CHAPTER 5 CIRCULAR MOTION

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1 CHAPTER 5 CIRCULAR MOTION and GRAVITATION

2 5.1 CENTRIPETAL FORCE It is known 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 this acclation is towad th cnt of th cicl and gin by. Along th adius a Fom Nwton s nd Law this acclation should b du to foc calld th cntiptal Foc. Cntiptal foc F ma m 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. Rmak Th cntiptal fing to th diction of th foc and not to a nw kind of focs. It is lik hoizontal o tical.

Exampl 5.1 Aflat (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 th ca will ha in od to ound th cu succssfully?

3 Exampl 5.1 Aflat (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 th ca will ha in od to ound th cu succssfully? 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 paalll to th oad. Th static fictional foc is th only foc that acts along th adius of th cu µ s N f s m µ s m N sinc th is no motion in th tical diction N mg µ s ( mg ) m µ sg 14 m/s 50.4 km/h f s

Ncosθ Nsinθ 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 oad s banking?

4 Ncosθ Nsinθ 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 oad s banking? Solution Not that th ca will mo aound a hoizontal cicl Th nomal foc N should b sold into two componnts: on towad th cnt of th cu (hoizontal), and th oth tical. Th cntiptal foc thn is th hoizontal componnt N sinθ m

5 sinc th is no motion in th tical diction mg N cosθ mg tanθ m N θ mg cosθ 1 tan 19.5 gr o

6 Exampl 5.3 A ball of mass 1 kg is attachd to on nd of asting 1mlong and is whild in ahoizontal 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. 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 To find th spd at th g of baking, w ha to substitut fo T by its baking alu, i.., TmaxR max.4 m/s m 1

7 5. NONUNIFORM CIRCULAR MOTION Whn th magnitud of th locity is not constant but chang with tim w ha th nonunifom cicula motion. 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 a a ˆ + a θ qˆ Unit cto along th adius Unit cto along th tangnt Th adial componnt,, is th cntiptal acclation dfind piously, and th tangntial componnt,, is th nw contibution du to th chang in th magnitud of th paticl's locity, so w will xpct

8 a θ d 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 ma F θ ma θ Th positi snss of and q will b chosn towad th cnt, and countclockwis spctily.

9 Exampl 5.4 A small body of mass m swings in atical cicl at th nd of a cod of lngth L as shown. If th spd T of th body whn th cod maks an angl with thtical is, find a) th adial and th tangntial componnts of th acclation at this mg point, b) th tnsion in th cod at th sam point. Solution Th f body diagam of th block is shown. mgsinθ mg has to b sold into a adial and a tangntial componnt Fo th adial componnt of th acclation w ha a To find th tangntial componnt of th acclation w us L θ

10 F θ ma θ a θ g sinθ mg sinθ maθ b) To find th tnsion w us F ma mgsinθ T mg cosθ m L T m + g cosθ L Not that at th bottom point θ0 th tnsion is maximum with T m + g L Whil at th top point θπ th tnsion is minimum with T m g L What happn if at th top point Lg T θ

12 N N 6 m mg 9 m mg Exampl 5.5 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 Th f body diagam of th hicl is shown at th two positions.

13 a) At point A, N is towad th cnt, whil mg is away fom th cnt. Applying th q. F ma N mg m 3 N m + g.3 10 N b) At point B mg is towad th cnt, whil N is away fom th cnt. Applying again th q. F ma mg N m Fo th hicl to b on tack, th nomal foc must ha a positi alu, that is, N > 0 mg N N mg N m g > 0 < g 9.39m

14 5.3 NEWTON S LAW OF GRAVITATION Ey 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 G m m 1 1 ˆ ˆ is a unit cto dictd fom m 1 to m and G is calld th gaitational constant with a alu G N.m /kg. 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

15 F G M m R wh M and R a th ath s mass and th ath s adius, spctily. At th cnt of th ath th gaitational foc on th body would b zo, why? Fom Nwton s scond law, and assuming a body of mass m at th sufac of th ath, w ha F G M m R mg g G M R R g M G If th body is at a distanc h abo th ath s sufac is thn M m F G M mg g G R + h R h ( ) ( ) + Thfo, g' dcas with incasing altitud. 4 kg

16 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. Solution Using F F G m m 1 ( ) ( 60 )( ) ( ) Exampl 5.7 Th bodis of mass kg, 4 kg, and 6 kg a aangd as shown. Calculat th total foc acting on th -kg mass by th oth two masss. m 3 6 kg 1 m F 13 F 1 m m 1 kg m1 m ( 11) G i i i N ( ) 1 Solution m 1 is actd upon by two focs: F F m m 6 N ( 11) j N () G j j 13 1 m 4 kg

17 Th total foc acting on m 1 is: F 1 F + F ( 1.33i+ 8.0 ) 10 N j Exampl 5.8 Calculat th magnitud of th acclation du to gaity at an altitud of 100 km. Solution Using th quation g g G M ( R + h) ( 11)( ) ( 6 5 ) m/s

18 5.4 SATELLITE MOTION Satllit is an objct obiting aound th ath. In satllit motion th gaitational foc btwn th satllit and th ath is th cntiptal foc. Now applying mm F m G m wh m is th mass of th satllit and is th adius of th satllit obit. Soling fo w gt GM Th piod of olution is τ π 3 τ π GM 3/ π GM

19 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 Using th quation ( GM )( ) m/s And fo th piod w us 3/ 11 3/ π π ( ) τ τ 1.55h GM

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