of &b!ng line. The weight makes a QOmpl.ete circle every O.SO f. What is the angl~ that the fishing lin~ lll.8kes with th~ vertical? [Rlnt. See Fig. S-20.]

!16. Coll!lider a train ~t roUD.dJ a curve with a radius of S70 m at upeed oft60 km/h (approximately 100mi/h). (a) Calcu· late 1he friction force nl:eded on a train passeDger of !lla!ll

7S tg if th~ tre.ck il not bank~ and the ttain d~ not tilt. (b) Cakulate the friction fotee on the pa.aenger If the train tiltl at an an,gle of s.o• toward ~ center of tho curve.

9'1. A car starts rollillg down a 1-ln-4 bill (1-in-4 means that for ~ 4 m ll'llveled alODg tho road, tho elevation dlango is 1m). How fast II it going when It rellil:b.es the bottom after traveling SSm? (a) Ignore fricliml. (b) Anume an eff~e coefficient o1 friction equal to 0.10.

General Problems 1:17

Page 7: Ch5 Problems

98. 'lb.o sides of a caoo mako an llllg!o ~ with the vertical. A smalllll.UI m Is plac:ed on the ladde of the cone alld the IXIIl.e, wilh ita point down, ia r:evolYed Ill a frequoDcy f (revolutionJ per secon.d) about its a,mme!Jy alii&. U the ooetficient of static fri.clion ill p..,, at what poaldo1111 on the cone can the m.8IIIJ be placed without sliding 011 the conn? (Givo tho m.uimum and mlnlm:um dl&tan.ca, r, l'rom the u:ls).

99. A 72-kg water a1:ier ill being accelerated by a ati boat on a flat ("glaasy") lake. The coefficient of kinetic l'rll:tion betwocm the llkier'a akill and tho water surface ia P.l: = 0.25 {Fis. S-55). (12) What ia the atier's accelerlllion if the rope pulliJJg d&CI wer behind the boat appliee a horiz<mtal. tenaion foiu of mapitudo Fr = 240N to tho mar (9 = o•)? (b) What Is the skier's horizontal acceleration If the rope pulliDg the wer ae!U a forllC of Fr = 2110 N on the akier at an upward angle 9 - 12"? (c) EJplain why the stie:r's aoc;elerlllion in part (b) ia gn:ater !han !hal in part (4).

~~~~~lL .::l:;~~~;;r ~~t=0.2S

FIGURE !1-55 Problem 99.

100. A baD of lll.ll6t m = 1.0 tg at !he end of a thin ami ofleusth r ~ 0.80 m revolvea In a vertical circle about po!ot 0, aa lhoWD in Fig. S-56. During tho time we ob:smvo it, the only fon::es acting on the ball are gravity and the tension In the c:ord.The motion ia cimllar but not UDiform ~ of 1he fonlO of gra'rity.'Iho ball inl:nlase8 in speed u it dosc:eDds and deulerates as ll risea on the other side of the drcle. At the moment the cord IDIIkea an an,glo 9 = 30• below tho horizontal, the ball's 11peed iJ 6.0m/a. At thia point, dotmnine the tangential accel­eration. tho radial acceleradon, and the ta:llion in the cord, llr· 'Illke 9 iDcreuing downward as .&own.

FIGUR£5-56 Problem 100.

AnlweiS to Exercises

A: (c).

B: Fn 1.1 inau11iclent to keep the box moving ftlr long. C: No-d&CI aoc;eleration iJ not conn ant (in direcli011). D: (4), it double& & (d).

1DL A car drivel at a c:omtant speed around a banked cireular traci. with a diameter of 127m. The motion of the car can be deiCribed in a cocn:dinato IJYlltom with ita origin at the uruer ot the circle. At a particular instant the au's aocel­eratlon In the horizontal plane i& glw:n by

i = { -1S.7i - 23.2j) mtr.

(4) What ill !he car's speed? (b) Where (x andy) ia the car at this Instant?

• Numerical/Computer • 101. (DI) The force of air re4inance (drag fo!'Qe) on a rapidly

falling body sadl u a skydiftl' baa the fonn Fo = -kif, so tlw Newton 'II SCIXlDd. law applied to IIIICh an object ill

md"=mg-/W' dl ,

where the downward direction Is taken to be poclti•e. (4) Use num.e.:riw imegrati011. (SeWOD l-9] to eslimllle (within 2%) the position, speed, and accelerUon.. from t - 0 up to t ~ l.S.O 1, for a 7S·kg akydlver who 11art11 from rest, IIIIIIDliiig k = 0.22 kg/m. (b) Show !hat tho diver ovcmtually readies a study speed, the temaiNI1 ~and e;,plaln why dliJ happen& (c) How long dora it take for the llkydiver to r:eacb 99.5% of tho terminalllpOOd?

•m (ID) 'lb.e coefficient of kinetic friction Il-k between two IJ1Il'faces ia not strictly independent of the vel()Qty of the object. A possible a.xpressi.O!I for Il-k for wood on wood is

0.20 P.k ~ {t + otmJW-'(

where 11 ia in m/a. A wooden block of maaa 8.0 kg iJ Ill rm on a wooden tloor, and a constant horizontal force of 41 N aca on the block. Use numeriul integration [S~on l-9] to ~rmino and graph (a) tho speed of the block, and (b) i!3 pocition, as a fandion oftime from 0 to S.O a. (c) DeteiDline tho porteD.t difference for tho speed and poaition Ill 5.0 s if Il-k is constant and equal to 0.20.

• 104. (DI) Aslume a net forte F ~ -mg - kVZ acta during the upward vertical moti011. of a 250-q ~. startiDg Ill the moment (t - 0) when the fuel baa burned out and the rocket has an upward 'fiCCd of 120m/& Let k ~ o.6S k¥Jm. Eltimate v and y at 1.0.. intervals for the upward motion only, and estimate the maximum heJ&ht readl.ed. Compare to free·fli8ht COD.ditionl without air re4inanoe (k = 0).

1': (4).

G: (c).

H: Ye&

I: {a) No change: (b) 4 times larger.

1H OiAPIER 5 Using Newton's Laws: Friction, Cira.llar Motion, Drag Forces