Lesson3 Th Po e of th Ci l Quadilateals and tiangles ae used to make eveyday things wok. Right tiangles ae the basis fo tigonometic atios elating angle measues to atios of lengths of sides. Anothe family of shapes that is boadly useful is the cicle and its thee-dimensional elatives, such as sphees, cylindes, and cicula cones. An impotant chaacteistic of a cicle is that it has otational symmety about its cente. Fo example, the hub of an automobile wheel is at the cente of a cicle. As the ca moves, it tavels smoothly because the cicula tie keeps the hub a constant distance fom the pavement. Motos often otate a cylindical dive shaft. The moe enegy output, the faste the dive shaft tuns. On an automobile engine, fo example, a belt connects thee pulleys, one on the cankshaft, one which dives the fan, and anothe which dives the altenato. When the engine is unning, the fan cools the adiato while the altenato geneates electical cuent. Fan Pulley 7.5 cm 10 cm Altenato Pulley 5 cm Cankshaft Pulley Thi k Ab Thi Si i The diamete measuements given in the diagam above ae fo a paticula fou-cylinde spots ca. a How does the speed of the cankshaft affect the speed of the fan? Of the altenato? b The idle speed of the cankshaft of a fou-cylinde spots ca is about 850 pm (evolutions pe minute). How fa, in centimetes, would a point on the edge of the fan pulley tavel in one minute? Do you think a simila point on the connected altenato pulley would tavel the same distance in one minute? Why o why not? c Descibe anothe situation in which tuning one pulley (o othe cicula object) tuns anothe. 412 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION 412 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION
Unit 1 Lesson3 LESSON OVERVIEW The Powe of the Cicle In the pevious two lessons, chaacteistics of the quadilateal and tiangle wee studied fom points of view that suppoted paticula applications. Uses of those shapes may not be obvious to the untained eye. In the case of the cicle, many of its uses ae moe evident because its shape is had to hide in anothe shape, but, fo example, a tiangle can hide in a quadilateal. This lesson intoduces many of the impotant ideas about cicles and thei chaacteistics. It begins with a study of pulleys and spockets, moves to linea and angula velocity, and ends with the study of the gaphs of the tigonometic functions and modeling peiodic motion. L Obj t To analyze a situation involving pulleys o spockets to detemine tansmission factos, angula velocity, and linea velocity To sketch the gaphs of the sine and cosine functions To detemine the peiod and amplitude of A sin Bx o A cos Bx To use sines and cosines to model peiodic phenomena To use adian and degee measues with tigonometic functions Maste 141 MASTER 141 Use with page 412. Think About This Situation Tanspaency Maste Fan Pulley Altenato Pulley The diamete measuements 7.5 cm given in the diagam ae fo 5 cm a paticula fou-cylinde spots ca. 10 cm Cankshaft How does the speed of Pulley the cankshaft affect the speed of the fan? Of the altenato? The idle speed of the cankshaft of a fou-cylinde spots ca is about 850 pm (evolutions pe minute). How fa, in centimetes, would a point on the edge of the fan pulley tavel in one minute? Do you think a simila point on the connected altenato pulley would tavel the same distance in one minute? Why o why not? Descibe anothe situation in which tuning one pulley (o othe cicula object) tuns anothe. U N I T 6 G E O METRIC FORM AND ITS FUNCTION LAUNCH full-class discussion This lesson could be launched with a two-minute bainstoming session, in which goups wite as many uses of cicula objects (including coss sections that ae cicula) as they can. Once the goup lists ae made, you might make a composite list on the boad. It is best if you take no moe than one o two fom a goup befoe moving to the next so that all goups can contibute. Fo seveal of the uses identified, you might ask students to descibe the chaacteistics of the cicle that ae cental to each use. Most of these chaacteistics pobably will elate to the complete otational symmety of the cicle. Let students know that they will study the cicle to see how that symmety plays an impotant pat in so many diffeent applications. Following the discussion on the applications of cicles, students should conside the cankshaft-pulley/altenato-pulley/fan-pulley setup pictued on the opening page. Ask students to speculate on how tuning one pulley will affect the othe pulleys. This discussion will help you intoduce the expeiments with linked spools in the fist pat of Investigation 1. See additional Teaching Notes on page T453J. LESSON 3 THE POWER OF THE CIRCLE T412
EXPLORE small-goup investigation INVESTIGATION 1 Follow That Dive! NOTE: Any cylindical shape, such as oatmeal boxes o nut cans, can be used with elastic bands as belts. Dowels inseted though the cente of the cylindes will allow fo easy tuning. The dowels could be attached to a boad at vaying intevals to povide a stationay position fo the cylindical shapes. Anothe way to povide stationay positions is to inset pointed objects such as pencils into dense foam. In this investigation students wok with models of pulleys and spockets and collect data that can be used to detemine the tansmission facto fom one pulley to anothe. Students also investigate angula velocity and, using the tansmission facto, detemine associated angula and linea velocities. This investigation is witten in two sections. In the fist section, students get an initial look at angula velocity given in evolutions pe unit of time fo a system of linked cicles o cylindes. Hee students exploe pulley systems and develop an undestanding of the tansmission facto that elates a vaiety of linked pulley systems. The second section asks students to make the tansition fom evolutions pe unit of time to undestanding and detemining linea velocity. When Investigation 1 has been completed, the students should undestand and be able to apply the concept of tansmission facto, in its vaious foms, to seveal mechanical situations. The tansmission facto fom cicle A to cicle B is given by any of the following atios: adius A d iamete A c icumfeence A angle tuned in B adius B diamete B cicumfeence B a ngle tuned in A Thus, if A is tuning 3 evolutions pe minute, B would tun 3 ( adius A adius B ) evolutions pe minute. Activities 1 3 equie students to wok in pais o goups of thee. The exta hands ae needed because distances and angles will need to be measued while the pulley system is being suspended. The linea data ae easie to collect if students wap a sting aound the spool (o use the thead on the spool itself) and measue how much comes off as the spool tuns. Angles can be measued faily easily if a baseline is maked on the spool. Setting this baseline to be eithe vetical o hoizontal will simplify the angle measuement poblem. Fo the best esults, students should use spools that diffe significantly in size. 1. a. The followe spool tuns in the same diection. b. The followe spool tuns moe than one complete evolution when the dive spool has a lage adius than the followe spool. The two spools tun the same amount when they have equal adii. The followe spool tuns less than one complete evolution when the followe adius is lage than the dive adius. c. The ubbe band advances 2π units; is the adius of the dive spool. d. Students should make a table and look fo a patten. The patten should be modeled adius of dive tun of dive spool by: tun of followe spool. This may not be appaent if students measuements ae inaccuate. It will be cleae when students use the adius of followe data in Pat g o in Activity 3. e. This is a linea elationship, and it should appea so when plotted. Students should be able to find a symbolic model to fit, such as F 2.5D o y 2.5x. At this stage, they may not see the connection with the adii. Some may see that the slope is d ; whee f d and f ae the adii of the dive and the followe, espectively. f. This pat should geneate discussion of measuement eos and diffeing tansmission. T413 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION
INVESTIGATION 1 Follow That Dive! In this investigation, you will exploe how otating cicula objects that ae connected can seve useful puposes. A simple way to investigate how the tuning of one cicle (the dive) is elated to the tuning of anothe (the followe) is to expeiment with thead spools and ubbe bands. The spools model the pulleys, spockets, o geas; the ubbe bands model the belts o chains connecting the cicula objects. Complete the fist two activities woking in pais. 1. Use two thead spools of diffeent sizes. If you use spools that still have thead be cetain the thead is secuely fastened. Put each spool on a shaft, such as a pencil, which pemits the spool to tun feely. Make a dive/followe mechanism by slipping a ubbe band ove the two spools and speading the spools apat so the ubbe band is taut enough to educe slippage. Choose one spool as the dive. a. Tun the dive spool. Descibe what happens to the followe spool. b. Tun the dive spool one complete evolution. Does the followe spool make one complete evolution, o does it make moe o fewe tuns? c. Tun the dive spool one complete evolution. How fa does the ubbe band advance? d. Design and cay out an expeiment that gives you infomation about how tuning the dive spool affects the amount of tun of the followe spool, when the spools have diffeent adii. Use whole numbe and factional tuns of the dive. Oganize you (dive tun amount, followe tun amount) data in a table. e. Plot you (dive tun amount, followe tun amount) data. Find an algebaic model that fits the data. f. Compae you scatteplot and model with those of othe pais of students. How ae they the same? How ae they diffeent? What might explain the diffeences? LESSON 3 THE POWER OF THE CIRCLE 413 LESSON 3 THE POWER OF THE CIRCLE 413
g. Examine the dive/followe spool data below. What patten would you expect to see in a plot of these data? What algebaic model do you suspect would fit these data? Dive/Followe Data Set 1 Dive Radius: 2.5 cm Followe Radius: 2 cm Dive Tun Amount Followe Tun Amount (in evolutions) (in evolutions) 0.5 0.6 1 1.3 2 2.5 3 3.8 5 6.2 8 10.0 10 12.5 12 15.0 Dive/Followe Data Set 2 Dive Radius: 1 cm Followe Radius: 1.5 cm Dive Tun Amount Followe Tun Amount (in evolutions) (in evolutions) 0 0.0 1 0.7 3 2.0 5 3.4 7 4.6 9 6.0 12 8.0 15 10.0 2. Revese the dive/followe oles of the two spools. How does tuning the dive affect the followe now? 3. Suppose the dive spool has a adius of 2 cm and the followe spool has a 1-cm adius. a. If the dive spool tuns though 90 degees, though how many degees will the followe spool tun? Suppot you position expeimentally o logically. b. In geneal, how will tuning the dive spool affect the followe spool? Povide evidence that you conjectue is tue, using data o easoning about the situation. c. How do the lengths of the adii of the spools affect the dive/followe elation? Answe as pecisely as possible. d. The numbe by which the tun o speed of the dive is multiplied to get the tun o speed of the followe is often called the tansmission facto fom dive to followe. What is the tansmission facto fo a dive with a 4-cm adius and a followe with a 2-cm adius? If you evese the oles, what is the tansmission facto? e. List two sets of dive/followe spool adii so that each set will have a tansmission facto of 3. Do the same fo tansmission factos of 3 2 and 4. 5 f. If the dive has adius 1 and the followe has adius 2, what is the tansmission facto? 414 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION 414 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION
EXPLORE continued 1. g. Students should expect a linea patten in a plot of this data. In both cases, the algebaic model y d x fits the data. Students may just suggest a linea f model. 2. The effect is just the opposite. If the oiginal followe tuned futhe than the oiginal dive, the new followe will now tun less than the new dive. Some students should be able to give numeical examples, such as The fist followe used to tun about 3 times as fa, the new one tuns about 1 3 as fa. In fact the tansmission facto fom B to A is the ecipocal of that fom A to B. 3. a. 180 Suppot statements will vay. Students may ague that the followe will tun twice as fa as the dive, so if the dive makes a 1 4 tun the followe makes a 1 2 tun. b. The followe will always tavel twice as fa. The cicumfeence of the dive spool is twice that of the followe spool, so the followe spool will go aound twice fo each single tun of the dive spool. c. The followe always tavels d times as fa. d. 4 2 o 2, 2 4 o 1 2 e. Facto Dive 1 Followe 1 Dive 2 Followe 2 3 3 1 6 2 3 2 3 2 6 4 4 4 5 8 101 5 f f. The tansmission facto is 1. 2 A full-class discussion of insights gained in Activities 1 3 would benefit many students. Some will have achieved basic insights, such as that the followe spool tuns fathe than the dive spool if the dive spool is lage, but might not have elated futhe to the atio of the adii. Some will undestand that the adii ae the key but not be able to complete the connection. Some will neatly connect this atio to the tansmission facto, but even with accuate gaphs, they might not see any connection between slope and the atio of the adii. Whateve stage they have eached, a class discussion will be helpful to bing out all these ideas. Since the vocabulay tansmission facto is not intoduced until Activity 3, the discussion should give students thei fist oppotunity to descibe what they found in thei expeiments and gaphs in Activity 1, using the new vocabulay. To be sue they undestand the concept, you might ask, What if the dive was 3 times as big as the followe? Fo one tun of the dive, how fa would the followe tun? What would the gaph look like? If the gaph had a slope of 4, what does that tell you about the spools? If the gaph had a slope of 0.5? If the tansmission facto is 3, what does that tell you about the spools? About the gaphs? LESSON 3 THE POWER OF THE CIRCLE T414
EXPLORE continued 4. a. Dive Followe b. If both dive and followe tun counteclockwise, thee would be no othe diffeences. Dive Followe c. The magnitude of the effect is the same as fo the model in Pat a, but the followe tuns in the opposite diection. Thus, the tansmission facto could be, whee the minus sign indicates that the wheels ae tuning in opposite diections. That diffeence is the only one necessay. 5. a. Dive Followe 5 4 1 2 Dive-to-followe tansmission facto is 5 o 1.25. 4 b. C d 2π 2π 5 10π C f 2π 4 8π C d Cf 2π 2π 5 5 Tansmission facto: 10π d 2πf 2π 4 8π 4 Since the pulleys ae attached by a belt, when the dive makes a full tun, it moves the belt a distance equal to its cicumfeence. The belt then tuns the followe that C same distance, which is d times the followe s cicumfeence. Cf c. Since a point on the cicle tavels the complete cicumfeence in each evolution it tavels, the distance is 10π o appoximately 31.4 inches. d. The point moves 50 times the cicumfeence of the dive. 50 C 1,570 in ches m in e. The tansmission facto is 1.25. Theefoe, the followe otates at 1.25 times the ate of the dive. 1.25 50 62.5 pm Some students may calculate as follows: 1,570 2π 4 in. min in. ev 62.5 pm f. C (evolutions in one minute) (2π 4) 62.5 1,571 inches The same distance is coveed by both pulleys. The dive has a lage cicumfeence but otates moe slowly than the followe. The followe has a smalle cicumfeence but otates moe quickly than the dive. T415 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION
4. In addition to designing a tansmission facto into a pulley system, you must also conside the diections in which the pulleys tun. a. In the spool/ubbe band systems you made, did the dive and followe tun in the same diection? Sketch a spool/ubbe band system in which tuning the dive clockwise tuns the followe clockwise also. Label the dive and followe. How would this system look if both the dive and followe wee to tun counteclockwise? b. Sketch a spool/ubbe band system in which tuning the dive spool clockwise tuns the followe counteclockwise. Make a physical model to check you thinking. c. Suggest a way to descibe the tansmission facto fo the system in Pat b. Should the tansmission facto diffe fom a system using the same spools tuning in the same diection? If so, how? If not, why not? 5. A clockwise dive/followe system has a dive with a 5-inch adius and a followe with a 4-inch adius. a. Sketch the system. What is the tansmission facto fo this system? b. What ae the cicumfeences of the two pulleys? How could you use the lengths of the cicumfeences to detemine the tansmission facto of the system? Explain why this is easonable. c. How fa does a point on the edge of the dive tavel in one evolution of the dive? d. If the dive is otating 50 evolutions pe minute (pm), how fa does the point in Pat c tavel in 1 minute? e. If the dive is tuning at 50 pm, how fast is the followe tuning? f. In one minute, how fa does a point on the cicumfeence of the followe tavel? Compae this esult with that in Pat d and explain you findings. 6. Wanda is iding he mountain bicycle using the cankset (also called the pedal spocket) with 42 teeth of equal size. The ea-wheel spocket being used has 14 teeth of a size equal to the cankset teeth. Chain Rea Spocket Cankset Pedal LESSON 3 THE POWER OF THE CIRCLE 415 LESSON 3 THE POWER OF THE CIRCLE 415
a. What does the teeth pe spocket infomation tell you about the cicumfeences of the two spockets? Tanslate the infomation about teeth pe spocket into a tansmission facto. b. Suppose Wanda is pedaling at 80 evolutions pe minute. What is the ate at which the ea spocket is tuning? Explain. What is the ate at which the ea wheel is tuning? Explain. c. The wheel on Wanda s mountain bike has a adius of about 33 cm. How fa does the bicycle tavel fo each complete evolution of the 14-tooth ea spocket? How fa does the bicycle tavel fo each complete evolution of the font spocket? If Wanda pedals 80 pm, how fa will she tavel in one minute? d. How long will Wanda need to pedal at 80 pm to tavel 2 kilometes? Ch kp i In this investigation, you exploed some of the featues of dive/followe mechanisms. a b c d What is the significance of the tansmission facto in the design of otating objects that ae connected? How can you use infomation about the adii of two connected pulleys, spools, o spockets to detemine the tansmission facto? Descibe the similaities and diffeences fo two belt-dive systems that have tansmission factos of 2 3 and 2. 3 If you know how fast a pulley is tuning, how can you detemine how fa a point on its cicumfeence tavels in a given amount of time? Be pepaed to shae you desciptions and thinking with the entie class. As you have seen, the tansmission facto fo otating cicula objects is positive when the two cicula objects tun in the same diection. When they tun in opposite diections, the tansmission facto is expessed as a negative value. Using negative numbes to indicate the diection opposite of an accepted standad diection is common in mathematics and science. 416 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION 416 UNIT 6 GEOMETRIC FORM AND ITS FUNCTION
EXPLORE continued Maste 142 6. This activity is an example in which the tansmission facto is used to detemine ates fo attached spockets given pedaling ates, and vice vesa. Students also look at how otations pe minute elates to distance taveled. Students may have a difficult time undestanding that the evolutions pe minute fo the inside and outside of a wheel will be the same (Pat b). You might suggest that students simulate this situation o obseve a bike wheel at home to veify fo themselves that this is the case. a. Since the teeth ae equally spaced, the atio of the teeth of the spockets is popotional to the atio of the cicumfeences of the spockets, which we know is popotional to the atio of the adii of the spockets. Thus, we can use teeth pe spocket just as if they wee the adii fo puposes of finding the tansmission facto. The tansmission facto of the cankset to ea-wheel spocket is 4 2 3. 14 b. Rea spocket ate 3 cankset ate 3 80 240 pm The tansmission facto gives the elation between the tuning ates. Thus, multiplying by the 3 gives the ea tuning ate. The ea wheel tuns at 240 pm also since it is attached to the ea spocket. c. Distance 2 π 33 66π cm 207 cm pe evolution of the ea spocket Distance 3 (distance taveled with one evolution of the back wheel) 3 66π cm 198π cm 622 cm pe evolution of the font spocket Distance 80 pm 198π cm 49,762.8 cm o about 498 metes in one minute d. 2 kilometes 2,000 metes. Since Wanda pedals 498 metes in 1 minute, she will 2,000 m need to pedal fo o appoximately 4.02 minutes. 49 8 m/min MASTER 142 Checkpoint Tanspaency Maste In this investigation, you exploed some of the featues of dive/followe mechanisms. What is the significance of the tansmission facto in the design of otating objects that ae connected? How can you use infomation about the adii of two connected pulleys, spools, o spockets to detemine the tansmission facto? Descibe the similaities and diffeences fo two belt-dive systems that have tansmission factos of 2 3 and 2. 3 If you know how fast a pulley is tuning, how can you detemine how fa a point on its cicumfeence tavels in a given amount of time? Be pepaed to shae you desciptions and thinking with the entie class. Use with page 416. U NIT 6 GEOMETRIC FORM AND ITS FUNCTION SHARE AND SUMMARIZE Checkpoint full-class discussion See Teaching Maste 142. a The tansmission facto descibes the elationship between the dive and the followe. A tansmission facto of b means that a single tun of the dive will cause b tuns in the followe. b If the dive is A with adius A and the followe is B with adius B, the tansmission c d facto fom A to B is A. B The adii of the pulleys ae in the same atio ( 2 3 ) so, fo evey two tuns of the dive, the followe will tun thee times. The diffeence is that fo 2, the pulleys tun in the 3 same diection, while fo 2, they tun in opposite diections. 3 Explanations may vay. The distance taveled by a point on the cicumfeence depends on two vaiables: R, the numbe of evolutions pe minute, and, the adius of the pulley. Some students may espond that the evolutions pe minute alone do not allow you to detemine distance since the evolutions pe minute on the inside and outside of a disk ae the same. Without knowing, you cannot say exactly how fa the point moves aound the cicumfeence. Othe students may explain in algebaic language that the adius is needed. If the pulley has adius, it olls 2π fo each evolution. If it makes R evolutions pe minute, it olls R 2π units. Ty to elicit both explanations fom the class. CONSTRUCTING A MATH TOOLKIT: Ask students to summaize the concepts in the fist potion of this investigation. Students should include how to find and use a tansmission facto, and how the ate at which a pulley is tuning can be used to detemine how fa a point on the edge of the pulley is tuning (Teaching Maste 186). LESSON 3 THE POWER OF THE CIRCLE T416