SECTION 1. Objectives. Solve problems involving centripetal acceleration. Solve problems involving centripetal force.

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SECTION 1 Plan and Pepae Peiew Vocabulay Latin Wod Oigins The wod centipetal is ombination of two pats, cente and petal.

The wod centipetal is used to descibe an object moing in icula path whose acceleation is diected towad the cente of the path. Teach Misconception Alet!

It is impotant to emphasize the distinction between centipetal (cente-seeking) and centifugal (cente-fleeing). To aoid einfocing this misconception, aoid using the tem centifugal.

1 SECTION 1 Plan and Pepae Peiew Vocabulay Latin Wod Oigins The wod centipetal is ombination of two pats, cente and petal. The second pat of the wod is deied fom the Latin wod petee, which means seeking. The wod centipetal is used to descibe an object moing in icula path whose acceleation is diected towad the cente of the path. Teach Misconception Alet! Some students will hae difficulty with teminology at this point because of the peious familiaity with the tem centifugal. It is impotant to emphasize the distinction between centipetal (cente-seeking) and centifugal (cente-fleeing). To aoid einfocing this misconception, aoid using the tem centifugal. Demonstation Centipetal Acceleation Pupose Show an example of centipetal acceleation. Mateials battey-opeated toy ca, 50 cm sting Pocedue Tie one end of the sting to the midpoint of the ca. Hold the othe end of the sting to the table o floo. Stat the ca. Hae students obsee the motion. Release the sting and allow the ca to moe in a staight line. Point out that een though the ca was taeling at constant speed, the ca was acceleating because the diection of motion continued to change. Show that the change in elocity was always pependicula to the diection of motion. Objecties Sole poblems inoling centipetal acceleation. Sole poblems inoling centipetal foce. Explain how the appaent existence of an outwad foce in cicula motion can be explained as inetia esisting the centipetal foce. FIGURE 1.1 SECTION 1 Cicula Motion Any point on a Feis wheel spinning about a fixed axis undegoes cicula motion. 224 Chapte 7 Diffeentiated Instuction Below Leel Cicula Motion Key Tem centipetal acceleation Ask students to explain how a line of dances moing in icle demonstates that tangential speed depends on the distance between the object and the cente of the cicula path. Also ask why jockeys maneue thei hoses duing a ace to get as close to the inside ail as possible. See if students can poide othe examples of how tangential speed depends on distance fom the cente. Centipetal Acceleation Conside a spinning Feis wheel, as shown in Figue 1.1. The cas on the otating Feis wheel ae said to be in cicula motion. Any object that eoles about a single axis undegoes cicula motion. The line about which the otation occus is called the axis of otation. In this case, it is a line pependicula to the side of the Feis wheel and passing though the wheel s cente. Tangential speed depends on distance. Tangential speed ( ) can be used to descibe the speed of an object in cicula motion. The tangential speed of a on the Feis wheel is the ca s speed along an imaginay line dawn tangent to the ca s cicula path. This definition can be applied to any object moing in cicula motion. When the tangential speed is constant, the motion is descibed as unifom cicula motion. The tangential speed depends on the distance fom the object to the cente of the cicula path. Fo example, conside a pai of hoses side-byside on aousel. Each completes one full cicle in the same time peiod, but the hose on the outside coes moe distance than the inside hose does, so the outside hose has a geate tangential speed. Centipetal acceleation is due to hange in diection. Suppose a on a Feis wheel is moing at onstant speed as the wheel tuns. Een though the tangential speed is constant, the ca still has an acceleation. To see why, conside the equation that defines acceleation: a = _ f i t f t i Acceleation depends on hange in the elocity. Because elocity is a ecto, acceleation can be poduced by hange in the magnitude of the elocity, hange in the diection of the elocity, o both. If the Feis wheel ca is moing at a constant speed, then thee is no change in the magnitude of the elocity ecto. Howee, think about the way the ca is moing. It is not taeling in a staight line. The elocity ecto is continuously changing diection. Untitled /9/2011 8:14:27 Joseph Sohm/Visions of Ameica/Cobis 224 Chapte 7

2 The acceleation of a Feis wheel ca moing in icula path and at constant speed is due to hange in diection. An acceleation of this natue is called entipetal acceleation. The magnitude of entipetal acceleation is gien by the following equation: Centipetal Acceleation = _ 2 (tangential speed) 2 centipetal acceleation = adius of cicula path What is the diection of centipetal acceleation? To answe this question, conside Figue 1.2(a). At time t i, an object is at point A and has tangential elocity i. At time t f, the object is at point B and has tangential elocity f. Assume that i and f diffe in diection but hae the same magnitudes. The change in elocity ( = f i ) can be detemined gaphically, as shown by the ecto tiangle in Figue 1.2(b). Note that when t is ey small, f will be almost paallel to i. The ecto will be appoximately pependicula to f and i and will be pointing towad the cente of the cicle. Because the acceleation is in the diection of, the acceleation will also be diected towad the cente of the cicle. Centipetal acceleation is always diected towad the cente of icle. In fact, the wod centipetal means cente seeking. This is the eason that the acceleation of an object in unifom cicula motion is called centipetal acceleation. Centipetal Acceleation Sample Poblem A A test ca moes at onstant speed aound icula tack. If the ca is 48.2 m fom the tack s cente and has entipetal acceleation of 8.05 m/s 2, what is the ca s tangential speed? ANALYZE Gien: = 48.2 m = 8.05 m/s 2 =? PREMIUM CONTENT centipetal acceleation the acceleation diected towad the cente of icula path FIGURE 1.2 Centipetal Acceleation (a) As the paticle moes fom A to B, the diection of the paticle s elocity ecto changes. (b) Fo shot time inteals, is diected towad the cente of the cicle. SOLVE Use the centipetal acceleation equation, and eaange to sole fo. = 2 _ t = = (8.05 m/s 2 ) (48.2 m) (a) (b) A i f i Inteactie Demo HMDScience.com B f Classoom Pactice Centipetal Acceleation The cylindical tub of a washing machine has a adius of 34 cm. Duing the spin cycle, the wall of the tub otates with a tangential speed of 5.5 m/s. Calculate the centipetal acceleation of the clothes sitting against the tub. Answe: 89 m/s 2 PROBLEM guide A Use this guide to assign poblems. SE = Student Edition Textbook PW = Sample Poblem Set I (online) PB = Sample Poblem Set II (online) Soling fo: SE Sample, 1 2; Ch. Rw. 8 9 PW 3 4 PB 4 6 Se 3; Ch. Rw. 50a PW 5 6 PB Sample, 1 3 SE 4 PW Sample, 1 2 PB 7 10 *Challenging Poblem = 19.7 m/s Continued titled Poblem Soling Deconstucting Poblems Use the following equations fo students who ae haing difficulty undestanding the algebaic simplifications used to deie the final equation. = 2 _ t = 2 ÇÇ = Ç 2 ÇÇ = Cicula Motion and Gaitation 225 5/9/2011 8:14:28 AM Cicula Motion and Gaitation 225

Teach continued Answes Pactice A 1. 2.5 m/s 2. 11 m/s 3. 1.5 m/s 2 4. 58.4 m Teaching Tip Tangential acceleation is coeed in moe detail in the featue Tangential Speed and Acceleation.

1 m, what is the gil s tangential speed? 2. As a young boy swings a yo-yo paallel to the gound and aboe his head, the yo-yo has entipetal acceleation of 250 m/s 2. If the yo-yo s sting is 0.

3 Teach continued Answes Pactice A m/s m/s m/s m Teaching Tip Tangential acceleation is coeed in moe detail in the featue Tangential Speed and Acceleation. Centipetal Acceleation (continued) 1. A ope attaches a tie to an oehanging tee limb. A gil swinging on the tie has entipetal acceleation of 3.0 m/s 2. If the length of the ope is 2.1 m, what is the gil s tangential speed? 2. As a young boy swings a yo-yo paallel to the gound and aboe his head, the yo-yo has entipetal acceleation of 250 m/s 2. If the yo-yo s sting is 0.50 m long, what is the yo-yo s tangential speed? 3. A dog sits 1.5 m fom the cente of a mey-go-ound. The mey-go-ound is set in motion, and the dog s tangential speed is 1.5 m/s. What is the dog s centipetal acceleation? 4. A aceca moing along icula tack has entipetal acceleation of 15.4 m/s 2. If the ca has a tangential speed of 30.0 m/s, what is the distance between the ca and the cente of the tack? Tangential acceleation is due to hange in speed. You hae seen that centipetal acceleation esults fom hange in diection. In cicula motion, an acceleation due to hange in speed is called tangential acceleation. To undestand the diffeence between centipetal and tangential acceleation, conside a taeling in a cicula tack. Because the ca is moing in icle, the ca has a centipetal component of acceleation. If the ca s speed changes, the ca also has a tangential component of acceleation. FIGURE 1.3 Foce on a Ball When a ball is whiled in icle, it is acted on by a foce diected towad the cente of the ball s cicula path. Centipetal Foce Conside a ball of mass m that is tied to a sting of length and that is being whiled in a hoizontal cicula path, as shown in Figue 1.3. Assume that the ball moes with constant speed. Because the elocity ecto,, continuously changes diection duing the motion, the ball expeiences entipetal acceleation that is diected towad the cente of motion. As seen ealie, the magnitude of this acceleation is gien by the following equation: = _ 2 The inetia of the ball tends to maintain the ball s motion in a staight path. Howee, the sting exets a foce that oecomes this tendency. The foces acting on the ball ae gaitational foce and the foce exeted by the sting, as shown in Figue 1.4(a) on the next page. The foce exeted by the sting has hoizontal and etical components. The etical component is equal and opposite to the gaitational foce. Thus, the hoizontal component is the net foce. This net foce is diected towad the cente of the cicle, as shown in Figue 1.4(b). The net foce that is diected towad the cente of an object s cicula path is called centipetal foce. Newton s second law can be applied to find the magnitude of this foce. = m 226 Chapte 7 Diffeentiated Instuction Inclusion Kinesthetic leanes may need help undestanding that Eath s gaitational foce causes a satellite in icula obit to change its diection but not its speed. Hae students imagine that a peson is unning on the side of a oad. Ask them to descibe what would happen if the unne suddenly gabbed a lamppost o small tee tunk. Be sue students ecognize that the foce was pependicula to the unne s fowad motion and caused the unne s diection to change. Students can ty this outdoos to feel the change in diection fo themseles. Untitled /9/2011 8:14: Chapte 7

4 led FIGURE 1.4 Centipetal Foce The net foce on a ball whiled in icle (a) is diected towad the cente of the cicle (b). (a) Fnet F sting = Fc The equation fo centipetal acceleation can be combined with Newton s second law to obtain the following equation fo centipetal foce: Centipetal Foce = _ m 2 (tangential speed) 2 centipetal foce = mass adius of cicula path Centipetal foce is simply the name gien to the net foce towad the cente of the cicula path followed by an object moing in unifom cicula motion. Any type of foce o combination of foces can poide this net foce. Fo example, fiction between a aceca s ties and a cicula tack is entipetal foce that keeps the ca in icula path. As anothe example, gaitational foce is entipetal foce that keeps the moon in its obit. Centipetal Foce Sample Poblem B A pilot is flying a small plane at 56.6 m/s in icula path with a adius of m. The centipetal foce needed to maintain the plane s cicula motion is N. What is the plane s mass? F g ANALYZE Gien: = 56.6 m/s = m = N Unknown: m =? SOLVE Use the equation fo centipetal foce. Reaange to sole fo m. Continued Poblem Soling = m 2 _ t (b) m = _ 2 = ( N) (188.5 m) (56.6 m/s) 2 m = 1110 kg Deconstucting Poblems Use the following equations to help students see how the equation is eaanged to sole fo mass. F C = _ m2 t F C = m 2 t _ F C = _ m2 t 2 t 2 t _ F C = m 2 t PREMIUM CONTENT m Inteactie Demo HMDScience.com Cicula Motion and Gaitation 227 Reality Check Help students elate the equation fo centipetal foce to the motion of the object. A shape tun means a geate change in diection, theefoe an object moing in a small cicle will equie a geate centipetal foce if the speed is kept constant. 5/9/2011 8:14:29 AM Demonstation Centipetal Foce Pupose Show that the hoizontal component of tension in a ope poides the centipetal foce fo a ball moing in a hoizontal cicle. Mateials 100 g mass (with hook), 30 cm sting, sping scale, potacto Pocedue Attach one end of the sting to the hook on the mass and attach the othe end to the sping scale. Use the potacto to measue the position at which the sting makes an angle of 30 with the etical. Then otate the sting with onstant speed so that the mass moes in icula path and the sting emains at the 30 angle with the etical. While maintaining this motion, hae a oluntee attempt to ead the sping scale. (Maintaining onstant angle can be difficult; you may wish to pactice this befoe demonstating to the class.) Point out that the scale eading equals the tension in the sting. The etical component balances the weight of the mass, while the hoizontal component of the tension holds the mass in icula path. Ask students to calculate the etical tension component and to compae it with the weight (F scale cosθ = mg) so that they can confim whethe the scale eading and angle ae coect. Then hae them calculate the centipetal foce fom the hoizontal component of tension (F scale sinθ = ). Repeat the demonstation at angles equal to 45 and 60. Ask students to pedict what change will occu in the foce as measued on the sping scale. Cicula Motion and Gaitation 227

5 Teach continued PROBLEM guide b Use this guide to assign poblems. SE = Student Edition Textbook PW = Sample Poblem Set I (online) PB = Sample Poblem Set II (online) Soling fo: m SE Sample, 1; Ch. Rw. 10a*, 11* PW 3 PB 7 10 SE 4 Ch. Rw. 10b, 39* PW Sample, 1 2 PB 5 7 SE 2 PW 4 PB 6 7 SE 3; Ch. Rw. 50b, 53 PW 5 PB Sample, 1 3 *Challenging Poblem Answes Pactice B kg m N m/s FIGURE 1.5 Centipetal Foce (continued) 1. A 2.10 m ope attaches a tie to an oehanging tee limb. A gil swinging on the tie has a tangential speed of 2.50 m/s. If the magnitude of the centipetal foce is 88.0 N, what is the gil s mass? 2. A bicyclist is iding at a tangential speed of 13.2 m/s aound icula tack. The magnitude of the centipetal foce is 377 N, and the combined mass of the bicycle and ide is 86.5 kg. What is the tack s adius? 3. A dog sits 1.50 m fom the cente of a mey-go-ound and eoles at a tangential speed of 1.80 m/s. If the dog s mass is 18.5 kg, what is the magnitude of the centipetal foce on the dog? 4. A 905 kg ca taels aound icula tack with icumfeence of 3.25 km. If the magnitude of the centipetal foce is 2140 N, what is the ca s tangential speed? Remoal of Centipetal Foce A ball that is on the end of a sting is whiled in a etical cicula path. If the sting beaks at the position shown in (a), the ball will moe etically upwad in fee fall. (b) If the sting beaks at the top of the ball s path, the ball will moe along a paabolic path. (a) (b) Centipetal foce is necessay fo cicula motion. Because centipetal foce acts at ight angles to an object s cicula motion, the foce changes the diection of the object s elocity. If this foce anishes, the object stops moing in icula path. Instead, the object moes along a staight path that is tangent to the cicle. Fo example, conside a ball that is attached to a sting and that is whiled in a etical cicle, as shown in Figue 1.5. If the sting beaks when the ball is at the position shown in Figue 1.5(a), the centipetal foce will anish. Thus, the ball will moe etically upwad, as if it has been thown staight up in the ai. If the sting beaks when the ball is at the top of its cicula path, as shown in Figue 1.5(b), the ball will fly off hoizontally in a diection tangent to the path. The ball will then moe in the paabolic path of a pojectile. Descibing a Rotating System To bette undestand the motion of a otating system, conside a taeling at high speed and appoaching an exit amp that cues to the left. As the die makes the shap left tun, the passenge slides to the ight and hits the doo. At that point, the foce of the doo keeps the passenge fom being ejected fom the ca. What causes the passenge to moe towad the doo? A popula explanation is that a foce must push the passenge outwad. This foce is sometimes called the centifugal foce, but that tem often ceates confusion, so it is not used in this textbook. Inetia is often misintepeted as a foce. The phenomenon is coectly explained as follows: Befoe the ca entes the amp, the passenge is moing in a staight path. As the ca entes the amp and taels along ued path, the passenge, because of inetia, tends to 228 Chapte 7 Diffeentiated Instuction Pe-AP Hae students extend the discussion of a on an exit amp by consideing how changing some of the aiables would affect the passenges. How would passenges of diffeent masses be affected as the moing ca entes the amp? Who would expeience geate foce, a heaie passenge o a lighte one? How would aying the speed of the ca affect the passenges? Students should explain thei answes in tems of inetia and mass. Untitled /9/2011 8:14: Chapte 7

6 moe along the oiginal staight path. This moement is in accodance with Newton s fist law, which states that the natual tendency of a body is to continue moing in a staight line. Howee, if a sufficiently lage centipetal foce acts on the passenge, the peson will moe along the same cued path that the ca does. The oigin of the centipetal foce is the foce of fiction between the passenge and the ca seat. If this fictional foce is not sufficient, the passenge slides acoss the seat as the ca tuns undeneath, as shown in Figue 1.6. Eentually, the passenge encountes the doo, which poides a lage enough foce to enable the passenge to follow the same cued path as the ca does. The passenge does not slide towad the doo because of some mysteious outwad foce. Instead, the fictional foce exeted on the passenge by the seat is not geat enough to keep the passenge moing in the same cicle as the ca. FIGURE 1.6 Path of Ca and Passenge The foce of fiction on the passenge is not enough to keep the passenge on the same cued path as the ca. Assess and Reteach Assess Use the Fomatie Assessment on this page to ealuate student mastey of the section. Reteach Fo students who need additional instuction, download the Section Study Guide. Response to Inteention To eassess students mastey, use the Section Quiz, aailable to pint o to take diectly online at HMDScience.com. SECTION 1 FORMATIVE ASSESSMENT Reiewing Main Ideas 1. What ae thee examples of cicula motion? 2. A gil on a spinning amusement pak ide is 12 m fom the cente of the ide and has entipetal acceleation of 17 m/s 2. What is the gil s tangential speed? 3. Use an example to descibe the diffeence between tangential and centipetal acceleation. 4. Identify the foces that contibute to the centipetal foce on the object in each of the following examples: a. a bicyclist moing aound a flat, cicula tack b. a bicycle moing aound a flat, cicula tack c. a aceca tuning one on a steeply banked cue 5. A 90.0 kg peson ides a spinning amusement pak ide that has a adius of 11.5 m. If the peson s tangential speed is 13.2 m/s, what is the magnitude of the centipetal foce acting on the peson? 6. Explain what makes a passenge in a tuning ca slide towad the doo. Citical Thinking 7. A olle coaste s passenges ae suspended upside down as it moes at a constant speed though a etical loop. What is the diection of the foce that causes the coaste and its passenges to moe in icle? What poides this foce? Answes to Section Assessment Cicula Motion and Gaitation 229 CNLESE586694_C07S1.indd Answes may ay m/s 3. Specific examples may ay, but all should indicate that an object moing in icula path undegoes centipetal acceleation because the object changes diection. Tangential acceleation occus when the object changes its speed aound the path. 4. a. the foces that the bicycle seat, handlebas, and pedals exet on the bicyclist b. fiction between the ties and the tack c. components of (1) the nomal foce fom the banked cue and (2) the fiction between the ties and the oad N 3/26/ :53:47 PM 6. When the ca tuns, the inetia of the passenge keeps the passenge moing in the same diection that the ca was initially moing. 7. towad the cente of the loop; the tack and gaity Cicula Motion and Gaitation 229

DYNAMICS OF UNIFORM CIRCULAR MOTION

DYNAMICS OF UNIFORM CIRCULAR MOTION Chapte 5 Dynamics of Unifom Cicula Motion Chapte 5 DYNAMICS OF UNIFOM CICULA MOTION PEVIEW An object which is moing in a cicula path with a constant speed is said to be in unifom cicula motion. Fo an object