Chapter. s r. check whether your calculator is in all other parts of the body. When a rigid body rotates through a given angle, all

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conveted to adians. Also, be sue to vanced to a new position (Fig. 7.2b). In this inteval, the line OP has moved check whethe you calculato is in all othe pats of the body.

1 conveted to adians. Also, be sue to vanced to a new position (Fig. 7.2b). In this inteval, the line OP has moved check whethe you calculato is in all othe pats of the body. When a igid body otates though a given angle, all though the angle with espect to the efeence line. The angle, measued in 92 Chapte degee o 7adian Rotational mode when Motion solvingand the pats the body otate though the same angle at the same time. Fo the compact poblems involving otation. adians, Law of is Gavity called the angula position and is analogous to the linea position vaiable x. the Likewise, axis of Potation has moved is at an the ac cente length of s measued the disc, O. along A point the cicumfeence P on the disc of is at a disk, distance fom the oigin and moves about O in a cicle of adius. We set up a O P the cicle. igue 7.4 An acceleating bicycle Refeence Rotational Motion and fixed In efeence Figue the7.3, line, as a point as shown on the in otating Figue disc 7.2a, moves and fom assume to that in at a time t t, it 0 the heel otates with (a) angula speed line point stats at P an is on tangle i that i efeence and ends line. Afte a time inteval t has elapsed, P has advanced to a new position (Fig. 7.2b). In this inteval, the line OP has moved Law of Gavity Chapte at an angle f The diffeence f t f i is called the i at time t i and (b) angula speed f t time t angula displacement. f. (a) though the angle with espect to the efeence line. The angle, measued in adians, is called the angula position and is analogous to the linea position vaiable x. Likewise, P has moved an ac length s measued along the cicumfeence of 7.1 ANGULAR SPEED AND ANGULAR ACCELERATION P An object s angula displacement,, is the diffeence in its final and initial O s P the angles: cicle. u Refeence line In Figue 7.3, as a point on the otating disc moves fom to in a time t, it O Refeence f i [7.2] stats at an angle i and ends at an angle f. The diffeence f i is called the line y (a) angula SI unit: displacement. adian (ad) vi vf Figue 7.2 (a) The point P on a otating compact P disc at t 0. (b) s = As the disc otates, sp moves though an ac length s. u x O (b) (b) u s Fo example, if a point on a disk is at An object s angula displacement,, i 4 ad and otates to angula position is the diffeence in its final and initial f 7 ad, the angula displacement is f i 7 ad 4 ad 3 ad. Note angles: that we use angula vaiables to descibe the otating disc because each point on the disc undegoes the same angula displacement f in any given time inteval. Refeence i [7.2] line Having defined angula displacements, it s natual to define an angula speed: SI unit: adian (ad) u = 1 ad 53.1 (a) (b) Aveage angula speed The aveage angula speed av of a otating igid object duing the time Fo inteval example, t is if defined a point as on the a angula disk is displacement at i 4 ad and divided otates by t: to angula position f 7 ad, the angula displacement is f i 7 ad 4 ad 3 ad. f i Note that we use angula vaiables av to descibe the otating disc because [7.3] each point (Fig. 7.4b). Just as a changing speed t on the disc undegoes the same angula f leads t i to tthe concept of an acceleation, a displacement in any given time inteval. changing SI Having angula unit: adian defined speed pe angula second leads to (ad/s) displacements, the concept it s of an natual angula to define acceleation. an angula speed: Figue 7.2 (a) The point P on a otating compact disc at t 0. (b) As the disc otates, P moves though an ac length s. Aveage angula acceleation Aveage angula speed An object s The aveage angula speed acceleation av of a otating av duing igid the object time duing inteval the ttime is defined as inteval the change t is defined its as angula the angula speed displacement divided by divided t: by t: i av f i av t f t i t t f t i t SI unit: adian pe second (ad/s) SI unit: adian pe second squaed (ad/s 2 ) f [7.3] [7.5] As with angula velocity, positive angula acceleations ae in the counteclockwise diection, negative angula acceleations in the clockwise diection. If the angula speed goes fom 15 ad/s to 9.0 ad/s in 3.0 s, then the aveage angula acceleation duing that time inteval is

2 t f t i t with that of the aveage linea speed, When a igid object otates about a fixed axis, as does the bicycle wheel, evey potion of the object has the same angula speed and the same angula acceleation. t t In these equations, takes the place i of v and takes the f place of x, so the equations diffe only in the names of the vaiables. In the same way, evey linea quantity we have encounteed so fa has a coesponding twin in otational motion. vi vf The pocedue used in Section 2.5 to develop the kinematic equations fo linea motion unde constant acceleation can be used to deive a simila set of equations fo otational motion unde constant angula acceleation. The esulting equations of otational kinematics, along with the coesponding equations fo linea motion, ae as follows: 7.2 ROTATIONAL MOTION UNDER CONSTANT ANGULAR ACCELERATION Linea Motion with a Constant (Vaiables: x and v) Rotational Motion about a Fixed Axis with Constant (Vaiables: and ) v v i at i t [7.7] 1 2 v av x f x i t f t i x t x v i t at 2 i t t 2 [7.8] v 2 v i 2 2a x 2 i 2 2 [7.9] Notice that evey tem in a given linea equation has a coesponding tem in the analogous otational equation. 1 2 Quick Quiz 7.3

3 Constant Speed

4 Changing Speed

7 7.3 RELATIONS BETWEEN ANGULAR AND LINEAR QUANTITIES tangential speed v t The tangential speed of a point on a otating object equals the distance of that point fom the axis of otation multiplied by the angula speed. evey point on the otating object has the same angula speed. a t The tangential acceleation of a point on a otating object equals the distance of that point fom the axis of otation multiplied by the angula acceleation.

9 of mass m that is tied to a sting of length and is t speed in a hoizontal cicula path, as illustated in F a c v2 its inetia, the tendency of the ball is to move in a st sting pevents motion along a staight line by exetin all a tension foce that a c 2 2 makes it follow the cicula 2 cted along the sting towad the cente of the cicle, as sho 7.4 CENTRIPETAL ACCELERATION suppoted by a fictionless table. Why does the ball mov ] 1/T, so the units o a a 2 t a 2 ying Newton s second law along the c adial diection yiel he net centipetal foce F Foces Causing Centipetal Acceleation c the sum of the adial compo v on a An given object can have object with a centipetal acceleation the only if centipetal some extenal acceleation: foce acts on it. F c ma c m v2 T m igue 7.11 A ball attached to a

5 of NEWTONIAN the Eath, GRAVITATION the expession F mg is valid. As shown in howeve, the fee-fall acceleation g vaies consideably with altitude ath. uiz 7.

12 eman mathematician and astonome Kal Fiedich Gauss, and is also tic fields, which we will encounte in Chapte 15. Gauss s law is a mathsult, tue because the foce falls off as an invese squae of the sepaan the paticles _07_p /22/04 10:49 AM Page 208 e suface 7.5 of NEWTONIAN the Eath, GRAVITATION the expession F mg is valid. As shown in howeve, the fee-fall acceleation g vaies consideably with altitude ath. uiz 7.8 If two paticles with masses m 1 and m 2 ae sepaated by a distance, then a gavitational foce acts along a line joining them, with magnitude given by F G m 1m 2 [7.20] 208 Chapte 7 Rotational Motion and 2 the Law of Gavity whee G kg 1 m 3 s 2 is a constant of popotionality called the constant of univesal gavitation. The gavitational foce is always Figue 7.18 (a) A schematic attactive. to the gound. Which of the following statements ae false? (a) The he ball exets on Eath is equal in magnitude to the foce that Eath exball. (b) The ball diagam undegoes of the Cavendish the appaatus same acceleation as Eath. (c) Eath hade on the fo ball measuing than G. the The ball smalle pulls sphees on Eath, so the ball falls while ins stationay. uiz 7.9 of mass m ae attacted to the lage sphees of mass M, and the od otates though a small angle. A light beam eflected fom a mio on the otating appaatus measues the angle of otation. (b) A student s two moons with Cavendish identical appaatus. mass. Moon 1 is in a cicula obit of adius is in a cicula obit of adius 2. The magnitude of the gavitational ed by the planet on Moon 2 is (a) fou times Mas lage (b) twice as lage e (d) half as lage (e) one-fouth as lage as the gavitational m foce exe planet on Moon 1. ent of the Gavitational Constant Mio Light souce Log into PhysicsNow at and go to Active Figue 7.17 to change the masses of the paticles and the sepaation between them to see the effect on the gavitational foce. m 1 TABLE 7.1 F 21 F Fee-Fall Acceleation g at Vaious Altitudes m 2 Altitude (km) a g (m/s 2 ) tional constant EXAMPLE G in Equation Billiads, was fist measued Anyone? in an impotant t by Heny Cavendish in His appaatus consisted of two small Goal Use vectos to find the net gavitational foce on an object ch of mass m, fixed to the ends of a light hoizontal od suspended by a m wie, as in Figue Poblem 7.18a (see (a) Thee page 208) kg Two billiad lage balls sphees, ae placed each on of a mass table at the cones of 0.13 ced nea the smalle a ight tiangle, sphees. as The shown attactive fom ovehead foce between in Figue the smalle Find the net gavitational sphees caused foce the on od the to cue otate ball in (designated a hoizontal as m plane and the wie to a 1) esulting fom the foces exeted All figues by ae thedistances above Eath s suface. angle though othe which two the balls. suspended (b) Find the od components otated was of the measued gavitational with foce a of m 2 on m m eflected fom a mio attached to the vetical suspension. (Such a Stategy (a) To find the net gavitational foce on the cue ball of mass m t of light is an effective technique fo amplifying the motion.) The ex- 1, we : F (a) (b) Coutesy of PASCO Scientific y m F 23 x

13 The gavitational potential enegy associated with an object of mass m at a distance fom the cente of Eath is PE G M Em [7.21] whee M E and R E ae the mass and adius of Eath, espectively, with R E. SI units: Joules ( J) PE 2 Escape Speed KE i PE i 1 2 mv 2 i GM Em R E 1 2 mv 2 esc GM Em 0 R E h = R E + h PE 1 R E Escape Speeds fo the Planets and the Moon Planet v e (km/s) Mecuy 4.3 Venus 10.3 Eath 11.2 Moon 2.3 Mas 5.0 Jupite 60.0 Satun 36.0 Uanus 22.0 Neptune 24.0 Pluto 1.1 v esc 2GM E R E Eath is about 11.2 km/s, whic

14 7.6 KEPLER S LAWS 1. All planets move in elliptical obits with the Sun at one of the focal points. 2. A line dawn fom the Sun to any planet sweeps out equal aeas in equal time intevals. 3. The squae of the obital peiod of any planet is popotional to the cube of the aveage distance fom the planet to the Sun. p q Keple s Fist Law Focus Focus (a) Planet (b) Sun

15 Keple s Second Law Sun S Figue 7.22 The two aeas swept Keple s Thid Law M p a c M pv 2 GM SM p 2 GM S 2 (2 /T) 2 T GM 3 K S 3 K S GM S S s2 /m3 ple s thid law fo a cicula obit. The

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects