Chapter 5 Applications of Newton s Laws

Transcription

1 Chapte 5 Application of Newton Law Conceptual Poblem Detemine the Concept Becaue the object ae peeding up (acceleating), thee mut be a net foce acting on them. The foce acting on an object ae the nomal foce exeted by the floo of the tuc, the weight of the object, the fiction foce; alo exeted by the floo of the tuc. Of thee foce, the only one that act in the diection of the acceleation (choen to be to the ight in the fee-body diagam) i the fiction foce. The foce of fiction between the object the floo of the tuc mut be the foce that caue the object to acceleate. * Detemine the Concept The foce acting on an object ae the nomal foce exeted by the floo of the tuc, the weight of the object, the fiction foce; alo exeted by the floo of the tuc. Of thee foce, the only one that act in the diection of the acceleation (choen to be to the ight in the fee-body diagam) i the fiction foce. Apply Newton nd law to the object to detemine how the citical acceleation depend on it weight. Taing the poitie x diection to be to the ight, apply Σ x ma x ole fo a x : f µ w µ mg ma x a x µ g Becaue a x i independent of m w, the citical acceleation ae the ame. 65

2 66 Chapte 5 3 Detemine the Concept The foce acting on the bloc ae the nomal foce n exeted by the incline, the weight of the bloc mg exeted by the eath, the tatic fiction foce f exeted by an extenal agent. We can ue the definition of µ the condition fo equilibium to detemine the elationhip between µ θ. Apply Apply x max to the bloc: f mginθ 0 () diection: ma in the y y y n mgcoθ 0 () Diide equation () by equation () to obtain: tanθ f n Subtitute fo f ( µ n ): µ n tan θ µ n (d) i coect. *4 Detemine the Concept The bloc i in equilibium unde the influence of n, mg, f ; i.e., + m g + n f 0 We can apply Newton nd law in the x diection to detemine the elationhip between f mg. x Apply 0 to the bloc: f mginθ 0 Sole fo f : f mginθ (d) i coect.

3 Application of Newton Law 67 5 Pictue the Poblem The foce acting on the ca a it ound a cue of adiu R at maximum peed ae hown on the fee-body diagam to the ight. The centipetal foce i the tatic fiction foce exeted by the oadway on the tie. We can apply Newton nd law to the ca to deie an expeion fo it maximum peed then compae the peed unde the two fiction condition decibed. Apply ma to the ca: om the y equation we hae: Expe f,max in tem of n in the x equation ole fo max : Expe ' max fo µ ' µ : x f, max m R y n mg 0 n mg max µ gr o max contant µ µ max ' max contant.707max 7% (b) i coect. max *6 Pictue the Poblem The nomal eaction foce n poide the centipetal foce the foce of tatic fiction, µ n, eep the cycle fom liding down the wall. We can apply Newton nd law the definition of f,max to deie an expeion fo min. Apply ma to the motocycle: x n m R

4 68 Chapte 5 y f mg 0 o the minimum peed: f f,max µ n Subtitute fo f, eliminate n between the foce equation, ole fo min : Aume that R 6 m µ 0.8 ole fo min : min min Rg µ ( 6m)( 9.8m/ ) m/ 30.9 m/h 7 Detemine the Concept A the ping i extended, the foce exeted by the ping on the bloc inceae. Once that foce i geate than the maximum alue of the foce of tatic fiction on the bloc, the bloc will begin to moe. Howee, a it acceleate, it will hoten the length of the ping, deceaing the foce that the ping exet on the bloc. A thi happen, the foce of inetic fiction can then low the bloc to a top, which tat the cycle oe again. One inteeting application of thi to the eal wold i the bowing of a iolin ting: The ting unde tenion act lie the ping, while the bow act a the bloc, o a the bow i dagged aco the ting, the ting peiodically tic fee itelf fom the bow. 8 Tue. The elocity of an object moing in a cicle i continually changing independently of whethe the object peed i changing. The change in the elocity ecto the acceleation ecto the net foce acting on the object all point towad the cente of cicle. Thi cente-pointing foce i called a centipetal foce. 9 Detemine the Concept A paticle taeling in a etical cicle expeience a downwad gaitational foce plu an additional foce that contain it to moe along a cicula path. Becaue the net foce acting on the paticle will ay with location along it tajectoy, neithe (b), (c), no (d) can be coect. Becaue the elocity of a paticle moing along a cicula path i continually changing, (a) cannot be coect. (e) i coect. *0 Detemine the Concept We can analyze thee demontation by dawing foce diagam fo each ituation. In both diagam, h denote h, g denote gaitational, m denote magnetic, n denote nomal.

5 (a) Demontation : Demontation : Application of Newton Law 69 (b) Becaue the magnet doen t lift the ion in the fit demontation, the foce exeted on the ion mut be le than it (the ion ) weight. Thi i till tue when the two ae falling, but the motion of the ion i not etained by the table, the motion of the magnet i not etained by the h. Looing at the econd diagam, the net foce pulling the magnet down i geate than it weight, implying that it acceleation i geate than g. The oppoite i tue fo the ion: the magnetic foce act upwad, lowing it down, o it acceleation will be le than g. Becaue of thi, the magnet will catch up to the ion piece a they fall. * Pictue the Poblem The fee-body diagam how the foce acting on the two object ome time afte bloc i dopped. Note that, while T T, T T. The only foce pulling bloc to the left i the hoizontal component of the tenion. Becaue thi foce i malle than the magnitude of the tenion, the acceleation of bloc, which i identical to bloc, to the ight (T T ) will alway be geate than the acceleation of bloc to the left. Becaue the initial ditance fom bloc to the pulley i the ame a the initial ditance of bloc to the wall, bloc will hit the pulley befoe bloc hit the wall. n Tue. The teminal peed of an object i gien by t ( mg b), whee b depend on the hape aea of the falling object a well a upon the popetie of the medium in which the object i falling. 3 Detemine the Concept The teminal peed of a y die i gien by ( ), whee b depend on the hape aea of the falling object a well a upon the popetie of the medium in which the object i falling. The y die oientation a he fall t mg b n

6 70 Chapte 5 detemine the uface aea he peent to the ai molecule that mut be puhed aide. (d) i coect. 4 Detemine the Concept In you fame of efeence (the acceleating efeence fame of the ca), the diection of the foce mut point towad the cente of the cicula path along which you ae taeling; that i, in the diection of the centipetal foce that eep you moing in a cicle. The fiction between you the eat you ae itting on upplie thi foce. The eaon you eem to be "puhed" to the outide of the cue i that you body inetia "want", in accodance with Newton law of inetia, to eep it moing in a taight line that i, tangent to the cue. *5 Detemine the Concept The centipetal foce that eep the moon in it obit aound the eath i poided by the gaitational foce the eath exet on the moon. A decibed by Newton 3 d law, thi foce i equal in magnitude to the foce the moon exet on the eath. (d) i coect. 6 Detemine the Concept The only foce acting on the bloc ae it weight the foce the uface exet on it. Becaue the loop-the-loop uface i fictionle, the foce it exet on the bloc mut be pependicula to it uface. Point A: the weight i downwad the nomal foce i to the ight. ee-body diagam 3 Point B: the weight i downwad, the nomal foce i upwad, the nomal foce i geate than the weight o that thei diffeence i the centipetal foce. ee-body diagam 4 Point C: the weight i downwad the nomal foce i to the left. ee-body diagam 5 Point D: both the weight the nomal foce ae downwad. ee-body diagam

7 Application of Newton Law 7 7 Pictue the Poblem Aume that the dag foce on an object i gien by the Newtonian fomula D CAρ, whee A i the pojected uface aea, i the object peed, ρ i the denity of ai, C a dimenionle coefficient. Expe the net foce acting on the falling object: Subtitute fo D unde teminal peed condition ole fo the teminal peed: mg ma net D mg ρ o mg T CAρ CA T 0 Thu, the teminal elocity depend on the atio of the ma of the object to it uface aea. o a oc, which ha a elatiely mall uface aea compaed to it ma, the teminal peed will be elatiely high; fo a lightweight, pead-out object lie a feathe, the oppoite i tue. Anothe iue i that the highe the teminal elocity i, the longe it tae fo a falling object to each teminal elocity. om thi, the feathe will each it teminal elocity quicly, fall at an almot contant peed ey oon afte being dopped; a oc, if not dopped fom a geat height, will hae almot the ame acceleation a if it wee in feefall fo the duation of it fall, thu be continually peeding up a it fall. An inteeting point i that the aeage dag foce acting on the oc will be lage than that acting on the feathe peciely becaue the oc aeage peed i lage than the feathe', a the dag foce inceae a. Thi i anotheeminde that foce i not the ame thing a acceleation. Etimation Appoximation *8 Pictue the Poblem The fee-body diagam how the foce on the Tecel a it low fom 60 to 55 mph. We can ue Newton nd law to calculate the aeage foce fom the ate at which the ca peed deceae the olling foce fom it definition. The dag foce can be infeed fom the aeage olling fiction foce the dag coefficient fom the defining equation fo the dag foce. (a) Apply x max to the ca to elate the aeage foce acting on it to it aeage elocity: ma a a m t

8 7 Chapte 5 Subtitute numeical alue ealuate a : mi m h mi 3.9 h m m ( 00 g) 58N a (b) Uing it definition, expe ealuate the foce of olling fiction: f µ µ mg olling olling n ( 0.0)( 00 g)( 9.8m/ ) 00 N olling Auming that only two foce ae acting on the ca in the diection of it motion, expe theielationhip ole fo ealuate the dag foce: + a dag dag a olling olling 58N 00 N 38N (c) Conet 57.5 mi/h to m/: Uing the definition of the dag foce it calculated alue fom (b) the aeage peed of the ca duing thi 5 mph inteal, ole fo C: Subtitute numeical alue ealuate C: mi 57.5 h mi.609 m 57.5 h mi 3 h 0 m 3600 m 5.7 m/ C A C dag dag ρ C ( 38N) ρ A 3 (.g/m )(.9m )( 5.7 m/) Pictue the Poblem We can ue the dimenion of foce elocity to detemine the dimenion of the contant b the dimenion of ρ,, to how that, fo n, Newton expeion i conitent dimenionally with oueult fom pat (b). In pat (d) (e), we can apply Newton nd law unde teminal elocity condition to find the teminal elocity of the y die nea the uface of the eath at a height of 8 m. (a) Sole the dag foce equation fo b with n : d b

9 Subtitute the dimenion of d ML implify to obtain: [] b T L T Application of Newton Law 73 M T the unit of b ae g/ (b) Sole the dag foce equation fo b with n : b d Subtitute the dimenion of d ML implify to obtain: [] b T L T M L the unit of b ae g/m (c) Expe the dimenion of [ ] [ ] ( ) Newton expeion: d ρπ L 3 ML T M L L T om pat (b) we hae: (d) Letting the downwad diection be the poitie y diection, apply y may to the y die: [ ] [ b ] d ML T mg ρπ M L t L T 0 Sole fo ealuate t : mg ( 56g)( 9.8m/ ) t ρπ 56.9 m/ π 3 (. g/m )( 0.3m) (e) Ealuate t at a height of 8 m: ( 56g)( 9.8m/ ) t π 3 ( 0.54 g/m )( 0.3m) 86.9 m/

10 74 Chapte 5 0 Pictue the Poblem om Newton nd law, the equation decibing the motion of falling aindop lage hailtone i mg d ma whee d ρπ b i the dag foce. Unde teminal peed condition (a 0), the dag foce i equal to the weight of the falling object. Tae the adiu of a aindop to be 0.5 mm the adiu of a golf-ball ized hailtone h to be cm. Uing b πρ, ealuate b b h : 3 3 b π (. g/m )( m) b h π 7 g/m 3 (. g/m )( 0 m) g/m Expe the ma of a phee in 4π m ρv tem of it olume denity: 3 3 ρ Uing ρ 0 3 g/m 3 ρ h 90 g/m 3, ealuate m m h : 4 m π m h ( m) ( 0 g/m ) π 7 3 g 3 3 ( 0 m) ( 90 g/m ) g Expe the elationhip between t the weight of a falling object unde teminal peed condition ole fo t : b mg t t mg b Ue numeical alue to ealuate t, t,h : 7 ( g)( 9.8m/ ) t, m/ g/m ( g)( 9.8m/ ) t,h m/ g/m

11 Application of Newton Law 75 iction * Pictue the Poblem The bloc i in equilibium unde the influence of n, mg, f ; i.e., + m g + n f 0 We can apply Newton nd law to detemine the elationhip between f, θ, mg. Uing it definition, expe the coefficient of inetic fiction: f µ () n Apply x max to the bloc: f mginθ ma x 0 becaue a x 0 Sole fo f : f mginθ Apply y may to the bloc: n mgcoθ ma y 0 becaue a y 0 Sole fo n : Subtitute in equation () to obtain: n mgcoθ mg inθ µ tanθ mg coθ (b) i coect. Pictue the Poblem The bloc i in equilibium unde the influence of n, mg, app, f ; i.e., n + m g + app + f 0 We can apply Newton nd law to detemine f. Apply x max to the bloc: app f ma x 0 becaue a x 0 Sole fo f : f app 0 N (e) i coect.

12 76 Chapte 5 *3 Pictue the Poblem Whethe the fiction foce i that due to tatic fiction o inetic fiction depend on whethe the applied tenion i geate than the maximum tatic fiction foce. We can apply the definition of the maximum tatic fiction to decide whethe f,max o T i geate. Calculate the maximum tatic fiction foce: f,max µ n µ w (0.8)(0 N) 6 N (a) Becaue f,max > T: f f T 5.0 N (b) Becaue T > f,max : f f µ w (0.6)(0 N).0 N 4 Pictue the Poblem The bloc i in equilibium unde the influence of the foce T, f, m g ; i.e., T + f + m g 0 We can apply Newton nd law to detemine the elationhip between T f. Apply x max to the bloc: T coθ f ma x 0 becaue a x 0 Sole fo f : f T coθ (b) i coect. 5 Pictue the Poblem Whethe the fiction foce i that due to tatic fiction o inetic fiction depend on whethe the applied tenion i geate than the maximum tatic fiction foce. Calculate the maximum tatic f,max µ n µ w

13 Application of Newton Law 77 fiction foce: (0.6)(00 g)(9.8 m/ ) 589 N Becaue f,max > app, the box doe not moe : app f 500 N 6 Pictue the Poblem Becaue the box i moing with contant elocity, it acceleation i zeo it i in equilibium unde the influence of app, n, w, f ; i.e., app + n + w + f 0 We can apply Newton nd law to detemine the elationhip between f mg. The definition of µ i: µ f n Apply y may to the box: n w ma y 0 becaue a y 0 Sole fo n : n w 600 N Apply x max to the box: Σ x app f ma x 0 becaue a x 0 Sole fo f : app f 50 N Subtitute to obtain µ : µ (50 N)/(600 N) Pictue the Poblem Aume that the ca i taeling to the ight let the poitie x diection alo be to the ight. We can ue Newton nd law of motion the definition of µ to detemine the maximum acceleation of the ca. Once we now the ca maximum acceleation, we can ue a contant-acceleation equation to detemine the leat topping ditance.

14 78 Chapte 5 (a) Apply x max to the ca: f,max µ n ma x () Apply y may to the ca ole fo n : n w ma y 0 o, becaue a y 0, n mg () Subtitute () in () ole fo a x,max µ g (0.6)(9.8m/ ) a x,max : 5.89 m/ (b) Uing a contant-acceleation equation, elate the topping ditance of the ca to it initial elocity it acceleation ole fo it diplacement: Subtitute numeical alue ealuate x: + a x 0 o, becaue 0, 0 x a x ( 30 m/) ( 5.89 m/ ) 76.4 m *8 Pictue the Poblem The fee-body diagam how the foce acting on the die wheel, the one we e auming uppot half the weight of the ca. We can ue the definition of acceleation apply Newton nd law to the hoizontal etical component of the foce to detemine the minimum coefficient of fiction between the oad the tie. (a) Becaue µ, > µ f will be geate if the wheel do not lip. (b) Apply x max to the ca: f µ n ma x () Apply y may to the ca ole fo n : n mg ma y Becaue a y 0, mg 0 mg n n ind the acceleation of the ca: ( 90 m/h)( 000 m/m) a x t.08m/

15 Application of Newton Law 79 Sole equation () fo µ : Subtitute numeical alue ealuate a x : ma mg a g x µ x (.08m/ ) µ 9.8m/ 9 Pictue the Poblem The bloc i in equilibium unde the influence of the foce hown on the fee-body diagam. We can ue Newton nd law the definition of µ to ole fo f n. (a) Apply y may to the bloc f mg ma y ole fo f : o, becaue a y 0, f mg 0 Sole fo ealuate f : f mg ( 5g)( 9.8m/ ) 49.N (b) Ue the definition of µ to expe n : n f,max µ Subtitute numeical alue ealuate n : 30 Pictue the Poblem The fee-body diagam how the foce acting on the boo. The nomal foce i the net foce the tudent exet in queezing the boo. Let the hoizontal diection be the x diection upwad the y diection. Note that the nomal foce i the ame on eithe ide of the boo becaue it i not acceleating in the hoizontal diection. The boo could be acceleating downwad. We can apply Newton nd law to elate the minimum foce equied to hold the boo in place to it ma to the coefficient of tatic fiction. In pat (b), we can poceed imilaly to elate the acceleation of the 49.N 0.4 n 3 N

16 80 Chapte 5 boo to the coefficient of inetic fiction. (a) Apply ma to the boo: x y,min µ ',,min,min 0 + µ, ',min mg 0 ' ' Noting that,min,min, ole the mg min y equation fo min : µ, + µ, Subtitute numeical alue ealuate min : (b) Apply y may with the boo acceleating downwad, to obtain: ( 0. g)( 9.8m/ ) 08 N min , y µ + µ mg ma, Sole fo a to obtain: µ a, + µ, m g Subtitute numeical alue g ealuate a: a ( 95 N) 4.7 m/ 9.8m/ 3 Pictue the Poblem A fee-body diagam howing the foce acting on the ca i hown to the ight. The fiction foce that the gound exet on the tie i the foce f hown acting up the incline. We can ue the definition of the coefficient of tatic fiction Newton nd law to elate the angle of the incline to the foce acting on the ca. Apply ma to the ca: in x f mg θ 0 () y mg coθ 0 () n Sole equation () fo f equation () fo n : f mg inθ

17 Application of Newton Law 8 n mg coθ Ue the definition of µ to elate f n : f mg inθ µ tanθ mg coθ n Sole fo ealuate θ : θ ( µ ) ( ) tan tan *3 Pictue the Poblem The fee-body diagam fo the two method ae hown to the ight. Method eult in the box being puhed into the floo, inceaing the nomal foce the tatic fiction foce. Method patially lift the box,, educing the nomal foce the tatic fiction foce. We can apply Newton nd law to obtain expeion that elate the maximum tatic fiction foce to the applied foce. (a) Method i pefeable a it educe n, theefoe, f. (b) Apply x max to the box: coθ f coθ µ n 0 Method : Apply y may to the bloc ole fo n : n mg inθ 0 n mg + inθ Relate f,max to n : f,max µ n µ (mg + inθ) () Method : Apply y may to the foce in the y diection ole fo n : n mg + inθ 0 n mg inθ Relate f,max to n : f,max µ n µ (mg inθ) () Expe the condition that mut be atified to moe the box by eithe method: Method : Subtitute () in (3) ole fo : f,max coθ (3) µ mg (4) coθ µ inθ

18 8 Chapte 5 Method : Subtitute () in (3) ole fo : µ mg (5) coθ + µ inθ Ealuate equation (4) (5) with θ 30 : ( 30 ) 50 N ( 30 ) 5 N Ealuate (4) (5) with θ 0 : ( ) ( 0 ) µ mg 94 N 0 33 Pictue the Poblem Daw a fee-body diagam fo each object. In the abence of fiction, the 3-g box will moe to the ight, the -g box will moe down. The fiction foce i indicated by f without ubcipt; it i f fo (a) f fo (b). o alue of µ le than the alue found in pat (a) equied fo equilibium, the ytem will acceleate the fall time fo a gien ditance can be found uing a contantacceleation equation. (a) Apply x max to the 3-g box: Apply y may to the 3-g box, ole fo n,3, ubtitute in (): T f 0 becaue a x 0 () n,3 m 3 g 0 becaue a y 0 T µ m 3 g 0 () Apply x max to the -g box: m g T 0 becaue a x 0 (3) Sole () (3) imultaneouly m ole fo µ : µ m 3 (b) The time of fall i elated to the acceleation, which i contant: x o, becaue 0 0, x a t ( ) 0 t + a t ( ) Sole fo t: t x a (4)

19 Application of Newton Law 83 Apply x max to each box: T µ m 3 g m 3 a (5) m g T m a (6) Add equation (5) (6) ole fo a: Subtitute numeical alue ealuate a: a a ( m µ m ) 3 3 m + m g [ g 0.3( 3g) ]( 9.8m/ ).6 m/ g + 3g Subtitute numeical alue in equation (4) ealuate t: t ( m).6 m/ Pictue the Poblem The application of Newton nd law to the bloc will allow u to expe the coefficient of inetic fiction in tem of the acceleation of the bloc. We can then ue a contant-acceleation equation to detemine the bloc acceleation. The pictoial epeentation ummaize what we now about the motion. A fee-body diagam howing the foce acting on the bloc i hown to the ight. Apply x max to the bloc: f µ n ma () Apply y may to the bloc ole fo n : n mg 0 becaue a y 0 n mg ()

20 84 Chapte 5 Subtitute () in () ole fo µ : µ a/g (3) Uing a contant-acceleation equation, elate the initial final elocitie of the bloc to it diplacement acceleation: Sole fo a to obtain: 0 + a x o, becaue 0, 0, x d, 0 + ad a d Subtitute fo a in equation (3) to obtain: µ gd *35 Pictue the Poblem We can find the peed of the ytem when it ha moed a gien ditance by uing a contantacceleation equation. Unde the influence of the foce hown in the fee-body diagam, the bloc will hae a common acceleation a. The application of Newton nd law to each bloc, followed by the elimination of the tenion T the ue of the definition of f, will allow u to detemine the acceleation of the ytem. Uing a contant-acceleation equation, elate the peed of the ytem to it acceleation diplacement; ole fo it peed: Apply ne t ma to the bloc whoe ma i m : Uing f µ n, ubtitute (3) in () to obtain: Apply x max to the bloc whoe ma i m : Add the lat two equation to eliminate T ole fo a to 0 + a x, becaue 0 0, a x () Σ x T f m gin30 m a () Σ y n, m gco30 0 (3) T µ m g co30 m gin30 m a m g T m a a ( m µ m co30 m in ) 30 m + m g

21 Application of Newton Law 85 obtain: Subtitute numeical alue ealuate a: a.6m/ Subtitute numeical alue in equation () ealuate : ( )( 0.3m) 0.835m/.6m/ (a) i coect. 36 Pictue the Poblem Unde the influence of the foce hown in the fee-body diagam, the bloc ae in tatic equilibium. While f can be eithe up o down the incline, the fee-body diagam how the ituation in which motion i impending up the incline. The application of Newton nd law to each bloc, followed by the elimination of the tenion T the ue of the definition of f, will allow u to detemine the ange of alue fo m. (a) Apply whoe ma i m : ma to the bloc Σ x T ± f,max m gin30 0 () Σ y n, m gco30 0 () Uing f,max µ n, ubtitute () in T ± µ m g co30 () to obtain: m g in 30 m a (3) Apply x max to the bloc whoe ma i m : m g T 0 (4) Add equation (3) (4) to eliminate T ole fo m : Ealuate (5) denoting the alue of m with the plu ign a m,+ the alue of m with the minu ign a m,- to detemine the ange of alue of m fo which the ytem i in tatic equilibium: m ( ± co30 + in 30 ) ( 4g) [ ± ( 0.4) co30 + in 30 ] m µ m, g m 0.64g m,- 3.39g 0.64g (5)

22 86 Chapte 5 (b) With m g, the impending motion i down the incline the tatic fiction foce i up the incline. Apply x max to the bloc whoe ma i m : Apply x max to the bloc whoe ma i m : Add equation (6) (7) ole fo ealuate f : T + f m gin30 0 (6) m g T 0 (7) f (m in30 m )g [(4 g)in30 g](9.8 m/ ) 9.8N 37 Pictue the Poblem Unde the influence of the foce hown in the fee-body diagam, the bloc will hae a common acceleation a. The application of Newton nd law to each bloc, followed by the elimination of the tenion T the ue of the definition of f, will allow u to detemine the acceleation of the ytem. inally, we can ubtitute fo the tenion in eithe of the motion equation to detemine the acceleation of the mae. Apply whoe ma i m : ma to the bloc Σ x T f m gin30 m a () Σ y n, m gco30 0 () Uing f µ n, ubtitute () in () to obtain: T µ m g co30 m g in 30 m a (3) Apply x max to the bloc whoe ma i m : Add equation (3) (4) to eliminate T ole fo a to obtain: Subtituting numeical alue ealuating a yield: m g T m a (4) a a ( m µ m co30 m in ) m/ m + m g

23 Application of Newton Law 87 Subtitute fo a in equation (3) to obtain: T 37.3N *38 Pictue the Poblem The tuc will top in the hotet poible ditance when it acceleation i a maximum. The maximum acceleation i, in tun, detemined by the maximum alue of the tatic fiction foce. The fee-body diagam how the foce acting on the box a the tuc bae to a top. Aume that the tuc i moing in the poitie x diection apply Newton nd law the definition of f,max to find the hotet topping ditance. Uing a contant-acceleation equation, elate the tuc topping ditance to it acceleation initial elocity; ole fo the topping ditance: min 0 + a x o, ince 0, x a 0 max Apply ne t ma to the bloc: Σ x f,max ma max () Σ y n mg 0 () Uing the definition of f,max, ole equation () () imultaneouly fo a: f,max µ n a max µ g (0.3)(9.8 m/ ).943 m/ Subtitute numeical alue ealuate x min : x ( 80 m/h) ( 000 m/m) ( h/3600) min (.943m/ ) 9.6m

24 88 Chapte 5 39 Pictue the Poblem We can find the coefficient of fiction by applying Newton nd law detemining the acceleation fom the gien alue of diplacement initial elocity. We can find the diplacement peed of the bloc by uing contant-acceleation equation. Duing it motion up the incline, the um of the inetic fiction foce a component of the object weight will combine to bing the object to et. When it i moing down the incline, the diffeence between the weight component the fiction foce will be the net foce. (a) Daw a fee-body diagam fo the bloc a it tael up the incline: Apply ma to the bloc: Σ x f mgin37 ma () Σ y n mg co37 0 () Subtitute f µ n n fom () in () ole fo µ : g in37 a µ g co37 a tan37 g co37 (3) Uing a contant-acceleation equation, elate the final elocity of the bloc to it initial elocity, acceleation, diplacement: + a x 0 Soling fo a yield: Subtitute numeical alue ealuate a: a 0 x a ( 5. m/) ( 4m/) ( 8m) 0.6 m/

25 Application of Newton Law 89 Subtitute fo a in (3) to obtain: 0.6m/ µ tan37 co ( 9.8m/ ) (b) Ue the ame contantacceleation equation ued aboe but with 0 to obtain: 0 + a x 0 Sole fo x to obtain: x 0 a Subtitute numeical alue ealuate x: x ( 4m/) ( 0.6 m/ ) 9.5m (c) When the bloc lide down the incline, f i in the poitie x diection: Σ x f mgin37 ma Σ y n mgco37 0 a g µ co37 in 37 Sole fo a a in pat (a): ( ).m/ Ue the ame contant-acceleation equation ued in pat (b) to obtain: + a x 0 Set 0 0 ole fo : a x Subtitute numeical alue ealuate : (.m/ )( 9.5m) 4.73m/ 40 Pictue the Poblem We can find the topping ditance by applying Newton nd law to the automobile then uing a contant-acceleation equation. The fiction foce the oad exet on the tie the component of the ca weight along the incline combine to poide the net foce that top the ca. The pictoial epeentation ummaize what we now about the motion of the ca. We can ue Newton nd law to detemine the acceleation of the ca a contant-acceleation equation to obtain it topping ditance.

26 90 Chapte 5 (a) Uing a contant-acceleation equation, elate the final peed of the ca to it initial peed, acceleation, diplacement; ole fo it diplacement: min 0 + a a max o, becaue x 0 max x min 0, Daw the fee-body diagam fo the ca going up the incline: Apply ma to the ca: Σ x f,max mgin5 ma () Σ y n mgco5 0 () Subtitute f,max µ n n fom () in () ole fo a: a max g µ ( co5 + in5 ) 9.7 m/ Subtitute numeical alue in the expeion fo x min to obtain: x ( 30m/) min ( 9.7 m/ ) 49.m (b) Daw the fee-body diagam fo the ca going down the incline:

27 Apply m a to the ca: Application of Newton Law 9 Σ x f,max mgin5 ma Σ y n mgco5 0 Poceed a in (a) to obtain a max : ( ) a max g µ co 5 in5 4.09m/ Again, poceed a in (a) to obtain the diplacement of the ca: x ( 30m/) 0 min amax ( 4.09 m/ ) 0m 4 Pictue the Poblem The fiction foce the oad exet on the tie poide the net foce that acceleate the ca. The pictoial epeentation ummaize what we now about the motion of the ca. We can ue Newton nd law to detemine the acceleation of the ca a contant-acceleation equation to calculate how long it tae it to each 00 m/h. (a) Becaue 40% of the ca weight i on it two die wheel the acceleating fiction foce act jut on thee wheel, the fee-body diagam how jut the foce acting on the die wheel. Apply ma to the ca: Σ x f,max ma () Σ y n 0.4mg 0 () Ue the definition of f,max in equation () eliminate n between the two equation to obtain: a 0.4µ g m/ ( 0.7)( 9.8m/ ) (b) Uing a contant-acceleation equation, elate the initial final 0 + a t

28 9 Chapte 5 elocitie of the ca to it acceleation the elaped time; ole fo the time: o, becaue 0 0 t t, t a Subtitute numeical alue ealuate t : t ( 00 m/h)( h/3600)( 000 m/m).75m/ 0. *4 Pictue the Poblem To hold the box in place, the acceleation of the cat box mut be geat enough o that the tatic fiction foce acting on the box will equal the weight of the box. We can ue Newton nd law to detemine the minimum acceleation equied. (a) Apply ma to the box: Σ x n ma min () Σ y f,max mg 0 () Subtitute µ n fo f,max in equation (), eliminate n between the two equation ole fo ealuate a min : µ n mg 0, µ(ma min ) mg 0 g 9.8m/ a min 6.4m/ µ 0.6 (b) Sole equation () fo f,max, ubtitute numeical alue ealuate f,max : f,max mg ( g)(9.8 m/ ) 9.6 N (c) If a i twice that equied to hold the box in place, f will till hae it maximum alue gien by: f,max 9.6 N (d) Becaue g µ i a, the box will not fall if a g. min µ 43 Pictue the Poblem Note that the bloc hae a common acceleation that the tenion in the ting act on both bloc in accodance with Newton thid law of motion. Let down the incline be the poitie x diection. Daw the feebody diagam fo each bloc apply Newton econd law of motion the definition of the inetic fiction foce to each bloc to obtain imultaneou

29 Application of Newton Law 93 equation in a x T. Daw the fee-body diagam fo the bloc whoe ma i m : Apply ma to the uppe bloc: Σ x f, + T + m ginθ m a x () Σ y n, m gcoθ 0 () The elationhip between f, n, i: f, µ, n, (3) Eliminate f, n, between (), (), (3) to obtain: µ, m gcoθ + T + m ginθ m a x (4) Daw the fee-body diagam fo the bloc whoe ma i m : Apply ma to the bloc: Σ x f, T + m ginθ m a x (5) Σ y n, m gcoθ 0 (6) The elationhip between f, n, i: f, µ, n, (7) Eliminate f, n, between (5), (6), (7) to obtain: µ, m gcoθ T + m ginθ m a x (8)

30 94 Chapte 5 Noting that T T T, add equation (4) (8) to eliminate T ole fo a x : a x µ m + m µ,, inθ coθ g m + m Subtitute numeical alue ealuate a x to obtain: (b) Eliminate a x between equation (4) (8) ole fo T T T to obtain: ax 0.965m/ whee the minu ign tell u that the acceleation i diected up the incline. m m T ( µ µ ),, co m + m g θ Subtitute numeical alue ealuate T: T 0.84 N *44 Pictue the Poblem The fee-body diagam how the foce acting on the two bloc a they lide down the incline. Down the incline ha been choen a the poitie x diection. T i the foce tanmitted by the tic; it can be eithe tenile (T > 0) o compeie (T < 0). By applying Newton nd law to thee bloc, we can obtain equation in T a fom which we can eliminate eithe by oling them imultaneouly. Once we hae expeed T, the ole of the tic will become appaent. (a) Apply ma to bloc : Apply ma to bloc : x y x y T + m g inθ f n, m g coθ 0, m a mg inθ T f, ma m g coθ 0 n, Letting T T T, ue the definition of the inetic fiction foce to eliminate f, n, between the equation fo bloc f, n, between the equation fo bloc to obtain: ma mg inθ + T µ mg coθ () ma m g inθ T µ m g coθ ()

31 Application of Newton Law 95 Add equation () () to eliminate T ole fo a: (b) Rewite equation () () by diiding both ide of () by m both ide of () by m to obtain. µ + µ m m a g inθ coθ m + m T a g inθ + m µ g coθ (3) T a g inθ µ g coθ m (4) Subtacting (4) fom (3) m m m m eaanging yield: T ( µ µ ) g coθ If µ µ, T 0 the bloc moe down the incline with the ame acceleation of g Ineting a tic between them can' t ( inθ µ coθ ). change thi; theefoe, the tic mut exet no foce on eithe bloc. 45 Pictue the Poblem The pictoial epeentation how the oientation of the two bloc on the inclined uface. Daw the fee-body diagam fo each bloc apply Newton nd law of motion the definition of the tatic fiction foce to each bloc to obtain imultaneou equation in θ c T. (a) Daw the fee-body diagam fo the lowe bloc: Apply ma to the bloc: Σ x m ginθ c f, T 0 () Σ y n, m gcoθ c 0 () The elationhip between f, n, i: f, µ, n, (3)

32 96 Chapte 5 Eliminate f, n, between (), (), (3) to obtain: m ginθ c µ, m gcoθ c T 0 (4) Daw the fee-body diagam fo the uppe bloc: Apply ma to the bloc: Σ x T + m ginθ c f, 0 (5) Σ y n, m gcoθ c 0 (6) The elationhip between f, n, i: f, µ, n, (7) Eliminate f, n, between (5), (6), (7) to obtain: Add equation (4) (8) to eliminate T ole fo θ c : T + m ginθ c µ, m gcoθ c 0 (8) θ tan c tan 5.0 µ,m + µ,m m + m ( 0.4)( 0.g) + ( 0.6)( 0.g) 0.g + 0.g (b) Becaue θ c i geate than the angle of epoe (tan (µ, ) tan (0.4).8 ) fo the lowe bloc, it would lide if T 0. Sole equation (4) fo T: T m g ( inθ µ θ ) C, co C Subtitute numeical alue ealuate T: T ( 0.g)( 9.8m/ ) in5 ( 0.4) [ co5 ] 0.8 N

33 Application of Newton Law Pictue the Poblem The pictoial epeentation how the oientation of the two bloc with a common acceleation on the inclined uface. Daw the fee-body diagam fo each bloc apply Newton nd law the definition of the inetic fiction foce to each bloc to obtain imultaneou equation in a T. (a) Daw the fee-body diagam fo the lowe bloc: Apply ma to the lowe bloc: Expe the elationhip between f, n, : Σ x m gin0 f, T m a () Σ y n, m gco0 0 () f, µ, n, (3) Eliminate f, n, between (), mg in 0 µ,m g co0 (), (3) to obtain: T m a (4) Daw the fee-body diagam fo the uppe bloc: Apply ma to the uppe bloc: Expe the elationhip between f, n, : Σ x T + m gin0 f, m a (5) Σ y n, m gco0 0 (6) f, µ, n, (7)

34 98 Chapte 5 Eliminate f, n, between (5), T + mg in 0 µ,mg co0 (), (7) to obtain: m a (8) Add equation (4) (8) to eliminate T ole fo a: µ m + µ m a g in 0 co0 m + m Subtitute the gien alue ealuate a: a m/ (b) Subtitute fo a in eithe equation (4) o equation (8) to obtain: T 0.46 N compeion. ; i.e., the od i unde *47 Pictue the Poblem The etical component of educe the nomal foce; hence, the tatic fiction foce between the uface the bloc. The hoizontal component i eponible fo any tendency to moe equal the tatic fiction foce until it exceed it maximum alue. We can apply Newton nd law to the box, unde equilibium condition, to elate to θ. (a) The tatic-fictional foce oppoe the motion of the object, the maximum alue of the tatic-fictional foce i popotional to the nomal foce N. The nomal foce i equal to the weight minu the etical component V of the foce. Keeping the magnitude contant while inceaing θ fom zeo eult in a deceae in V thu a coeponding deceae in the maximum tatic-fictional foce f max. The object will begin to moe if the hoizontal component H of the foce exceed f max. An inceae in θ eult in a deceae in H. A θ inceae fom 0, the deceae in N i lage than the deceae in H, o the object i moe moe liely to lip. Howee, a θ appoache 90, H appoache zeo no moement will be initiated. If i lage enough if θ inceae fom 0, then at ome alue of θ the bloc will tat to moe. (b) Apply ma to the bloc: Σ x coθ f 0 () Σ y n + inθ mg 0 () Auming that f f,max, eliminate f n between equation () () ole fo : µ mg coθ + µ inθ

35 Application of Newton Law 99 Ue thi function with mg 40 N to geneate the table hown below: θ (deg) (N) The following gaph of (θ) wa plotted uing a peadheet pogam (N) theta (degee) om the gaph, we can ee that the minimum alue fo occu when θ 3. Rema: An altenatie to manually plotting a a function of θ o uing a peadheet pogam i to ue a gaphing calculato to ente gaph the function. 48 Pictue the Poblem The fee-body diagam how the foce acting on the bloc. We can apply Newton nd law, unde equilibium condition, to elate to θ then et it deiatie with epect to θ equal to zeo to find the alue of θ that minimize. (a) Apply ma to the bloc: Σ x coθ f 0 () Σ y n + inθ mg 0 ()

36 300 Chapte 5 Auming that f f,max, eliminate f n between equation () () ole fo : µ mg (3) coθ + µ inθ To find θ min, diffeentiate with epect to θ et the deiatie equal to zeo fo extema of the function: d dθ ( coθ + µ inθ ) ( µ mg) dθ ( coθ + µ inθ ) µ mg( inθ + µ coθ ) ( coθ + µ inθ ) d µ mg d 0 fo extema ( coθ + µ inθ ) dθ ( coθ + µ inθ ) Sole fo θ min to obtain: θ min tan µ (b) Ue the efeence tiangle hown below to ubtitute fo coθ inθ in equation (3): min + µ µ mg + µ + µ µ + µ µ mg + µ mg µ + µ (c) The coefficient of An analyi identical to the one aboe how hould apply to eep the bloc moing hould be applied at an angle gien by θ min tan µ. inetic fiction i le than the coefficient of Theefoe, once the bloc i will deceae, o the angle can be deceaed. tatic fiction. that the minimum foce one moing the coefficient of fiction 49 Pictue the Poblem The etical component of inceae the nomal foce the tatic fiction foce between the uface the bloc. The hoizontal component i eponible fo any tendency to moe equal the tatic fiction foce until it exceed it maximum alue. We can apply Newton nd law to the box, unde equilibium condition, to elate to θ.

37 Application of Newton Law 30 (a) A θ inceae fom zeo, inceae the nomal foce exeted by the uface the tatic fiction foce. A the hoizontal component of deceae with inceaing θ, one would expect to continue to inceae. (b) Apply ma to the bloc: Σ x coθ f 0 () Σ y n inθ mg 0 () Auming that f f,max, eliminate f n between equation () () ole fo : µ mg (3) coθ µ inθ Ue thi function with mg 40 N to geneate the table hown below. θ (deg) (N) ey lage The gaph of a a function of θ, plotted uing a peadheet pogam, confim ou pediction that continue to inceae with θ (N) theta (degee) (a) om the gaph we ee that: θ 0 min

38 30 Chapte 5 (b) Ealuate equation (3) fo θ 0 to obtain: µ mg co0 µ in 0 µ mg (c) You hould eep the angle at 0. Rema: An altenatie to the ue of a peadheet pogam i to ue a gaphing calculato to ente gaph the function. 50 Pictue the Poblem The foce acting on each of thee mae ae hown in the feebody diagam below. m epeent the ma of the 0-g ma m that of the 00-g ma. A decibed by Newton 3 d law, the nomal eaction foce n, the fiction foce f, ( f, ) act on both mae but in oppoite diection. Newton nd law the definition of inetic fiction foce can be ued to detemine the aiou foce the acceleation called fo in thi poblem. (a) Daw a fee-body diagam howing the foce acting on the 0-g ma: Apply ma to thi ma: Σ x f, m a () Σ y n, m g 0 () Sole equation () fo f, : f, m a (0 g)(4 m/ ) 80.0 N (b) Daw a fee-body diagam howing the foce acting on the 00-g ma: Apply x max to the 00-g object ealuate net : ( 00g)( 6m/ ) 600 N net m a

39 Application of Newton Law 303 Expe in tem of net f, : net + f, 600 N + 80 N 680 N (c) When the 0-g ma fall off, the 680-N foce act jut on the 00-g ma it acceleation i gien by Newton nd law: a m 680 N 00g 6.80 net m/ 5 Pictue the Poblem The foce acting on each of thee bloc ae hown in the feebody diagam to the ight. m epeent the ma of the 60-g bloc m that of the 00-g bloc. A decibed by Newton 3 d law, the nomal eaction foce n, the fiction foce f, ( f, ) act on both object but in oppoite diection. Newton nd law the definition of inetic fiction foce can be ued to detemine the coefficient of inetic fiction acceleation of the 00-g bloc. (a) Apply ma to the 60-g bloc: Apply x max to the 00-g bloc: Uing equation (), expe the elationhip between the inetic fiction foce f, f, : Σ x f, m a () Σ y n, m g 0 () f, m a (3) f, f, f µ n, µ m g (4) Subtitute equation (4) into equation ma µ () ole fo µ : m g Subtitute numeical alue ealuate µ : 30 N µ ( 60g)( 3m/ ) ( 60g)( 9.8m/ ) 0.38 (b) Subtitute equation (4) into µ mg a equation (3) ole fo a : m

40 304 Chapte 5 Subtitute numeical alue ealuate a : a ( 0.38)( 60g)( 9.8m/ ).40 m/ 00g *5 Pictue the Poblem The acceleation of the tuc can be found by applying Newton nd law of motion. The fee-body diagam fo the tuc climbing the incline with maximum acceleation i hown to the ight. (a) Apply ma to the tuc when it i climbing the incline: Sole equation () fo n ue the definition of f,max to obtain: Subtitute equation (3) into equation () ole fo a: Subtitute numeical alue ealuate a: Σ x f,max mgin ma () Σ y n mgco 0 () f,max µ mgco (3) ( co in ) a g a µ ( 9.8m/ ) ( 0.85) 6. m/ [ co in ] (b) When the tuc i decending the incline with maximum acceleation, the tatic fiction foce point down the incline; i.e., it diection i eeed on the BD. Apply x max to the tuc unde thee condition: Subtitute equation (3) into equation (4) ole fo a: Subtitute numeical alue ealuate a: f,max mgin ma (4) ( co + in ) a g a µ ( 9.8m/ ) ( 0.85) 0. m/ [ co + in ]

41 Application of Newton Law Pictue the Poblem The foce acting on each of the bloc ae hown in the feebody diagam to the ight. m epeent the ma of the -g bloc m that of the 4-g bloc. A decibed by Newton 3 d law, the nomal eaction foce n, the fiction foce f, ( f, ) act on both object but in oppoite diection. Newton nd law the definition of the maximum tatic fiction foce can be ued to detemine the maximum foce acting on the 4-g bloc fo which the -g bloc doe not lide. (a) Apply ma to the -g bloc: Σ x f,,max m a max () Σ y n, m g 0 () Apply ma to the 4-g bloc: Σ x f,,max m a max (3) Σ y n, n, - m g 0 (4) Uing equation (), expe the elationhip between the tatic fiction foce f,, max f,,max : f,,max f,,max µ m g (5) Subtitute (5) in () ole fo a max : Sole equation (3) fo max : Subtitute numeical alue ealuate max : a max µ g (0.3)g.94 m/ max mamax + µ m g max ( 4g)(.94 m/ ) + ( 0.3)( g) ( 9.8m/ ) 7.7 N (b) Ue Newton nd law to expe the acceleation of the bloc moing a a unit: Subtitute numeical alue ealuate a: a a m + m ( 7.7 N) g + 4g.47 m/

42 306 Chapte 5 Becaue the fiction foce ae an action-eaction pai, the fiction foce acting on each bloc i gien by: f m a ( g)(.47 m/ ).94 N (c) If max, then m lip on m the fiction foce (now inetic) i gien by: f f µ m g Ue x max to elate the acceleation of the -g bloc to the net foce acting on it ole fo a : Ue x max to elate the acceleation of the 4-g bloc to the net foce acting on it: f µ m g m a a µ g (0.)g µ m g m a.96 m/ Sole fo a : a µ m g m Subtitute numeical alue ealuate a : a ( ) ( 0.)( g)( 9.8m/ ) 7.7 N 4g 7.87 m/ 54 Pictue the Poblem Let the poitie x diection be the diection of motion of thee bloc. The foce acting on each of the bloc ae hown, fo the tatic fiction cae, on the fee-body diagam to the ight. A decibed by Newton 3 d law, the nomal eaction foce n, the fiction foce f, ( f, ) act on both object but in oppoite diection. Newton nd law the definition of the maximum tatic fiction foce can be ued to detemine the maximum acceleation of the bloc whoe ma i m. (a) Apply ma to the -g bloc: Σ x f,,max m a max ()

43 Application of Newton Law 307 Σ y n, m g 0 () Apply ma to the 4-g bloc: Uing equation (), expe the elationhip between the tatic fiction foce f,, max f,, max : Σ x T f,,max m a max (3) Σ y n, n, m g 0 (4) f,,max f,,max µ m g (5) Subtitute (5) in () ole fo a max : a max µ g (0.6)g 5.89 m/ (b) Ue x max to expe the acceleation of the bloc moing a a unit: Apply x max to the object whoe ma i m 3 : T (m + m ) a max (6) m 3 g T m 3 a max (7) Add equation (6) (7) to eliminate T then ole fo ealuate m 3 : m 3 ( m + ) ( 0.6)( 0g + 5g) µ m µ.5g 0.6 (c) If m 3 30 g, then m will lide on m the fiction foce (now inetic) i gien by: Ue x max to elate the acceleation of the 30-g bloc to the net foce acting on it: f f µ m g m 3 g T m 3 a 3 (8) Noting that a a 3 that the fiction foce on the body whoe ma i m i due to inetic fiction, add equation (3) (8) ole fo ealuate the common acceleation: a a 3 g ( m µ m ) 3 m ( 9.8m/ ) 30g ( 0.4)( 5g) 6.87 m/ + m 3 [ ] 0g + 30g With bloc liding on bloc, the f µ m g m a ( )

44 308 Chapte 5 fiction foce acting on each i inetic equation () (3) become: Sole equation ( ) fo ealuate a : T f T µ m g m a (3 ) a µ g 3.9 m/ ( 0.4)( 9.8m/ ) Sole equation (3 ) fo T: T m a + µ m g Subtitute numeical alue ealuate T: T ( 0g)( 6.87 m/ ) + ( 0.4)( 5g)( 9.8m/ ) 88.3N 55 Pictue the Poblem Let the diection of motion be the poitie x diection. The fee-body diagam how the foce acting on both the bloc (M) the counteweight (m). While T T, T T. By applying Newton nd law to thee bloc, we can obtain equation in T a fom which we can eliminate the tenion. Once we now the acceleation of the bloc, we can ue contant-acceleation equation to detemine how fa it moe in coming to a momentay top. (a) Apply ma to the bloc on the incline: Apply ma to the counteweight: Letting T T T uing the definition of the inetic fiction foce, eliminate f n between the equation fo the bloc on the incline to obtain: x y T Mg inθ f n Mg coθ 0 Ma x mg T ma () T Mg inθ µ Mg coθ Ma () Eliminate T fom equation () () by adding them ole fo a: a m M ( inθ + µ coθ ) g m + M

45 Application of Newton Law 309 Subtitute numeical alue ealuate a: 550g a ( 600 g)( in co0 ) ( ) 550g + 600g 9.8m/ 0.63m/ (b) Uing a contant-acceleation equation, elate the peed of the bloc at the intant the ope bea to it acceleation diplacement a it lide to a top. Sole fo it diplacement: The bloc had been acceleating up the incline fo 3 befoe the ope boe, o it ha an initial peed of : om equation () we can ee that, when the ope bea (T 0) : Subtitute in equation (3) ealuate x: (c) When the bloc i liding down the incline, the inetic fiction foce will be up the incline. Expe the bloc acceleation: f i + a x o, becaue f 0, i x (3) a (0.63 m/ )(3 ) m/ ( inθ + µ co ) [ in co0 ] a g θ ( 9.8m/ ) ( ) 3.5m/ whee the minu ign indicate that the bloc i being acceleated down the incline, although it i till liding up the incline. x ( m/) ( 3.5m/ ) m ( inθ µ co ) [ in0 0.5 co0 ] a g θ ( 9.8m/ ) ( ) 0.54 m/ 56 Pictue the Poblem If the 0-g bloc i not to lide on the bacet, the maximum alue fo mut be equal to the maximum alue of f will poduce the maximum acceleation of thi bloc the bacet. We can apply Newton nd law the definition of f,max to fit calculate the maximum acceleation then the maximum alue of. (a) (b) Apply ma to the 0-g bloc when it i expeiencing it maximum acceleation: Σ x f,max m a,max () Σ y n, m g 0 ()

46 30 Chapte 5 Expe the tatic fiction foce acting on the 0-g bloc: Eliminate f,max n, fom equation (), () (3) to obtain: Apply x max to the bacet to obtain: f,max µ n, (3) µ m g m a,max (4) µ m g m a,max (5) Becaue a,max a,max, denote thi acceleation by a max. Eliminate fom equation (4) (5) ole fo a max : Subtitute numeical alue ealuate a max : a a max max mg µ m + m ( 0.4)( 0g)( 9.8m/ ).57 m/ ( ) 5g + 0g Sole equation (4) fo max : µ m g m a m ( µ g a ) Subtitute numeical alue ealuate : max max ( 0g) [( 0.4)( 9.8m/ ).57 m/ ] 3.5N *57 Pictue the Poblem The fee-body diagam how the foce acting on the bloc a it i moing up the incline. By applying Newton nd law, we can obtain expeion fo the acceleation of the bloc up down the incline. Adding ubtacting thee equation, togethe with the data found in the noteboo, will lead to alue fo g V µ. Apply a i i m to the bloc when it i moing up the incline: Uing the definition of f, eliminate n between the two equation to obtain: a up x y f n mg mg V V inθ ma coθ 0 µ g coθ g inθ () V V up

47 Application of Newton Law 3 When the bloc i moing down the incline, f i in the poitie x diection, it acceleation i: a down µ g coθ g inθ () V V Add equation () () to obtain: Sole equation (3) fo g V : Detemine θ fom the figue: a up + a g inθ (3) down V aup + adown g V (4) inθ 0.73glapp θ tan glapp Subtitute the data fom the noteboo in equation (4) to obtain: g V.73glapp/plipp +.4glapp/plipp in glapp/plipp Subtact equation () fom equation () to obtain: a a down up V µ g coθ Sole fo µ : Subtitute numeical alue ealuate µ : µ a a down g V up coθ.4glapp/plipp µ ( 8.4glapp/plipp ).73glapp/plipp co *58 Pictue the Poblem The fee-body diagam how the bloc liding down the incline unde the influence of a fiction foce, it weight, the nomal foce exeted on it by the inclined uface. We can find the ange of alue fo m fo the two ituation decibed in the poblem tatement by applying Newton nd law of motion to, fit, the condition unde which the bloc will not moe o lide if puhed, econdly, if puhed, the bloc will moe up the incline. (a) Aume that the bloc i liding down the incline with a contant elocity with no hanging weight (m 0) apply ma to x y f n + Mg inθ 0 Mg coθ 0

48 3 Chapte 5 the bloc: Uing f µ n, eliminate n between the two equation ole fo the net foce acting on the bloc: If the bloc i moing, thi net foce mut be nonnegatie : µ Mg coθ + Mg inθ net ( µ coθ + inθ ) Mg 0 Thi condition equie that: µ tanθ tan Becaue µ 0., thi condition i atified : To find the maximum alue, note that the maximum poible alue fo the tenion in the ope i mg. o the bloc to moe down the incline, the component of the bloc weight paallel to the incline minu the fictional foce mut be geate than o equal to the tenion in the ope: m 0 min Mginθ µ Mgcoθ mg Sole fo m max : m M ( inθ µ coθ ) max Subtitute numeical alue ealuate m max : m max ( 00g) [ in8 ( 0.) co8 ].9 g The ange of alue fo m i: (b) If the bloc i being dagged up the incline, the fictional foce will point down the incline, : Sole fo ealuate m min : If the bloc i not to moe unle puhed: Sole fo ealuate m max : The ange of alue fo m i: 0 m.9g Mg inθ + µ Mg coθ < mg m min > M (inθ + µ coθ) (00 g)[in8 + (0.)co8 ] 49.9g Mg inθ + µ Mg coθ > mg m max < M (inθ + µ coθ) (00 g)[in8 + (0.4)co8 ] 68.9g 49.9g m 68.9 g

49 Application of Newton Law Pictue the Poblem The fee-body diagam how the foce acting on the 0.5 g bloc when the acceleation i a minimum. Note the choice of coodinate ytem i conitent with the diection of. Apply Newton nd law to the bloc ole the eulting equation fo a min a max. (a) Apply ma to the 0.5-g bloc: Unde minimum acceleation, f f,max. Expe the elationhip between f,max n : Σ x n inθ f coθ ma () Σ y n coθ + f inθ mg 0 () f,max µ n (3) Subtitute f,max fo f in equation () ole fo n : n mg coθ + µ inθ Subtitute fo n in equation () ole fo a a min : Subtitute numeical alue ealuate a min : a a min min inθ µ coθ g coθ + µ inθ in35 ( ) ( 0.8) 9.8m/ 0.67 m/ co35 + ( 0.8) co35 in35 Teat the bloc incline a a ingle object to detemine min : min m tot a min (.5 g)( 0.67 m/ ).57 N To find the maximum acceleation, eee the diection of f apply ma to the bloc: Σ x n inθ + f coθ ma (4) Σ y n coθ f inθ mg 0 (5) Poceed a aboe to obtain: Subtitute numeical alue ealuate a max : a a max max inθ + µ coθ g coθ µ inθ in35 + ( ) ( 0.8) 9.8m/ 33.5m/ co35 ( 0.8) co35 in35

50 34 Chapte 5 Teat the bloc incline a a ingle object to detemine max : max m tot a max (.5 g)(33.5 m/ ) 83.8 N (b) Repeat (a) with µ 0.4 to obtain: min 5.75 N max 37.5 N 60 Pictue the Poblem The inetic fiction foce f i the poduct of the coefficient of liding fiction µ the nomal foce n the uface exet on the liding object. By applying Newton nd law in the etical diection, we can ee that, on a hoizontal uface, the nomal foce i the weight of the liding object. Note that the acceleation of the bloc i oppoite it diection of motion. (a) Relate the foce of inetic fiction to µ the nomal foce 0. f µ mg n 4 ( ) acting on the liding wooden object: Subtitute 0 m/ ealuate f : f 0.00g ( )( 9.8m/ ) 4 ( ( 0m/) ) 03N (b) Subtitute 0 m/ ealuate f : f 0.00g ( )( 9.8m/ ) 4 ( ( 0m/) ) 90.5 N 6 Pictue the Poblem The pictoial epeentation how the bloc liding fom left to ight coming to et when it ha taeled a ditance x. Note that the diection of the motion i oppoite that of the bloc acceleation. The acceleation topping ditance of the bloc can be found fom contant-acceleation equation. Let the diection of motion of the liding bloc be the poitie x diection. Becaue the uface i hoizontal, the nomal foce acting on the liding bloc i the bloc weight.

51 Application of Newton Law 35 (a) Uing a contant-acceleation equation, elate the bloc topping ditance to it initial peed acceleation; ole fo the topping ditance: + a x 0 o, becaue 0, 0 x () a Apply x max to the liding bloc, intoduce Konecny empiical expeion, ole fo the bloc acceleation: net, a m x 0.4 f m ( mg) m m 0.9 n Ealuate a with m 0 g: ( 0.4) [( 0g)( 9.8m/ )] a.60 m/ 0g 0.9 Subtitute in equation () ealuate the topping ditance when 0 0 m/: (b) Poceed a in (a) with m 00 g to obtain: x a ( 0m/) (.60 m/ ) 9.m ( 0.4) [( 00g)( 9.8m/ )].m/ 00g 0.9 ind the topping ditance a in (a): ( 0m/) x (.m/ ) 3.7 m *6 Pictue the Poblem The inetic fiction foce f i the poduct of the coefficient of liding fiction µ the nomal foce n the uface exet on the liding object. By applying Newton nd law in the etical diection, we can ee that, on a hoizontal uface, the nomal foce i the weight of the liding object. We can apply Newton nd law in the hoizontal (x) diection to elate the bloc acceleation to the net foce acting on it. In the peadheet pogam, we ll find the acceleation of the bloc fom thi net foce (which i elocity dependent), calculate the inceae in the bloc peed fom it acceleation the elaped time add thi inceae to it peed at end of the peiou time inteal, detemine how fa it ha moed in thi time inteal, add thi ditance to it peiou poition to find it cuent poition. We ll alo calculate the poition of the bloc x, unde the aumption that µ 0., uing a contant-acceleation equation.

52 36 Chapte 5 The peadheet olution i hown below. The fomula ued to calculate the quantitie in the column ae a follow: Cell omula/content Algebaic om C9 C8+$B$6 t + t D9 D8+9*$B$6 + a t E9 $B$5 ($B$3)*($B$)*$B$5/ µ mg E0/$B$5 net / m G9 G9+D0*$B$6 x + t K9 0.5*5.9*I0^ at L9 J0-K0 x (+$B$4*D9^)^ ( ) x A B C D E G H I J g 9.8 m/^ Coeff 0. 3 Coeff.30E Ma 0 g 5 Applied 70 N oce 6 Time tep t x x x x Net foce a x muaiable mucontant t

53 Application of Newton Law The diplacement of the bloc a a function of time, fo a contant coefficient of fiction (µ 0.) i hown a a olid line on the gaph fo a aiable coefficient of fiction, i hown a a dotted line. Becaue the coefficient of fiction deceae with inceaing paticle peed, the paticle tael lightly fathe when the coefficient of fiction i aiable mu aiable mu contant x (m) t () The elocity of the bloc, with aiable coefficient of inetic fiction, i hown below (m/) t ()

54 38 Chapte 5 63 Pictue the Poblem The fee-body diagam how the foce acting on the bloc a it moe to the ight. The inetic fiction foce will low the bloc, eentually, bing it to et. We can elate the coefficient of inetic fiction to the topping time ditance by applying Newton nd law then uing contantacceleation equation. (a) Apply ma to the bloc of wood: x y f n ma mg 0 Uing the definition of f, eliminate n between the two equation to obtain: Ue a contant-acceleation equation to elate the acceleation of the bloc to it diplacement it topping time: a µ g () ( ) x () 0 t + a t Relate the initial peed of the bloc, 0, to it diplacement topping ditance: 0 + x a t t t ince 0. 0 (3) Ue thi eult to eliminate 0 in equation (): Subtitute equation () in equation ( ) x (4) t a x µ g t (4) ole fo µ : ( ) Subtitute fo x.37 m t 0.97 to obtain: µ (.37 m) ( 9.8m/ )( 0.97) (b) Ue equation (3) to find 0 : x (.37 m) 0 t m/

55 Application of Newton Law 39 *64 Pictue the Poblem The fee-body diagam how the foce acting on the bloc a it lide down an incline. We can apply Newton nd law to thee foce to obtain the acceleation of the bloc then manipulate thi expeion algebaically to how that a gaph of a/coθ eu tanθ will be linea with a lope equal to the acceleation due to gaity an intecept whoe abolute alue i the coefficient of inetic fiction. (a) Apply ma to the bloc a it lide down the incline: Subtitute µ n fo f eliminate n between the two equation to obtain: Diide both ide of thi equation by coθ to obtain: Note that thi equation i of the fom y mx + b: x y a g mg inθ f n ma mg coθ 0 ( inθ µ coθ ) a g tanθ gµ coθ Thu, if we gaph a/coθ eu tanθ, we hould get a taight line with lope g y-intecept gµ. (b) A peadheet olution i hown below. The fomula ued to calculate the quantitie in the column ae a follow: Cell omula/content Algebaic om C7 θ D7 a E7 TAN(C7*PI()/80) π tan θ 80 7 D7/COS(C7*PI()/80) a π co θ 80 C D E 6 theta a tan(theta) a/co(theta)

56 30 Chapte A gaph of a/coθ eu tanθ i hown below. om the cue fit (Excel Tendline.6 m/ wa ued), g 9.77 m/ µ m/ The pecentage eo in g fom the commonly accepted alue of 9.8 m/ i 9.8m/ 9.77 m/ % 9.8m/ a /co(theta) y 9.768x R tan(theta)

57 Application of Newton Law 3 Motion Along a Cued Path 65 Pictue the Poblem The fee-body diagam howing the foce acting on the tone i upeimpoed on a etch of the tone otating in a hoizontal cicle. The only foce acting on the tone ae the tenion in the ting the gaitational foce. The centipetal foce equied to maintain the cicula motion i a component of the tenion. We ll ole the poblem fo the geneal cae in which the angle with the hoizontal i θ by applying Newton nd law of motion to the foce acting on the tone. Apply ma to the tone: Σ x Tcoθ ma c m / () Σ y Tinθ mg 0 () Ue the ight tiangle in the diagam to elate, L, θ : Eliminate T between equation (), () (3) ole fo : Expe the elocity of the tone in tem of it peiod: Eliminate between equation (4) (5) ole fo θ : Subtitute numeical alue ealuate θ : Lcoθ (3) glcotθ coθ (4) π (5) t e θ in gte 4π L θ in (c) i coect. ( 9.8m/ )(.) 4π ( 0.85m) 5.8

58 3 Chapte 5 66 Pictue the Poblem The fee-body diagam howing the foce acting on the tone i upeimpoed on a etch of the tone otating in a hoizontal cicle. The only foce acting on the tone ae the tenion in the ting the gaitational foce. The centipetal foce equied to maintain the cicula motion i a component of the tenion. We ll ole the poblem fo the geneal cae in which the angle with the hoizontal i θ by applying Newton nd law of motion to the foce acting on the tone. Apply ma to the tone: Σ x Tcoθ ma c m / () Σ y Tinθ mg 0 () Ue the ight tiangle in the diagam to elate, L, θ: Lcoθ (3) Eliminate T between equation (), (), (3) ole fo : Subtitute numeical alue ealuate : glcotθ coθ ( 9.8m/ )( 0.8m) 4.50 m/ cot 0 co0 67 Pictue the Poblem The fee-body diagam howing the foce acting on the tone i upeimpoed on a etch of the tone otating in a hoizontal cicle. The only foce acting on the tone ae the tenion in the ting the gaitational foce. The centipetal foce equied to maintain the cicula motion i a component of the tenion. We ll ole the poblem fo the geneal cae in which the angle with the etical i θ by applying Newton nd law of motion to the foce acting on the tone. (a) Apply ma to the tone: Σ x Tinθ ma c m / ()

59 Application of Newton Law 33 Σ y Tcoθ mg 0 () Eliminate T between equation () () ole fo : Subtitute numeical alue ealuate : g tanθ ( 0.35m)( 9.8m/ ).4m/ tan30 (b) Sole equation () fo T: T mg coθ Subtitute numeical alue ealuate T: T ( 0.75g)( 9.8m/ ) 8.50 N co30 *68 Pictue the Poblem The etch how the foce acting on the pilot when he plane i at the lowet point of it die. n i the foce the aiplane eat exet on he. We ll apply Newton nd law fo cicula motion to detemine n the adiu of the cicula path followed by the aiplane. (a) Apply y may to the pilot: n mg ma c Sole fo ealuate n : n mg + ma c m(g + a c ) m(g + 8.5g) 9.5mg (9.5) (50 g) (9.8 m/ ) 4.66N (b) Relate he acceleation to he elocity the adiu of the cicula ac ole fo the adiu: a c a c Subtitute numeical alue ealuate : [( 345m/h)( h/3600)( 000 m/m) ] ( ) m/ 0m

60 34 Chapte 5 69 Pictue the Poblem The diagam how the foce acting on the pilot when he plane i at the lowet point of it die. n i the foce the aiplane eat exet on he. We ll ue the definition of centipetal acceleation centipetal foce apply Newton nd law to calculate thee quantitie the nomal foce acting on he. (a) He acceleation i centipetal gien by: a c, upwad Subtitute numeical alue ealuate a c : a c 3 [( 80 m/h)( h/3600)( 0 /m)] 300m 8.33m/, upwad (b) The net foce acting on he at the bottom of the cicle i the foce eponible fo he centipetal acceleation: net ma c ( 65g)( 8.33m/ ) 54N, upwad (c) Apply y may to the pilot: n mg ma c Sole fo n : n mg + ma c m(g + a c ) Subtitute numeical alue ealuate n : n (65 g)(9.8 m/ m/ ).8N, upwad 70 Pictue the Poblem The fee-body diagam fo the two object ae hown to the ight. The hole in the table change the diection the tenion in the ting (which poide the centipetal foce equied to eep the object moing in a cicula path) act. The application of Newton nd law the definition of centipetal foce will lead u to an expeion fo a a function of m, m, the time T fo one eolution.

61 Application of Newton Law 35 Apply x max to both object ue the definition of centipetal acceleation to obtain: Becaue we can eliminate both of them between thee equation to obtain: Expe the peed of the object in tem of the ditance it tael each eolution the time T fo one eolution: Subtitute to obtain: m g 0 m a c m / m g m π T m o m 4π T 0 g m 4π T g m 0 0 Sole fo: mgt 4π m *7 Pictue the Poblem The fee-body diagam how the foce acting on each bloc. We can ue Newton nd law to elate thee foce to each othe to the mae acceleation of the bloc. Apply x max to the bloc T T m whoe ma i m : L Apply x max to the bloc T m whoe ma i m : L + L Relate the peed of each bloc to thei common peiod thei ditance fom the cente of the ( L L ) πl π + T T

62 36 Chapte 5 cicle: Sole the fit foce equation fo T, ubtitute fo, implify to obtain: T [ m ( L + L )] π T Subtitute fo T in the fit foce equation to obtain: T [ m( L + L ) + ml ] π T *7 Pictue the Poblem The path of the paticle it poition at - inteal ae hown. The diplacement ecto ae alo hown. The elocity ecto fo the aeage elocitie in the fit econd inteal ae along 0, epectiely, ae hown in the lowe diagam. point towad the cente of the cicle. Ue the diagam to the ight to find : in.5 (4 cm) in cm ind the aeage elocity of the paticle along the chod: Uing the lowe diagam the fact that the angle between i 45, expe in tem of ( ): Ealuate uing a a : a / t (3.06 cm)/( ) 3.06 cm/ in.5 (3.06 cm/)in.5.34 cm/ Now we can detemine a / t:.34 cm/ a.34 cm/

63 Application of Newton Law 37 ind the peed ( ) of the paticle along it cicula path: π T ( 4cm) π 8 3.4cm/ Calculate the adial acceleation of the paticle: a c ( 3.4cm/) 4cm.46cm/ Compae a c a by taing thei atio: a.46cm/ a.34cm/ o c a c. 05a Pictue the Poblem The diagam to the ight ha the fee-body diagam fo the child upeimpoed on a pictoial epeentation of he motion. The foce he fathe exet i the angle it mae with epect to the diection we e choen a the poitie y diection i θ. We can infe he peed fom the gien infomation concening the adiu of he path the peiod of he motion. Applying Newton nd law a it decibe cicula motion will allow u to find both the diection magnitude of. Apply ma to the child: Σ x inθ m / Σ y coθ mg 0 Eliminate between thee equation ole fo θ : Expe in tem of the adiu peiod of the child motion: Subtitute fo in the expeion fo θ to obtain: θ tan π T θ tan g π gt 4

64 38 Chapte 5 Subtitute numeical alue ealuate θ : Sole the y equation fo : 4 ( 0.75m) ( 9.8m/ )(.5) π θ tan mg coθ 53.3 Subtitute numeical alue ealuate : ( 5g)( 9.8m/ ) 40 N co Pictue the Poblem The diagam to the ight ha the fee-body diagam fo the bob of the conical pendulum upeimpoed on a pictoial epeentation of it motion. The tenion in the ting i the angle it mae with epect to the diection we e choen a the poitie x diection iθ. We can findθ fom the y equation the infomation poided about the tenion. Then, by uing the definition of the peed of the bob in it obit applying Newton nd law a it decibe cicula motion, we can find the peiod T of the motion. Apply ma to the pendulum bob: Uing the gien infomation that 6mg, ole the y equation fo θ: Σ x coθ m / Σ y inθ mg 0 θ in mg in mg mg With 6mg, ole the x equation fo : 6g coθ Relate the peiod T of the motion to the peed of the bob the adiu of the cicle in which it moe: π T π 6g coθ om the diagam, one can ee that: Lcoθ

65 Application of Newton Law 39 Subtitute fo in the expeion fo the peiod to obtain: T π L 6g Subtitute numeical alue 0.5m T π ealuate T: 6 ( 9.8m/ ) Pictue the Poblem The tatic fiction foce f i eponible fo eeping the coin fom liding on the tuntable. Uing Newton nd law of motion, the definition of the peiod of the coin motion, the definition of the maximum tatic fiction foce, we can find the magnitude of the fiction foce the alue of the coefficient of tatic fiction fo the two uface. (a) Apply ma to the coin: If T i the peiod of the coin motion, it peed i gien by: x f m y n mg 0 π T Subtitute fo in the foce equation 4π m f implify to obtain: T Subtitute numeical alue ealuate f : f 4π ((0.g)( 0.m) ( ) N (b) Detemine n fom the y equation: n mg If the coin i about to lide at 6 cm, f f,max. Sole fo µ in tem of f,max n : 4π m f,max µ T mg n 4π gt

66 330 Chapte 5 Subtitute numeical alue ealuate µ : µ 4π ( 0.6m) ( 9.8m/ )( ) Pictue the Poblem The foce acting on the tetheball ae hown upeimpoed on a pictoial epeentation of the motion. The hoizontal component of T i the centipetal foce. Applying Newton nd law of motion oling the eulting equation will yield both the tenion in the cod the peed of the ball. (a) Apply ma to the tetheball: x T in 0 m y T co0 mg 0 Sole the y equation fo T: Subtitute numeical alue ealuate T: T T mg co0 ( 0.5g)( 9.8m/ ).6N co0 (b) Eliminate T between the foce equation ole fo : g tan 0 Note fom the diagam that: Subtitute fo in the expeion fo to obtain: Lin0 glin 0 tan 0 Subtitute numeical alue ealuate : ( 9.8m/ )( ).m/.m in0 tan 0

67 Application of Newton Law 33 *77 Pictue the Poblem The diagam include a pictoial epeentation of the eath in it obit about the un a foce diagam howing the foce on an object at the equato that i due to the eath otation, R, the foce on the object due to the obital motion of the eath about the un, o. Becaue thee ae centipetal foce, we can calculate the acceleation they equie fom the peed adii aociated with the two cicula motion. Expe the adial acceleation due to the otation of the eath: a R R R Expe the peed of the object on the equato in tem of the adiu of the eath R the peiod of the eath otation T R : R πr T R Subtitute fo R in the expeion 4π R ar fo a R to obtain: T R Subtitute numeical alue ealuate a R : a R 4 π ( 6370 m)( 000 m/m) ( 4 h) h m/ g Expe the adial acceleation due to the obital motion of the eath: Expe the peed of the object on the equato in tem of the eath-un ditance the peiod of the eath motion about the un T o : a o o π o T o

68 33 Chapte 5 Subtitute fo o in the expeion 4π ao fo a o to obtain: T o Subtitute numeical alue ealuate a c : a o 4π ( 365d) (.5 0 m) 4h 3600 d h 3 m/ g 78 Pictue the Poblem The mot ignificant foce acting on the eath i the gaitational foce exeted by the un. Moe ditant o le maie object exet foce on the eath a well, but we can calculate the net foce by conideing the adial acceleation of the eath in it obit. Similaly, we can calculate the net foce acting on the moon by conideing it adial acceleation in it obit about the eath. (a) Apply ma to the eath: Expe the obital peed of the eath in tem of the time it tae to mae one tip aound the un (i.e., it peiod) it aeage ditance fom the un: on eath π T m Subtitute fo to obtain: on eath 4π m T Subtitute numeical alue ealuate on eath : 4 ( g)( m) N 4 on eath (b) Poceed a in (a) to obtain: π 4h d d h 8 ( g)( m).00 0 N 4 0 on moon π 4h d d h

69 Application of Newton Law Pictue the Poblem The emicicula wie of adiu 0 cm limit the motion of the bead in the ame manne a would a 0-cm ting attached to the bead fixed at the cente of the emicicle. The hoizontal component of the nomal foce the wie exet on the bead i the centipetal foce. The application of Newton nd law, the definition of the peed of the bead in it obit, the elationhip of the fequency of a cicula motion to it peiod will yield the angle at which the bead will emain tationay elatie to the otating wie. Apply ma to the bead: x n inθ m y n coθ mg 0 Eliminate n fom the foce equation to obtain: The fequency of the motion i the ecipocal of it peiod T. Expe the peed of the bead a a function of the adiu of it path it peiod: tan θ π T g Uing the diagam, elate to L θ : Linθ Subtitute fo in the expeion fo tanθ ole fo θ : Subtitute numeical alue ealuate θ : θ co gt 4π L θ co ( 9.8m/ )( 0.5) 4π ( 0.m) 5.6

70 334 Chapte 5 80 Pictue the Poblem Note that the acceleation of the bead ha two component, the adial component pependicula to, a tangential component due to fiction that i oppoite to. The application of Newton nd law will eult in a diffeential equation with epaable aiable. It integation will lead to an expeion fo the peed of the bead a a function of time. Apply ma to the bead in the adial tangential diection: Expe f in tem of µ n : Subtitute fo n f in the tangential equation to obtain the diffeential equation: n m d t f mat m dt f µ n d µ dt Sepaate the aiable to obtain: d µ dt Expe the integal of thi equation with the limit of integation being fom 0 to on the left-h ide fom 0 to t on the ight-h ide: Ealuate thee integal to obtain: 0 µ d' dt' ' 0 µ t 0 t Sole thi equation fo : 0 µ 0 + t

71 Application of Newton Law Pictue the Poblem Note that the acceleation of the bead ha two component the adial component pependicula to, a tangential component due to fiction that i oppoite to. The application of Newton nd law will eult in a diffeential equation with epaable aiable. It integation will lead to an expeion fo the peed of the bead a a function of time. (a) In Poblem 8 it wa hown that: 0 µ 0 + t Expe the centipetal acceleation of the bead: a c 0 µ + 0 t (b) Apply Newton nd law to the bead: n m d t f mat m dt Eliminate n f to ewite the adial foce equation ole fo a t : (c) Expe the eultant acceleation in tem of it adial tangential component: a t µ µ a a a a t c + a c + µ c ( µ a ) c + a c

72 336 Chapte 5 Concept of Centipetal oce *8 Pictue the Poblem The diagam depict a eat at it highet lowet point. Let t denote the top of the loop b the bottom of the loop. Applying Newton nd law to the eat at the top of the loop will etablih the alue of m /; thi can then be ued at the bottom of the loop to detemine n,b. Apply ma to the eat at the top of the loop: Apply ma to the eat at the bottom of the loop: mg + n,t mg ma m / n,b mg m / Sole fo n,b ubtitute fo m / to obtain: n,b 3mg (d) i coect. 83 Pictue the Poblem The peed of the olle coate i imbedded in the expeion fo it adial acceleation. The adial acceleation i detemined by the net adial foce acting on the paenge. We can ue Newton nd law to elate the net foce on the paenge to the peed of the olle coate. Apply adial maadial to the paenge: Sole fo : mg + 0.4mg m /. 4 g Subtitute numeical alue ealuate : ( )(.0m).4 9.8m/.8m/

73 Application of Newton Law Pictue the Poblem The foce the paenge exet on the amet of the ca doo i the adial foce equied to maintain the paenge peed aound the cue i elated to that peed though Newton nd law of motion. Apply x max to the foce acting on the paenge: m Sole thi equation fo : m Subtitute numeical alue ealuate : ( 80m)( 0 N) 70g 5.9 m/ (a) i coect. *85 Pictue the Poblem The foce acting on the bicycle ae hown in the foce diagam. The tatic fiction foce i the centipetal foce exeted by the uface on the bicycle that allow it to moe in a cicula path. n + f mae an angle θ with the etical diection. The application of Newton nd law will allow u to elate thi angle to the peed of the bicycle the coefficient of tatic fiction. (a) Apply ma to the bicycle: Relate n f to θ : m x f y n mg 0 m f tan θ mg n g

74 338 Chapte 5 Sole fo : Subtitute numeical alue ealuate : g tanθ ( 0m)( 9.8m/ ) 7.5m/ tan5 (b) Relate f to µ n : f f,max µ mg Sole fo µ ubtitute fo f to obtain: µ f mg g Subtitute numeical alue ealuate µ µ ( 7.5m/) ( 0m)( 9.8m/ ) Pictue the Poblem The diagam how the foce acting on the plane a it flie in a hoizontal cicle of adiu R. We can apply Newton nd law to the plane eliminate the lift foce in ode to obtain an expeion fo R a a function of θ. Apply ma to the plane: x lift in θ m R y coθ mg lift 0 Eliminate lift between thee equation to obtain: tan θ Rg Sole fo R: R g tanθ Subtitute numeical alue ealuate R: m h 480 h 3600 R tan40 ( 9.8m/ ).6 m

75 Application of Newton Law Pictue the Poblem Unde the condition decibed in the poblem tatement, the only foce acting on the ca ae the nomal foce exeted by the oad the gaitational foce exeted by the eath. The hoizontal component of the nomal foce i the centipetal foce. The application of Newton nd law will allow u to expe θ in tem of,, g. Apply ma to the ca: x y n n inθ m coθ mg 0 Eliminate n fom the foce equation to obtain: tan θ g Sole fo θ : θ tan g Subtitute numeical alue ealuate θ: [( 90 m/h)( h 3600)( 000 m/m) ] θ tan ( 60m)( 9.8m/ ).7 *88 Pictue the Poblem Both the nomal foce the tatic fiction foce contibute to the centipetal foce in the ituation decibed in thi poblem. We can apply Newton nd law to elate f n then ole thee equation imultaneouly to detemine each of thee quantitie.

76 340 Chapte 5 (a) Apply ma to the ca: x y n n inθ + coθ f f coθ m inθ mg 0 Multiply the x equation by inθ the y equation by coθ to obtain: f inθ coθ + n in θ m inθ co θ f inθ coθ mg coθ 0 n Add thee equation to eliminate f : n mg coθ m inθ Sole fo n : n mg coθ + m inθ m g coθ + inθ Subtitute numeical alue ealuate n : n ( ) ( ) ( 85m/h) ( 000 m/m) ( h/3600) 800g 9.8m/ co N 50m in0 (b) Sole the y equation fo f : f n coθ mg inθ Subtitute numeical alue ealuate f : f ( 8.5N) co0 ( 800g)( 9.8m/ ).59N in0 (c) Expe µ,min in tem of f n : µ,min f n Subtitute numeical alue ealuate µ,min : µ.59 N 8.5N, min 0.93

77 Application of Newton Law Pictue the Poblem Both the nomal foce the tatic fiction foce contibute to the centipetal foce in the ituation decibed in thi poblem. We can apply Newton nd law to elate f n then ole thee equation imultaneouly to detemine each of thee quantitie. (a) Apply ma to the ca: Multiply the x equation by inθ the y equation by coθ : x y n n inθ + f coθ m coθ f inθ mg 0 f inθ coθ + n in θ m inθ co θ f inθ coθ mg coθ 0 n Add thee equation to eliminate f : n mg coθ m inθ Sole fo n : n mg coθ + m inθ m g coθ + inθ Subtitute numeical alue ealuate n : n ( ) ( ) ( 38m/h) ( 000 m/m) ( h/3600) 800g 9.8m/ co N 50m in0 (b) Sole the y equation fo f : f n coθ mg inθ mg n cotθ inθ Subtitute numeical alue ealuate f :

78 34 Chapte 5 f ( 7.83 N) cot0 ( 800g)( 9.8m/ ) 777 N in0 The negatie ign tell u that f point upwad along the inclined plane athe than a hown in the foce diagam. *90 Pictue the Poblem The fee-body diagam to the left i fo the ca at et. The tatic fiction foce up the incline balance the downwad component of the ca weight peent it fom liding. In the fee-body diagam to the ight, the tatic fiction foce point in the oppoite diection a the tendency of the moing ca i to lide towad the outide of the cue. Apply ma to the ca that i at et: Subtitute f f,max µ n in equation () ole fo ealuate the maximum allowable alue of θ: Apply ma to the ca that i moing with peed : Subtitute f µ n in equation (3) (4) implify to obtain: Subtitute numeical alue into (5) y n coθ + f inθ mg 0 () x n inθ f coθ 0 () θ tan ( ) tan ( 0.08) 4.57 µ y n coθ f inθ mg 0 (3) x n inθ + f coθ m (4) ( coθ µ inθ ) mg n (5) n µ o m (6) ( c θ + inθ ) n mg

79 Application of Newton Law 343 (6) to obtain: Eliminate n ole fo: Subtitute numeical alue ealuate : 0.595n m 0.60g ( 60 m/h h/ m/m) 76m ( ) m/ 9 Pictue the Poblem The fee-body diagam to the left i fo the caounding the cue at the minimum (not liding down the incline) peed. The tatic fiction foce up the incline balance the downwad component of the ca weight peent it fom liding. In the fee-body diagam to the ight, the tatic fiction foce point in the oppoite diection a the tendency of the ca moing with the maximum afe peed i to lide towad the outide of the cue. Application of Newton nd law the imultaneou olution of the foce equation will yield min max. Apply ma to a ca taeling aound the cue when the coefficient of tatic fiction i zeo: Diide the fit of thee equation by the econd to obtain: min x n inθ m y n coθ mg 0 tan θ g o θ tan g Subtitute numeical alue ealuate the baning angle:

80 344 Chapte 5 ( 40 m/h) ( 000 m/m) ( h/3600 ) ( 30m)( 9.8m/ ) θ tan.8 Apply ma to the ca min x n inθ f co θ m taeling aound the cue at minimum peed: y n coθ + f inθ mg 0 Subtitute f f,max µ n in the foce equation implify to obtain: Ealuate thee equation fo θ.8 µ 0.3: Eliminate n between thee two equation ole fo min : Subtitute numeical alue ealuate min : Apply ma to the ca taeling aound the cue at maximum peed: Subtitute f f,max µ n in the foce equation implify to obtain: n ( µ co θ inθ ) m coθ + µ inθ n ( ) mg 0.0 n m.038 n mg min min 0. 06g min 0.06 min ( 30m)( 9.8m/ ) 5.59 m/ 0.m/h max x n inθ + f co θ m y n coθ f inθ mg 0 n co m ( µ θ + inθ ) ( coθ µ inθ ) mg n max Ealuate thee equation fo θ.8 µ 0.3: n m n mg max

81 Application of Newton Law 345 Eliminate n between thee two equation ole fo max : max g Subtitute numeical alue ealuate max : max ( 0.843)( 30m)( 9.8m/ ) 5.6 m/ 56.m/h Dag oce 9 Pictue the Poblem We can apply Newton nd law to the paticle to obtain it equation of motion. Applying teminal peed condition will yield an expeion fo b that we can ealuate uing the gien numeical alue. Apply y may to the paticle: mg b may When the paticle eache it teminal peed t a y 0: Sole fo b to obtain: Subtitute numeical alue ealuate b: mg b t 0 mg b t b 3 ( 0 g)( 9.8m/ ) m/ g/ 93 Pictue the Poblem We can apply Newton nd law to the Ping-Pong ball to obtain it equation of motion. Applying teminal peed condition will yield an expeion fo b that we can ealuate uing the gien numeical alue. Apply Pong ball: ma y y to the Ping- When the Ping-Pong ball eache it teminal peed t a y 0: Sole fo b to obtain: mg b ma y mg b t mg b t 0

82 346 Chapte 5 Subtitute numeical alue ealuate b: b 3 (.3 0 g)( 9.8m/ ).79 0 ( 9m/) 4 g/m *94 Pictue the Poblem Let the upwad diection be the poitie y diection apply Newton nd law to the y die. (a) Apply die: ma y y to the y d mg ma y o, becaue a y 0, d mg () Subtitute numeical alue ealuate d : (b) Subtitute d b t in equation () to obtain: ( 60g)( 9.8m/ ) 589 N d b t mg Sole fo b: mg b t d t Subtitute numeical alue b 589 N ealuate b: ( 5m/) 0.94 g/m 95 Pictue the Poblem The fee-body diagam how the foce acting on the ca a it decend the gade with it teminal elocity. The application of Newton nd law with a 0 d equal to the gien function will allow u to ole fo the teminal elocity of the ca. Apply x max to the ca: mg inθ d max o, becaue t a x 0, mg inθ d 0 Subtitute fo d to obtain: mg inθ 00 N (. N / m ) 0 t

83 Application of Newton Law 347 Sole fo t : t mg inθ 00 N. N / m Subtitute numeical alue ealuate t : t ( 800g)( 9.8m/ ). N 4.5m/ 88. m/h in 6 00 N / m 96 Pictue the Poblem Let the upwad diection be the poitie y diection apply Newton nd law to the paticle to obtain an equation fom which we can find the paticle teminal peed. (a) Apply ma pollution paticle: Sole fo t to obtain: y y to a mg 6 πη ma y o, becaue a y 0, mg 6πη t 0 t mg 6πη Expe the ma of a phee in tem of it olume: m ρ V 4π ρ 3 3 Subtitute fo m to obtain: t ρg 9η Subtitute numeical alue ealuate t : t 5 3 ( m) ( 000 g/m )( 9.8m/ ) 5 9(.8 0 N /m ) 0.4 cm/ (b) Ue ditance equal aeage peed time the fall time to find the time to fall 00 m at.4 cm/: t 4 0 cm cm/.5h *97 Pictue the Poblem The motion of the centifuge will caue the pollution paticle to migate to the end of the tet tube. We can apply Newton nd law Stoe law to deie an expeion fo the teminal peed of the edimentation paticle. We can then ue thi teminal peed to calculate the edimentation time. We ll ue the cm ditance

84 348 Chapte 5 fom the cente of the centifuge a the aeage adiu of the pollution paticle a they ettle in the tet tube. Let R epeent the adiu of a paticle the adiu of the paticle cicula path in the centifuge. Expe the edimentation time in tem of the edimentation peed t : t ediment x t Apply adial maadial to a pollution paticle: 6πη R ma t c Expe the ma of the paticle in tem of it adiu R denity ρ: m ρv 4 3 π R ρ 3 Expe the acceleation of the pollution paticle due to the motion of the centifuge in tem of thei obital adiu peiod T: a c π T 4π T 3 Subtitute fo m a c implify 3 4π 6π ρ R 4 6πη Rt to obtain: 3 π R ρ T 3T 3 Sole fo t : ind the peiod T of the motion fom the numbe of eolution the centifuge mae in econd: Subtitute numeical alue ealuate t : T t t 8π ρ R 9ηT min/e 800e / min min/e 60/min /e 8π ( 000 g/m )( 0.m)( 0 m) 5 3 ( 0 N /m )( 75 0 ).08m/ ind the time it tae the paticle to moe 8 cm a they ettle in the tet tube: t ediment x 38.5m 8cm 08cm/

85 Application of Newton Law 349 In Poblem 96 it wa hown that the ate of fall of the paticle in ai i.4 cm/. ind the time equied to fall 8 cm in ai unde the influence of gaity: t ai x 3.3 8cm.4cm/ ind the atio of the two time: t ai / t ediment 00 Eule Method 98 Pictue the Poblem The fee-body diagam how the foce acting on the baeball ometime afte it ha been thown downwad but befoe it ha eached it teminal peed. In ode to ue Eule method, we ll need to detemine how the acceleation of the ball aie with it peed. We can do thi by applying Newton nd law to the ball uing it teminal peed to expe the contant in the acceleation equation in tem of the ball teminal peed. We can then ue n+ n + an t to find the peed of the ball at any gien time. Apply Newton nd law to the ball to obtain: mg b d m dt Sole fo d/dt to obtain: d b g dt m When the ball eache it teminal peed: Subtitute to obtain: Expe the poition of the ball to obtain: Letting a n be the acceleation of the ball at time t n, expe it peed when t t n + : b 0 t d dt x b g g m m t n+ g t n+ x + n n+ n + an t whee n a n g t + n t

86 350 Chapte 5 t i an abitaily mall inteal of time. A peadheet olution i hown below. The fomula ued to calculate the quantitie in the column ae a follow: Cell omula/content Algebaic om A0 B9+$B$ t + t B0 B9+0.5*(C9+C0)*$B$ n+ + n xn+ xn + t C0 C9+D9*$B$ n+ n + a n t D0 $B$4*( C0^/$B$5^) n a n g t A B C D t 0.5 x0 0 m m/ 4 a0 9.8 m/^ 5 t 4.67 m/ 6 7 t x a 8 () (m) (m/) (m/^) om the table we can ee that the peed of the ball afte 0 i appoximately 4.4 m/. We can etimate the uncetainty in thi eult by haling t ecalculating the peed of the ball at t 0. Doing o yield (0 ) 4.3 m/, a diffeence of about 0.0%. The gaph how the elocity of the ball thown taight down a a function of time.

87 Application of Newton Law 35 Ball Thow n Staight Dow n (m/) t () Reet t to 0.5 et 0 0. Ninety-nine pecent of 4.67 m/ i appoximately 4.3 m/. Note that the ball will each thi peed in about 0.5 that the ditance it tael in thi time i about 3m. The following gaph how the ditance taeled by the ball dopped fom et a a function of time. Ball Dopped om Ret x (m) t () *99 Pictue the Poblem The fee-body diagam how the foce acting on the baeball afte it ha left you h. In ode to ue Eule method, we ll need to detemine how the acceleation of the ball aie with it peed. We can do thi by applying Newton nd law to the baeball. We can then ue n+ n + an t xn+ xn + n t to find the peed

88 35 Chapte 5 poition of the ball. Apply y may to the baeball: Sole fo d/dt: Unde teminal peed condition ): ( t Subtitute to obtain: Letting a n be the acceleation of the ball at time t n, expe it poition peed when t t n + : d b mg m dt whee fo the upwad pat of the flight of the ball fo the downwad pat of the flight. d dt g b m b 0 g + t m b g m t d dt g g g + t t ( + ) t yn+ yn + n n n+ n + an t whee + n n an g t t i an abitaily mall inteal of time. A peadheet olution i hown below. The fomula ued to calculate the quantitie in the column ae a follow: Cell omula/content Algebaic om D D0+$B$6 t + t E E E0 $B$4* (+E0*ABS(E0)/($B$5^))*$B$6 n+ n + an t 0 0 y 0 y y *(E0+E)*$B$6 ( ) t n+ n n n G0 0 y 0 G $E$0*D 0.5*$B$4*D^ 0t gt

89 Application of Newton Law 353 A B C D E G 4 g 9.8 m/^ 5 t 4.7 m/ 6 t t y y no dag om the table we can ee that, afte 3.5, the ball eache a height of about 60.4m. It eache it pea a little ealie at about 3.3, it height at t 3.3 i 60.6m. The ball hit the gound at about t going up. 7 o it pend a little longe coming down than The olid cue on the following gaph how y(t) when thee i no dag on the baeball the dotted cue how y(t) unde the condition modeled in thi poblem.

90 354 Chapte 5 y (m) x with dag 30 x with no dag t () 00 Pictue the Poblem The pictoial epeentation how the bloc in it initial poition againt the compeed ping, late a the ping acceleate it to the ight, finally when it ha eached it maximum peed at x f 0. In ode to ue Eule method, we ll need to detemine how the acceleation of the bloc aie with it poition. We can do thi by applying Newton nd law to the box. We can then ue n+ n + an t xn+ xn + n t to find the peed poition of the bloc. Apply x max Sole fo a n : 0.3m x n ma to the bloc: ( ) n a m m ( 0.3 ) n x n Expe the poition peed of the bloc when t t n + : xn+ xn + n t n+ n + an t whee an ( 0.3m x n ) m t i an abitaily mall inteal of time. A peadheet olution i hown below. The fomula ued to calculate the quantitie in the column ae a follow:

91 Application of Newton Law 355 Cell omula/content Algebaic om A0 A9+$B$ t + t B0 B9+C0*$B$ x + t C0 C9+D9*$B$ + a t D0 ($B$4/$B$5)*(0.3 B0) m n n n n ( 0.3 ) A B C D t x0 0 m m/ 4 50 N/m 5 m 0.8 g 6 7 t x a 8 () (m) (m/) (m/^) x n om the table we can ee that it too about 0.00 fo the ping to puh the bloc 30 cm that it wa taeling about.44m/ at that time. We can etimate the uncetainty in thi eult by haling t ecalculating the peed of the ball at t 0. Doing o yield (0.00 ).4 m/, a diffeence of about.%.

92 356 Chapte 5 Geneal Poblem 0 Pictue the Poblem The foce that act on the bloc a it lide down the incline ae hown on the fee-body diagam to the ight. The acceleation of the bloc can be detemined fom the ditance--time infomation gien in the poblem. The application of Newton nd law to the bloc will lead to an expeion fo the coefficient of inetic fiction a a function of the bloc acceleation the angle of the incline. Apply ma to the bloc: Σ x mginθ f ma Σ y n mg 0 Set f µ n, n between the two equation, ole fo µ : µ g inθ a g coθ Uing a contant-acceleation equation, elate the ditance the bloc lide to it liding time: ( t) 0 x 0 t + a whee 0 Sole fo a: Subtitute numeical alue ealuate a: ind µ fo a m/ θ 8 : a a µ x ( t) (.4m) ( 5.) 0.775m/ ( 9.8m/ ) in8 ( 9.8m/ ) m/ co8

93 Application of Newton Law Pictue the Poblem The fee-body diagam how the foce acting on the model aiplane. The peed of the plane can be calculated fom the data concening the adiu of it path the time it tae to mae one eolution. The application of Newton nd law will gie u the tenion in the ting. (a) Expe the peed of the aiplane in tem of the cicumfeence of the cicle in which it i flying it peiod: π T Subtitute numeical alue ealuate : π ( 5.7 m) m/ (b) Apply x max to the model aiplane: m Subtitute numeical alue ealuate : 0.7 m/ 5.7m ( 0.4g) ( ) 8.03N *03 Pictue the Poblem The fee-body diagam how the foce acting on the box. If the tudent i puhing with a foce of 00 N the box i on the ege of moing, the tatic fiction foce mut be at it maximum alue. In pat (b), the motion i impending up the incline; theefoe the diection of f,max i down the incline. (a) Apply ma to the box: f + mg inθ 0 x y n mg coθ 0

94 358 Chapte 5 Subtitute f f,max µ n, eliminate n between the two equation, ole fo µ : µ tanθ mg coθ Subtitute numeical alue µ tan30 ealuate µ : ( 800 N) N co30 (b) ind f,max fom the x-diection foce equation: Subtitute numeical alue ealuate f,max : f f, max,max mg inθ ( 800 N) 00 N in30 00 N If the bloc i on the ege of liding up the incline, f,max mut act down the incline. The x-diection foce equation become: f, max + mg inθ 0 Sole the x-diection foce equation fo : Subtitute numeical alue ealuate : mg inθ + f,max ( 800 N) in N 600 N 04 Pictue the Poblem The path of the paticle i a cicle if i a contant. Once we hae hown that it i, we can calculate it alue fom it component. The diection of the paticle motion can be detemined by examining two poition of the paticle at time that ae cloe to each othe. (a) (b) Expe the magnitude of in tem of it component: + x y Ealuate with x 0 m co ωt y 0 m inωt: [( 0m) coωt] + [( 0m) inωt] 0.0m ( ωt + in ωt) 00 co m

95 (c) Ealuate x y at t 0 : x ( 0m) co0 ( 0m) in 0 0 y Application of Newton Law 359 0m Ealuate x y at t t, whee t i mall: ( 0m) coω t ( 0m) x co0 0m y ( 0m) inω t y whee y i poitie the motion i clocwie (d) Diffeentiate with epect to time to obtain : d / dt [ m]j ˆ [ (0ω inωt) m ] iˆ + ( 0ω coωt ) Ue the component of to find it peed: x + y [( 0ω inωt) m] + [( 0ω coωt) m] ( 0m) ω ( 0m)( ) 0.0 m/ (e) Relate the peiod of the paticle motion to the adiu of it path it peed: T π ( 0m) π 0m/ π 05 Pictue the Poblem The fee-body diagam how the foce acting on the cate of boo. The inetic fiction foce oppoe the motion of the cate up the incline. Becaue the cate i moing at contant peed in a taight line, it acceleation i zeo. We can detemine by applying Newton nd law to the cate, ubtituting fo f, eliminating the nomal foce, oling fo the equied foce. Apply ma to the cate, with both a x a y equal to zeo, to the cate: x coθ f mg inθ 0 y n inθ mg coθ 0

96 360 Chapte 5 Subtitute µ n fo f eliminate n to obtain: ( inθ + µ coθ ) mg coθ µ inθ Subtitute numeical alue ealuate : ( 00g)( 9.8m/ ) in30 + ( 0.5) co30 ( co30 ) ( 0.5) in30.49 N 06 Pictue the Poblem The fee-body diagam how the foce acting on the object a it lide down the inclined plane. We can calculate it peed at the bottom of the incline fom it acceleation diplacement find it acceleation fom Newton nd law. Uing a contant-acceleation equation, elate the initial final elocitie of the object to it acceleation diplacement: ole fo the final elocity: Apply ma to the liding object: Sole the y equation fo n uing f µ n, eliminate both n f fom the x equation ole fo a: Subtitute equation () in equation () ole fo : + a x 0 Becaue 0 0, a x () x f + mg inθ ma y n mg coθ 0 ( inθ µ coθ ) a g () g( inθ µ co ) x θ Subtitute numeical alue ealuate : ( ) in 30 ( 0.35) ( co30 )( 7m) 6.7 m/ 9.8m/ (d) i coect.

97 Application of Newton Law 36 *07 Pictue the Poblem The fee-body diagam how the foce acting on the bic a it lide down the inclined plane. We ll apply Newton nd law to the bic when it i liding down the incline with contant peed to deie an expeion fo µ in tem of θ 0. We ll apply Newton nd law a econd time fo θ θ ole the equation imultaneouly to obtain an expeion fo a a a function of θ 0 θ. Apply m a to the bic when it i liding with contant peed: Sole the y equation fo n uing f µ n, eliminate both n f fom the x equation ole fo µ : Apply ma to the bic when θ θ : Sole the y equation fo n, ue f µ n to eliminate both n f fom the x equation, ue the expeion fo µ obtained aboe to obtain: x f + mg inθ 0 0 y n mg coθ 0 0 µ tanθ 0 x f + mg inθ ma y n mg coθ 0 a g ( inθ tanθ θ ) 0 co 08 Pictue the Poblem The fact that the object i in tatic equilibium unde the influence of the thee foce mean that Dawing the coeponding foce tiangle will allow u to elate the foce to the angle between them though the law of ine the law of coine.

98 36 Chapte 5 (a) Uing the fact that the object i in tatic equilibium, edaw the foce diagam connecting the foce head-to-tail: Apply the law of ine to the tiangle: in( π θ ) in( π θ ) in( π θ ) Ue the tigonometic identity in(π α) inα to obtain: inθ 3 inθ 3 3 inθ (b) Apply the law of coine to the tiangle: Ue the tigonometic identity co(π α) coα to obtain: co ( π ) θ + θ co Pictue the Poblem We can calculate the acceleation of the paenge fom hi/he peed that, in tun, i a function of the peiod of the motion. To detemine the longet peiod of the motion, we focu ou attention on the ituation at the ey top of the ide when the eat belt i exeting no foce on the ide. We can ue Newton nd law to elate the peiod of the motion to the acceleation peed of the ide. (a) Becaue the motion i at contant peed, the acceleation i entiely adial i gien by: Expe the peed of the motion of the ide a a function of the adiu of the cicle the peiod of it motion: a c π T

99 Application of Newton Law 363 Subtitute in the expeion fo a c to 4π ac obtain: T Subtitute numeical alue ealuate a c : a c 4 π ( 5m) ( ) 49.3m/ (b) Apply ma to the paenge when he/he i at the top of the cicula path ole fo a c : mg ma a c g c Relate the acceleation of the motion to it adiu peed ole fo : g g Expe the peiod of the motion a a function of the adiu of the cicle the peed of the paenge ole fo T m : T π m π g Subtitute numeical alue ealuate T m : T 5m 9.8m/ m π 4.49 Rema: The ide i weightle unde the condition decibed in pat (b). *0 Pictue the Poblem The pictoial epeentation to the ight how the cat it load on the inclined plane. The load will not lip poided it maximum acceleation i not exceeded. We can find that maximum acceleation by applying Newton nd law to the load. We can then apply Newton nd law to the cat-pluload ytem to detemine the tenion in the ope when the ytem i expeiencing it maximum acceleation.