CHAPTER 5 CIRCULAR MOTION AND GRAVITATION

84 CHAPTER 5 CIRCULAR MOTION AND GRAVITATION

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 85 In th pious chapt w discussd Nwton's laws of motion and its application in simpl dynamics poblms. In this chapt w continu ou study of dynamics and th applications of Nwton's laws, spcially on cicula motion. Nwton's law of gaitation is also addssd which is th cntal law in plant and satllit motion. 5.1 CENTRIPETAL FORCE In sction 3.3 w ha showd that, if a paticl mos with constant spd in a cicula path of adius, it acquis a cntiptal acclation du to th chang in th diction of th paticl's locity. Th diction of th cntiptal acclation is always towad th cnt of th path and its magnitud is gin by a =. (5.1) Th subscipt fing to th adial componnt of th acclation. Accoding to Nwton's scond law this acclation should b a sult of an applid foc acting on th paticl towad th cnt of th path, and should ha a magnitud of F = ma m. (5.) = Bcaus of its diction such a foc is calld th cntiptal foc. Both th cntiptal acclation and th cntiptal foc a cto quantitis whos magnituds a constant but whos dictions a always changing so as to point towad th cnt of th cicula path. It should b notd that any foc in natu can b tatd as a cntiptal foc if it acts on a paticl in a diction towad th cnt of th cicula path followd by th paticl. Th fictional

86 foc is th cntiptal foc whn a ca ounding a cu, and th tnsion is th cntiptal foc whn you whil a ball, tid to a sting, in a hoizontal cicl. Rmak: Th wod cntiptal fing to a spcific diction of th foc and not to a nw kind of focs. It is lik hoizontal o tical. Exampl 5.1 A flat (unbankd) cu on a highway has a adius of 100 m. If th cofficint of static-fiction btwn th tis and th oad is 0., what is th maximum spd with which th ca will ha in od to ound th cu succssfully?. f s R Figu 5.1 Exampl 5.1. Solution H th a th focs acting on th ca: Th wight and th nomal foc act ppndicula to th plan of motion, and th static fictional foc which must b paalll to th oad. Hnc th cntiptal foc on th ca is th foc of static fiction, so w ha f = µ sn m, R s = but, sinc th is no motion in th tical diction w can wit N= mg. Soling th two quations fo w gt = µ s gr = 50.4km/h.

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 87 Exampl 5. A cicula cu of a oad is dsignd fo taffic moing at 60 km/h without dpnding on th fiction. If th adius of th cu is 80 m, what is th coct angl of banking of th oad. N N cosθ N sinθ (a) θ (b) mg Figu 5. Exampl 5., with th f body diagam. Solution In th bankd oads, th nomal foc N should b sold into two componnts: on towad th cnt of th cu (hoizontal), and th oth tical as shown in Figu 5.(b). Th cntiptal foc will b thn th componnt N sinθ, i.., N sin θ = m, R and, sinc th is no motion in th tical diction w ha N cos θ = mg. Substituting fo N fom th scond quation into th fist quation, w ha

88 mg tan θ = m R o θ gr 1 o = tan =19.5 Exampl 5.3 A ball of mass 1 kg is attachd to on nd of a sting 1 m long and is whild in a hoizontal cicl, as shown in Figu 5.3. Find th maximum spd th ball can attain without baking th sting. Th baking stngth of th sting is 500 N. 1 m 1 kg Figu 5.3 Exampl 5.3. Solution Th only two focs acting on th ball a th wight and th tnsion. Sinc th wight is nomal to th plan of th cicl, th cntiptal foc in this cas is th tnsion, so w can wit T = m. R To find th spd at th g of baking, w ha to substitut fo T by its baking alu, i.., T 500 1 1 max max = = = m R.4 m/s 5. NONUNIFORM CIRCULAR MOTION

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 89 In th pious sction w ha considd th cicula motion with constant spd (unifom cicula motion). Whn th magnitud of th locity is not constant but ath chang with tim w ha th nonunifom cicula motion. Now what will happn if th locity changs both in magnitud and in diction. Th chang in th spd will add anoth contibution to th acclation. Rsoling th acclation cto into two ppndicula componnts: adial componnt and tangntial componnt, w can wit a = a ˆ + a qˆ, (5.3) θ wh ˆ is a unit cto dictd along th adius of th cicula path, and q is anoth unit cto tangnt to th path. th adial componnt, a, is th cntiptal acclation dfind piously, and th tangntial componnt, a θ, is th nw contibution du to th chang in th magnitud of th paticl's locity, so w will xpct d aθ =. (5.4) dt Rmak: In applying Nwton's scond law fo th cicula motion, th coodinat axs will b th adius-axis and th tangnt-axis, so all th applid focs ha to b sold accodingly. Th law now ads F =, and F θ = maθ. (5.5) ma Th positi snss of ˆ and q will b chosn towad th cnt, and countclockwis spctily.

90 Exampl 5.4 A small body of mass m swings in a tical cicl at th nd of a cod of lngth L as shown in Figu 5.4. If th spd of th body whn th cod maks an angl θ with th tical is, find a) th adial and th tangntial componnts of th acclation at this point, b) th tnsion in th cod at th sam point. L θ T mg cosθ mg sinθ mg Figu 5.4 Exampl 5.4. Solution Th wight ha to b sold as shown in th f-body diagam of th systm. As it is cla fom th diagam, th adial componnt is a) a =, R L = and th tangntial componnt is Fθ a θ = = g sinθ. m b) Sinc F = ma, w ha o T mg cos θ = m L T = m( g cosθ + ) L

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 91 B 6 m 9 m A N N mg mg Figu 5.5 Exampl 5.5. Exampl 5.5 A hicl of mass 350 kg mos on a oll-coast as shown in Figu 5.5. a) If th spd of th hicl at point A is 18 m/s, what is th nomal foc th tack xts on th hicl? b) What is th maximum spd fo th hicl to main on tack at point B? Solution a) Examining th f-body diagam of th hicl at point A w s that N is towad th cnt, whil mg is away fom th cnt. Applying th quation

9 W obtain F = ma = m N mg = m, o N = m + g =.3 10 b) Fo th hicl to b on tack, th nomal foc must ha a positi alu, that is, N > 0. Now fom th f body diagam of th hicl at point b w wit 3 N mg N = m, o N = m g This lads to > 0 < g So w gt max = g = 9.39 m/s

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 93 5.3 NEWTON S LAW OF GRAVITATION Th law stats that y paticl in th unis attacts y oth paticl with a foc that is dictly popotional to th poduct of thi masss and insly popotional to th squa of th distanc btwn thm. Thus th gaitational foc xtd on a paticl of mass m 1 by a paticl of mass m is F 1 m1m = G ˆ, (5.6) 1 wh 1 is th distanc btwn th two paticls and ˆ is a unit cto dictd fom m 1 to m. Th unisal constant G is calld th gaitational constant with a alu, in SI units of G = 6.67 10-11 N.m /kg. It can b shown that th foc xtd by any homognous sph is th sam as if th nti mass of th sph is concntatd at its cnt. Thfo, th foc xtd by th ath on a small body of mass m, a distanc fom its cnt, is M m F = G, >R (5.7) wh M and R a th ath s mass and th ath s adius, spctily. This foc is dictd towad th cnt of th ath. Insid th ath, th foc would dcas as appoaching th 1 cnt ath than incasing as. At th cnt of th ath th gaitational foc on th body would b zo, why? Fo fly falling body th only foc acting is th gaitational foc of th ath and th acclation poducd is th

94 acclation du to gaity, g. Now, fom Nwton s scond law, and assuming th body to b at th sufac of th ath, w ha o M m F = G = mg, (5.8) R M g = G. (5.9) R Th mass of th ath can b calculatd using Equation (5.9) as M Rg 4 = = 596. 10 kg, G with R =6370 km Th foc acting on a paticl at a distanc h abo th ath s sufac is, fom Equation (5.6) and Equation (5.8) o M m F G = mg, (5.10) g = ( R + h) = G. (5.11) ( R + h) M Thfo, g' dcas with incasing altitud. Exampl 5.6 Two bodis of mass 60 kg, and 80 kg a placd m apat. Calculat th gaitational foc xtd by on body on th oth.

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 95 Solution Fom Equation (5.5) w ha F m m = G 1 = 8 = 8 10 N. ( ) ( )( ) 11 60 80 6.67 10 ( ) Exampl 5.7 Th bodis of mass kg, 4 kg, and 6 kg a aangd as shown in Figu 5.6. Calculat th total foc acting on th -kg mass by th oth two masss. Solution Th foc xtd on th -kg mass by th 4-kg mass is 11 4 ( 6.67 10 ) i ( ) F mm4 4 = G i = 4 = 133. 10 10 i N, and th foc xtd by th 6-kg is 11 6 ( 6.67 10 ) j () F mm6 6 = G j= 1 6 = 8.0 10 10 jn. 6 kg 1 m Thfo, th total foc acting on th -kg mass du to th 4-kg and th 6-kg masss is th cto sum of F 4 and F 6 : y kg x m 4 kg Figu 5.6 (Exampl 5.7). F = F 4 + F 6 10 ( 1.33i+ 8.0 ) 10 N = j

96 Exampl 5.8 Calculat th magnitud of th acclation du to gaity at an altitud of 100 km. Solution Fom Equation (5.10) w ha g = G = ( R + h) M 11 4 ( 6.67 10 )( 5.98 10 ) 6 5 ( 6.37 10 + 1 10 ) = 9.5 m/s. This mans that g' is dcasd by a 3%. 5.4 SATELLITE MOTION Fom sction 3. w show that if on launch a pojctil fom th sufac of th ath with a ath small locity, th tajctoy will b a paabola poidd that th ai sistanc is nglctd. By incasing th locity of pojction w can incas th siz of th tajctoy, and abo a ctain citical alu of th locity, th tajctoy will miss th ath and th pojctil has bcom an ath satllit. In this cas th foc acting on th pojctil is no long 1 constant but ais insly as, with is th adius of th satllit's obit. It tuns out that und such an attacti foc, th pojctil s path may b a cicl, an llips, a paabola, o a hypbola. Th cicula path will b considd fo simplicity. W ha land that a paticl in a unifom cicula motion has a cntiptal acclation gin by a =, with is its spd

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 97 and is th adius of th cicula path. In satllit motion th gaitational foc (Equation 5.7) is th foc that poids such acclation, that is M m G = m, wh m is th mass of th satllit and is th adius of th satllit obit. Soling fo w gt GM =. (5.1) Th piod of olution is τ = π By substituting fo fom Equation 5.1, w gt 3 3 / π τ = π = (5.13) GM GM It should b cla that th pious considations a also applicabl to th motion of ou moon aound th ath and th motion of th plants aound th sun. Exampl 5.9 If on want to plac a communication satllit into a cicula obit of adius 6800 km. What must b its spd, and its piod? Solution Fom Equation (5.11) w obtain

98 = GM 11 4 ( 6.67 10 )( 5.98 10 ) = 6.8 10 =7.66 10 3 m/s. 6 Th piod is, fom Equation (5.13), τ = π 3/ GM = π 6.67 10 6 ( 6.8 10 ) 11 3/ 5.98 10 4 =1.55 h. 5.6 KEPLER S LAWS Th fact that th plants mo about th sun in such a way that th aal locitis a constant was found by Johanns Kpl in 1609. Kpl studid th data of his tach Tycho Bah and ntually fomulatd th following th laws applid to th sola systm. 1. Th law of obits: Each plant mos in an llips with th sun as a focus.. Th law of aas: Th adius cto dawn fom th sun to any plant swps out qual aas in qual tims. 3. Th law of piods: Th squa of th piod of olution of any plant is popotional to th cub of th majo axis of th obit.

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 99 PROBLEMS 5.1 A -kg mass mos in a cicl with a spd of 4 m/s. If th adius of th cicl is 0.5 m, what is th cntiptal foc acting on th mass? 5. A -kg mass is attachd to a light sting otats in cicula motion on a hoizontal, fictionlss tabl. Th adius of th cicl is 1.0 m, and th sting can suppot a maximum foc of 40 N. What is th maximum spd th mass can ha bfo th sting baks? 5.3 A ca ounds an unbankd cu with a adius of 50 m. If th cofficint of static fiction btwn th tis and th oad is 0.6, what is th maximum spd th ca can ha in od not to slid duing th ounding. 5.4 A 00-g mass on a fictionlss tabl is attachd to a hanging block of mass 00 g 0.6 m 800 g by a cod though a hol in th tabl as in Figu 5.7. Th suspndd block mains in quilibium whil th 800 g mass ols on th sufac of th tabl in a cicl of adius 0.6 m. a) What is th tnsion in th cod? Figu 5.7 Poblm 5.4. b) What is th spd of th mass? c) What is th cntiptal foc acting on th mass? 5.5 A ca moing at 50 km/h want to tun a 15 o -bankd cu with adius of 40 m on a ainy day (fiction is nglctd). Would th ca mak th tun succssfully? If not with what spd must it mo?

100 5.6 A small mass is placd 0.5 m fom th cnt of a otating, hoizontal tabl that otats with a constant spd of 1.5 m/s. Th mass is in th g of sliiping with spct to th tuntabl. What is th cofficint of static fiction btwn th mass and th tabl? 5.7 A ca tals o a hill, which can b gadd as an ac of a cicl of adius 150 m, as in Figu 5.8. What is th maximum spd th ca can ha without laing th oad at th top of th hill? 150 m Figu 5.8 Poblm 5.7. 5.8 A coin is placd insid an opn baskt, which in tun allowd to otat in a tical cicl of adius 1. m. What is th minimum spd of th baskt at th top of th cicl if th coin not to fall off? 5.9 A 30-kg child sits in a conntional swing of lngth.5 m. Th tnsion in ach chain that suppot th sat of th swing at th lowst point is 300 N, and th mass of th sat is 4 kg. a) What is th child s spd at th lowst point? b) What is th foc acting on th child by th sat? 5.10 A 0.-kg pndulum bob passs though th lowst point with a spd of 6 m/s. What is th tnsion in th cod of th pndulum if it is 1 m long? 5.11 A small ball of mass m is suspndd fom a sting of lngth L that maks an angl θ with th tical. Th ball ols in a hoizontal cicl with constant spd (conical pndulum), as in Figu 5.9. Find th spd of th ball.

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 101 L θ Figu 5.9 Poblm 5.11. 5.1 An objct tid to th nd of a sting is whild in a tical cicl of adius R. What is th minimum spd blow which th sting would bcom loos at th highst point? 5.13 A Fis whl ا رجوحة الدولاب) ) with adius 0 m otats at 8 m/s. Find th appant wight of a 40-kg boy at a) th top of th Fis whl, b) th bottom of whl. p 0 m Figu 5.10 Poblm 5.14. 5.14 A skat mos on an igula tack as shown in Figu 5.10. Point p in th tack is at th top of an ac of a cicl of adius 0m. What is th maximum spd of th skat at point p to main in th tack?

10 5.15 A pson nts a Roto of adius 3 m, as shown in Figu 5.11. If th cofficint of static fiction btwn th pson and th wall is 0.4, what is th minimum spd with which th Roto must otat such that th pson is saf fom falling? 5.16 A small block of mass m is placd insid a con that is otating about its axis, as in Figu 5.1. If th insid wall of th con is fictionlss, what is th spd of th con to kp th mass fom sliding down? 3 m Figu 5.11 Poblm 5.15. m θ h 5.17 A 3-kg mass is connctd to a tical od by mans of two Figu 5.1 Poblm 5.16. masslss stings, as in Figu 5.13. Th stings a taut, and fom two sids of quilatal tiangl of lngth m. Th od otats about its axis and th mass m otats in a hoizontal plan. If th tnsion in th upp sting is m 3 kg 10 N, a) daw th f-body diagam of m th mass, b) calculat th tnsion in th low sting, c) calculat th spd of th mass. Figu 5.13 Poblm 5.17. 5.18 A distanc of 0.5 m. spaats two paticls of mass 00 kg, and 500 kg What is th gaitational foc 1 m 0 kg Figu 5.14Poblm 5.19.

CHAPTER 5 CIRCULAR MOTION AND GRAVITATION 103 xtd by on paticl on th oth? 5.19 Fou idntical balls ach of mass 0 kg a locatd at th cons of a squa of sid 1 m, as in Figu 5.13. Calculat th total foc acting on on ball fom th oth th balls. 5.0 What would b th wight of a 90-kg man at th top of a hill of hight 500 km? 5.1 A satllit of mass 400 kg is in a cicula obit of adius 6.1 10 6 m about th ath. Calculat a) Th piod of its olution, b) Th gaitational foc acting on it.