Uniform Circular Motion

Transcription

1 11.2 Unifo Cicula Motion Have you eve idden on a ide like the one shown in the photogaph? Fo a distance, it ight not look exciting, but the sensations could supise you. Eveyone lines up aound the oute edge and the ide slowly begins to tun. Not vey exciting yet, but soon, the ide is spinning quite fast and you feel as though you ae being pessed tightly against the wall. The otations begin to ake you feel disoiented and you stoach stats to feel a little queasy. Then, suddenly, the floo dops away, but you stay helplessly stuck to the wall. Just as you ealize that you ae not going to fall, the entie ide begins to tilt. At one point duing each otation, you find youself looking towad the gound, which is alost diectly in font of you. You do not feel as though you ae going to fall, though, because you ae liteally stuck to the wall. SECTION Analyze, pedict, and explain unifo cicula otion. Explain foces involved in unifo cicula otion in hoizontal and vetical planes. Investigate elationships between peiod and fequency of an object in unifo cicula otion. KEY TERMS OUTCOMES unifo cicula otion centipetal acceleation centipetal foce Figue 11.5 If this ide stopped tuning, the people would stat to fall. What featue of cicula otion pevents people fo falling when the ide is in otion and they ae facing the gound? What is unique about oving in a cicle that allows you to appaently defy gavity? What causes people on the Round Up to stick to the wall? As you study this section, you will be able to answe these questions and any oe. Centipetal Acceleation Auseent pak ides ae only one of a vey lage nube of exaples of cicula otion. Motos, geneatos, vehicle wheels, fans, ai in a tonado o huicane, o a ca going aound a cuve ae othe exaples of cicula otion. When an object is oving in a cicle and its speed the agnitude of its velocity is Chapte 11 Pojectiles and Cicula Motion MHR 551

2 v P 1 O v 1 θ θ v v2 v = v 2 v 1 v1 Figue 11.6 The diection of the change in velocity is found by defining the vecto v 1 and then adding v 2 and v 1. Place the tail of v 1 at the tip of v 2 and daw the esultant vecto, v, fo the tail of v 2 to the tip of v 1. 2 Q v2 constant, it is said to be oving with unifo cicula otion. The diection of the object s velocity is always tangent to the cicle. Since the diection of the otion is always changing, the object is always acceleating. Figue 11.6 shows the how the velocity of the object changes when it is undegoing unifo cicula otion. As an object oves fo point P to point Q, its velocity changes fo v 1 to v 2. Since the diection of the acceleation is the sae as the diection of the change in the velocity, you need to find v o v 2 v 1. Vectos and v 1 and v 2 ae subtacted gaphically unde the cicle. To develop an equation fo centipetal acceleation, you will fist need to show that the tiangle OPQ is siila to the tiangle foed by the velocity vectos, as shown in the following points. 1 = 2 because they ae adii of the sae cicle. Theefoe, tiangle OPQ is an isosceles tiangle. v 1 = v 2 because the speed is constant. Theefoe, the tiangle foed by v 1, v 2, and v is an isosceles tiangle. 1 v 1 and 2 v 2 because the adius of a cicle is pependicula to the tangent to the point whee the adius contacts the cicle. θ = θ v because the angle between coesponding ebes of sets of pependicula lines ae equal. Since the angles between the equal sides of two isosceles tiangles ae equal, the tiangles ae siila. Now use the two siila tiangles to find the agnitude of the acceleation. Since the deivation involves only agnitudes, oit vecto notations. The atios of the coesponding sides of siila tiangles ae equal. Thee is no need to distinguish between the sides 1 and 2 o v 1 and v 2, because the adii ae equal and the agnitudes of the velocities ae equal. = v v The object tavelled fo point P to point Q in the tie inteval t. Theefoe, the agnitude of the object s displaceent along the ac fo P to Q is The length of the ac fo point P to point Q is alost equal to. As the angle becoes vey sall, the lengths becoe oe nealy identical. d = v t = v t Substitute this value of into the fist equation. v t = v v 552 MHR Unit 5 Foce, Motion, Wok, and Enegy

3 Divide both sides of the equation by t. Recall the definition of acceleation. Substitute a into the equation fo v t. Multiply both sides of the equation by v. v = v v t a = v t v = a v a = v2 The agnitude of the acceleation of an object oving with unifo cicula otion is a = v 2 /. To deteine its diection, again inspect the tiangle foed by the velocity vectos in Figue The acceleation is changing constantly, so iagine a vecto v 2 as close to v 1 as possible. The angle θ is exteely sall. In this case, v is alost exactly pependicula to both v 1 and v 2. Since v 1 and v 2 ae tangent to the cicle and theefoe ae pependicula to the associated adii of the cicle, the acceleation vecto points diectly towad the cente of the cicle. Descibing the acceleation vecto in a typical Catesian coodinate syste would be exteely difficult, because the diection is always changing and, theefoe, the agnitude of the x- and y-coponents would always be changing. It is uch siple to specify only the agnitude of the acceleation, which is constant fo unifo cicula otion, and to note that the diection is always towad the cente of the cicle. To indicate this, physicists speak of a cente-seeking acceleation o centipetal acceleation, which is denoted as a c, without a vecto notation. Math Link Matheaticians have developed a unique syste fo defining coponents of vectos such as foce, acceleation, and velocity fo oveent on cuved paths, even when the agnitude of the velocity is changing. Any cuve can be teated as an ac of a cicle. So, instead of using the x- and y-coponents of the typical Catesian coodinate syste, the vectos ae divided into tangential and adial coponents. The tangential coponent is the coponent of the vecto that is tangent to the cuved path at the point at which the object is oentaily located. The adial coponent is pependicula to the path and points to the cente of the cicle defined by the ac o cuved section of the path. Radial coponents ae the sae as centipetal coponents. CENTRIPETAL ACCELERATION Centipetal acceleation is the quotient of the squae of the velocity and the adius of the cicle. Quantity Sybol SI unit centipetal a c 2 (etes pe second squaed) s acceleation velocity (agnitude) v (etes pe second) s adius (of cicle) (etes) Unit Analysis ete second 2 = ( ete ) 2 second ete a c = v2 ( ) 2 s = 2 s 2 = s 2 Note: The diection of the centipetal acceleation is always along a adius pointing towad the cente of the cicle. Chapte 11 Pojectiles and Cicula Motion MHR 553

4 Fg Ff F F c v v F c a c a c F c a c a c a c v F c F c Figue 11.7 A foce acting pependicula to the diection of the velocity is always equied in ode fo any object to ove continuously along a cicula path. v v Centipetal Foce Accoding to Newton s laws of otion, an object will acceleate only if a foce is exeted on it. Since an object oving with unifo cicula otion is always acceleating, thee ust always be a foce exeted on it in the sae diection as the acceleation, as illustated in Figue If at any instant the foce is withdawn, the object will stop oving along the cicula path and will poceed to ove with unifo otion, that is, in a staight line that is tangent to the cicula path on which it had been oving. Since the foce causing a centipetal acceleation is always pointing towad the cente of the cicula path, it is called a centipetal foce. The concept of centipetal foce diffes geatly fo that of othe foces that you have encounteed. It is not a type of foce such as fiction o gavity. It is, instead, a foce that is equied in ode fo an object to ove in a cicula path. A centipetal foce can be supplied by any type of foce. Fo exaple, as illustated in Figue 11.8, gavity povides the centipetal foce that keeps the Moon on a oughly cicula path aound Eath, fiction povides a centipetal foce that causes a ca to ove in a cicula path on a flat oad, and the tension in a sting tied to a ball will cause the ball to ove in a cicula path when you twil it aound. In fact, two diffeent types of foce could act togethe to povide a centipetal foce. Figue 11.8 Any foce that is diected towad the cente of a cicle can povide a centipetal foce. You can deteine the agnitude of a centipetal foce equied to cause an object to tavel in a cicula path by applying Newton s second law to a ass oving with a centipetal acceleation. 554 MHR Unit 5 Foce, Motion, Wok, and Enegy

5 Wite Newton s second law. Wite the equation descibing centipetal acceleation. Substitute into Newton s second law. Oit vecto notations because the foce and acceleation always point towad the cente of the cicula path. F = a a c = v2 F c = v2 The equation fo the centipetal foce equied to cause a ass oving with a velocity v to follow a cicula path of adius is suaized in the following box. CENTRIPETAL FORCE The agnitude of the centipetal foce is the quotient of the ass ties the squae of the velocity and the adius of the cicle. F c = v2 Quantity Sybol SI unit centipetal foce F c N (newtons) ass kg (kilogas) velocity v (etes pe second) s adius of cicula path (etes) Unit Analysis (newtons) = N = kg( ) 2 s ( ( kiloga etes second etes 2 kg = s2 = kg s 2 ) 2 = N ) MODEL PROBLEMS Centipetal Foce in a Hoizontal and a Vetical Plane 1. A ca with a ass of 2135 kg is ounding a cuve on a level oad. If the adius of cuvatue of the oad is 52 and the coefficient of fiction between the ties and the oad is 0.70, what is the axiu speed at which the ca can ake the cuve without skidding off the oad? v Ff not enough fiction continued Chapte 11 Pojectiles and Cicula Motion MHR 555

6 continued fo pevious page Fae the Poble Make a sketch of the otion of the ca and the foces acting on it. The foce of fiction ust povide a sufficient centipetal foce to cause the ca to follow the cuved oad. The agnitude of foce equied to keep the ca on the oad depends on the velocity of the ca, its ass, and the adius of cuvatue of the oad. Since is in the denoinato of the expession fo centipetal foce, as the adius becoes salle, the aount of foce equied becoes geate. Since v is in the nueato, as the velocity becoes lage, the foce equied to keep the ca on the oad becoes geate. v F c = F f Identify the Goal The axiu speed, v, at which the ca can ake the tun Vaiables and Constants Known Iplied Unknown = 2135 kg = 52 µ = 0.70 g = 9.81 s 2 F f v F N Stategy Set the fictional foce equal to the centipetal foce. Since the ca is oving on a level oad, the noal foce of the oad is equal to the weight of the ca. Substitute g fo F N. Solve fo the velocity. Substitute in the nueical values and solve. Calculations F f = F c µf N = v2 µg = v2 ( ) v 2 = µg v = µg v = (0.70)(52 ) (9.81 ) s 2 v = s 2 If the ca is going faste than 19 /s, it will skid off the oad. v = s v 19 s Validate A adius of cuvatue of 52 is a shap cuve. A speed of 19 /s is equivalent to 68 k/h, which is a high speed at which to take a shap cuve. The answe is easonable. The units cancelled popely to give etes pe second fo velocity. 556 MHR Unit 5 Foce, Motion, Wok, and Enegy

7 2. You ae playing with a yo-yo with a ass of 225 g. The full length of the sting is 1.2. You decide to see how slowly you can swing it in a vetical cicle while keeping the sting fully extended, even when the yo-yo is at the top of its swing. (a) Calculate the iniu speed at which you can swing the yo-yo while keeping it on a cicula path. (b) At the speed that you deteine in pat (a), find the tension in the sting when the yo-yo is at the side and at the botto of its swing. Fae the Poble Daw fee-body diagas of the yo-yo at the top, botto, and one side of the swing. FT Fg FT FT Fg Fg At the top of the swing, both tension and the foce of gavity ae acting towad the cente of the cicle. If the equied centipetal foce is less than the foce of gavity, the yo-yo will fall away fo the cicula path. If the equied centipetal foce is geate than the foce of gavity, the tension in the sting will have to contibute to the centipetal foce. Theefoe, the sallest possible velocity would be the case whee the equied centipetal foce is exactly equal to the foce of gavity. At the side of the swing, the foce of gavity is pependicula to the diection of the equied centipetal foce and theefoe contibutes nothing. The centipetal foce ust all be supplied by the tension in the sting. At the botto of the swing, the foce of gavity is in the opposite diection fo the equied centipetal foce. Theefoe, the tension in the sting ust balance the foce of gavity and supply the equied centipetal foce. Identify the Goal The iniu speed, v, at which the yo-yo will stay on a cicula path The tension, F T, in the sting when the yo-yo is at the side of its cicula path The tension, F T, in the sting when the yo-yo is at the botto of its cicula path continued Chapte 11 Pojectiles and Cicula Motion MHR 557

8 continued fo pevious page Vaiables and Constants Known Iplied Unknown = 225 kg = 1.2 g = 9.81 s 2 v in F T(side) F T(botto) Stategy Set the foce of gavity on the yo-yo equal to the centipetal foce and solve fo the velocity. Substitute nueical values and solve. A negative answe has no eaning in this application. Calculations F g = F c g = v2 ( ) g = v 2 v = g ( v = 9.81 ) s 2 (1.2 ) v = s 2 v =±3.431 s (a) The iniu speed at which the yo-yo can ove is 3.4 /s. Set the foce of tension in the sting equal to the centipetal foce. Inset nueical values and solve. (b): Side When the yo-yo is at the side of its swing, the tension in the sting is 2.2 N. Set the centipetal foce equal to the vecto su of the foce of tension in the sting and the gavitational foce. Solve fo the foce due to the tension in the sting. Substitute nueical values and solve. v 3.4 s F T = F c F T = v2 F T = (225 g)( 1kg 1000 g) ( s kg F T = s 2 F T 2.2 N F c = F T + F g v 2 = F T g F T = v2 + g F T = (225 g)( 1kg 1000 g) ( ) s 1.2 ( 1kg + (225 g) )(9.81 ) 1000 g s 2 (b): Botto When the yo-yo is at the botto of its swing, the tension in the sting is 4.4 N. kg 2 s 2 F T = F T = N F T 4.4 N ) 2 kg s MHR Unit 5 Foce, Motion, Wok, and Enegy

9 Validate The foce of gavity (weight) of the yo-yo is 2.2 N. At the top of the swing, the weight supplies the entie centipetal foce and the speed of the yo-yo is deteined by this value. At the side of the swing, the tension ust povide the centipetal foce and the poble was set up so that the centipetal foce had to be equal to the weight of the yo-yo, o 2.2 N. At the botto of the swing, the tension ust suppot the weight (2.2 N) and, in addition, povide the equied centipetal foce (2.2 N). You would theefoe expect that the tension would be twice the weight of the yo-yo. The units cancel popely to give newtons fo foce. PRACTICE PROBLEMS 15. A boy is twiling a 155 g ball on a 1.65 sting in a hoizontal cicle. The sting will beak if the tension eaches 208 N. What is the axiu speed at which the ball can ove without beaking the sting? 16. An electon (ass kg) obits a hydogen nucleus at a adius of at a speed of /s. Find the centipetal foce acting on the electon. What type of foce supplies the centipetal foce? 17. A stone of ass 284 g is twiled at a constant speed of 12.4 /s in a vetical cicle of adius Find the tension in the sting (a) at the top and (b) at the botto of the evolution. (c) What is the axiu speed the stone can have if the sting will beak when the tension eaches 33.7 N? 18. You ae diving a 1654 kg ca on a level oad suface and stat to ound a cuve at 77 k/h. If the adius of cuvatue is 129, what ust be the fictional foce between the ties and the oad so that you can safely ake the tun? 19. A stunt dive fo a ovie needs to ake a 2545 kg ca begin to skid on a lage, flat, paking lot suface. The foce of fiction between his ties and the concete suface is N and he is diving at a speed of 24 /s. As he tuns oe and oe shaply, what adius of cuvatue will he each when the ca just begins to skid? Centipetal Foce vesus Centifugal Foce You leaned in Chapte 5 that a centifugal foce is a fictitious foce. Now that you have leaned about centipetal foces, you can undestand oe clealy why a centifugal foce is classed as fictitious. Analyze the otion of and the foce on a peson who is iding the Round Up. Iagine that Figue 11.9 is a view of the Round Up ide fo above and at soe instant you ae at point A on the ide. At that oent, you velocity ( v ) is tangent to the path of the ide. If no foce was acting on you at all, you would soon be located at point B. Howeve, the solid cylindical stuctue of the ide exets a noal foce on you, pushing you to point C. Thee is no foce pushing you outwad, just a centipetal foce pushing you towad the cente of the cicula ide. F N C θ B (no foce) Figue 11.9 Assue that the Round Up ide is otating at a constant speed and you ae at point A. Afte a shot tie inteval, in the absence of a foce acting on you, you would ove to point B, adially outwad fo point C. A centipetal foce is equied to change the diection of you velocity and place you at point C. v A you Chapte 11 Pojectiles and Cicula Motion MHR 559

10 PROBEWARE atlphysics If you school has pobewae equipent, visit the Intenet site above and follow the links fo an in-depth activity on cicula otion. Descibing Rotational Motion When an object is constantly otating, physicists soeties find it oe convenient to descibe the otion in tes of the fequency the nube of coplete otations pe unit tie o the peiod the tie equied fo one coplete otation instead of the velocity of the object. You can expess the centipetal acceleation and the centipetal foce in these tes by finding the elationship between the agnitude of the velocity of an object in unifo cicula otion and its fequency and peiod. Wite the definition of velocity. Since peiod and fequency ae scala quantities, oit vecto notations. v = d t The distance that an object tavels in one otation is the cicufeence of the cicle. The tie inteval fo one cycle is the peiod, T. Substitute the distance and peiod into the equation fo velocity, v. Substitute the above value fo v into the equation fo centipetal acceleation, a, and siplify. Substitute the above value fo a into the equation fo centipetal foce and siplify. The fequency is the invese of the peiod. Substitute the above value fo the peiod into the equation fo centipetal acceleation and siplify. Substitute the above value fo acceleation into the equation fo the centipetal foce. d = 2π t = T v = 2π T a c = v2 a c = a c = ( 2π T 4π 2 2 T 2 a c = 4π2 T 2 ) 2 F c = a c ( 4π F c = 2 ) T 2 F c = 4π2 T 2 f = 1 T o T = 1 f a c = 4π2 ( 1 ) 2 f a c = 4π 2 f 2 F c = (4π 2 f 2 ) F c = 4π 2 f MHR Unit 5 Foce, Motion, Wok, and Enegy

11 INVESTIGATION 11-B Veifying the Cicula Motion Equation TARGET SKILLS Pefoing and ecoding Analyzing and intepeting Counicating esults You have seen the deivation of the equation fo cicula otion and solved pobles by using it. Howeve, it is always had to accept a theoetical concept until you test it fo youself. In this investigation, you will obtain expeiental data fo unifo cicula otion and copae you data to the theoy. Poble How well does the equation descibe actual expeiental esults? Equipent laboatoy balance foce pobewae o stopwatch ball on the end of a stong sting glass tube (15 c long with fie-polished ends, wapped in tape) ete stick 12 etal washes tape pape clips CAUTION Wea ipact-esistant safety goggles. Also, do not stand close to othe people and equipent while doing this activity. Pocedue Altenative A: Using Taditional Appaatus 1. Measue the ass of the ball. 2. Choose a convenient adius fo swinging the ball in a cicle. Use the pape clip o tape as a ake, as shown in the diaga at the top of the next colun, so you can keep the ball cicling within you chosen adius. 3. Measue the ass of one washe. 4. Fasten thee washes to the fee end of the sting, using a bent pape clip to hold the in place. Swing the sting at a velocity that will aintain the chosen adius. Measue the tie fo seveal evolutions and use it to calculate the peiod of otation. glass tube, wapped with tape etal washes pape clip stong sting 5. Calculate the gavitational foce on the washes (weight), which ceates tension in the sting. This foce povides the centipetal foce to keep the ball oving on the cicula path. 6. Repeat fo at least fou oe adii. Altenative B: Using Pobewae 1. Measue the ass of the ball. tetheed ball o #4 two-hole ubbe stoppe bent pape clip 2. Attach the fee end of the sting to a swivel on a foce pobe, as shown in the diaga on the next page. 3. Set the softwae to collect foce-tie data appoxiately 50 ties pe second. Stat the ball otating at constant velocity, keeping the adius at the pope value, and collect data fo at least 10 evolutions. continued Chapte 11 Pojectiles and Cicula Motion MHR 561

12 continued fo pevious page tetheed ball 4. Exaination of the gaph will show egula vaiations fo which you can calculate the peiod of one evolution, as well as the aveage foce. Foce (N) Repeat fo at least five diffeent adii. Analyze and Conclude glass tube, wapped with tape C-clap Tie (s) one evolution stong sting fishing swivel foce pobe to copute foce pobe 1. Fo each adius, calculate and ecod in you data table the velocity of the ball. Use the peiod and the distance the ball tavels in one evolution (the cicufeence of its cicula path). 2. Fo each adius, calculate and ecod in you data table v2. 3. Gaph F c against v2. Each adius will poduce one data point on you gaph. 4. Daw the best-fit line though you data points. How can you tell fo the position of the points whethe the elationship being tested, F c = v2, actually descibes the data easonably well? 5. Calculate the slope of the line. What does the slope tell you about the validity of the atheatical elationship? 6. Identify the ost likely souces of eo in the expeient. That is, what facet of the expeient ight have been ignoed, even though it could have a significant effect on the esults? Apply and Extend Based on the expeience you have gained in this investigation and the theoy that you have leaned, answe the following questions about cicula otion. Suppot you answes in each case by descibing how you would expeientally deteine the answe to the question and how you would use the equations to suppot you answe. 7. How is the equied centipetal foce affected when eveything else eains the sae but the fequency of otation inceases? 8. How is the equied centipetal foce affected when eveything else eains the sae but the peiod of otation inceases? 9. If the adius of the cicula path of an object inceases and the fequency eains the sae, how will the centipetal foce change? 10. How can you keep the velocity of the object constant while the adius of the cicula path deceases? 562 MHR Unit 5 Foce, Motion, Wok, and Enegy

13 Banked Cuves Have you eve wondeed why aiplanes tilt o bank so uch when they tun, as the aiplanes in the photogaph ae doing? Now that you have leaned that a centipetal foce is equied in ode to follow a cuved path o tun, you pobably ealize that banking the aiplane has soething to do with ceating a centipetal foce. Land vehicles can use fiction between the ties and the oad suface to obtain a centipetal foce, but ai fiction (o dag) acts opposite to the diection of the otion of the aiplane and cannot act pependicula to the diection of otion. What foce could possibly be used to povide a centipetal foce fo an aiplane? When an aiplane is flying staight and hoizontally, the design of the wings and the flow of ai ove the ceates a lift foce (L) that keeps the aiplane in the ai, as shown in Figue The lift ust be equal in agnitude and opposite in diection to the weight of the aiplane in ode fo the aiplane to eain on a level path. When an aiplane banks, the lift foce is still pependicula to the wings. The vetical coponent of the lift now ust balance the gavitational foce, while the hoizontal coponent of the lift povides a centipetal foce. The fee-body diaga on the ight-hand side of Figue helps you to see the elationship of the foces oe clealy. Figue When an aiplane follows a cuved path, it ust tilt o bank to geneate a centipetal foce. L L L path of aiplane L y L x Fg Figue When a pilot banks an aiplane, the foces of gavity and lift ae not balanced. The esultant foce is pependicula to the diection that the aiplane is flying, thus ceating a centipetal foce. Cas and tucks can use fiction as a centipetal foce. Howeve, the aount of fiction changes with oad conditions and can becoe vey sall when the oads ae icy. As well, fiction causes wea and tea on ties and causes the to wea out faste. Fo these easons, the enginees who design highways whee speeds ae high and lage centipetal foces ae equied incopoate anothe souce of a centipetal foce banked cuves. Banked cuves on a oad function in a way that is siila to the banking of aiplanes. Chapte 11 Pojectiles and Cicula Motion MHR 563

14 Figue shows you that the noal foce of the oad on a ca povides a centipetal foce when the oad is banked, since a noal foce is always pependicula to the oad suface. You can use the following logic to develop an equation elating the angle of banking to the speed of a vehicle ounding a cuve. Since an angle is a scala quantity, oit vecto notations and use only agnitudes. Assue that you wanted to know what angle of banking would allow a vehicle to ove aound a cuve with a adius of cuvatue at a speed v, without needing any fiction to supply pat of the centipetal foce. Since a ca does not ove in a vetical diection, the vetical coponent of the noal foce ust be equal in agnitude to the foce of gavity. The hoizontal coponent of the noal foce ust supply the centipetal foce. Divide the second equation by the fist and siplify. F N cos θ = F g F N cos θ = g F N sin θ = F c F N sin θ = v2 F N sin θ v2 F N cos θ = g sin θ v2 cos θ = g tan θ = v2 g Figue When you look at a coss section of a ca ounding a cuve, you can see that the only two foces in a vetical plane that ae acting on the ca ae the foce of gavity and the noal foce of the oad. The esultant foce is hoizontal and pependicula to the diection in which the ca is oving. This esultant foce supplies a centipetal foce that causes the ca to follow a cuved path. C θ C F N sin θ θ F N θ g F N cos θ Notice that the ass of the vehicle does not affect the aount of banking that is needed to dive safely aound a cuve. A seitaile and tuck could take a cuve at the sae speed as a otocycle without elying on fiction to supply any of the equied centipetal foce. Apply what you have leaned about banking to the following pobles. 564 MHR Unit 5 Foce, Motion, Wok, and Enegy

15 Conceptual Poble A conical pendulu swings in a cicle, as shown in the diaga. Show that the fo of the equation elating the L θ angle that the sting of the h pendulu akes with the F T vetical to the speed of the pendulu bob is identical to the equation fo the banking of cuves. The pendulu has g a length L, an angle θ with the vetical, a foce of tension F T in the sting, a weight g, and swings in a cicula path of adius. The plane of the cicle is a distance h fo the ceiling fo which the pendulu hangs. MODEL PROBLEM Banked Cuves and Centipetal Foce Canadian Indy acing ca dive Paul Tacy set the speed ecod fo tie tials at the Michigan Intenational Speedway (MIS) in the yea Tacy aveaged k/h in the tie tials. The ends of the 3 k oval tack at MIS ae banked at 18.0 and the adius of cuvatue is 382. (a) At what speed can the cas ound the cuves without needing to ely on fiction to povide a centipetal foce? (b) Did Tacy ely on fiction fo soe of his equied centipetal foce? Fae the Poble The noal foce of a banked cuve povides a centipetal foce to help cas tun without equiing an excessive aount of fiction. Fo a given adius of cuvatue and angle of banking, thee is one speed at which the noal foce povides pecisely the aount of centipetal foce that is needed. Identify the Goal (a) The speed, v, fo which the noal foce povides exactly the equied aount of centipetal foce fo diving aound the cuve (b) Whethe Tacy needed fiction to povide an additional aount of centipetal foce Vaiables and Constants Known Iplied Unknown = 382 g = 9.81 s 2 v θ = 18.0 v PT = k h continued Chapte 11 Pojectiles and Cicula Motion MHR 565

16 Stategy Wite the equation that elates angle of banking, speed, and adius of cuvatue, and solve fo speed, v. Substitute the nueical values and solve. Calculations tan θ = v2 g v 2 = g tan θ v = g tan θ v = (382 ) (9.81 ) s 2 (tan 18.0 ) v = s 2 v = s (a) A vehicle diving at 34.9 /s could ound the cuve without needing any fiction fo centipetal foce. Convet the velocity in /s into k/h. v 34.9 s v = ( s v = k h )( )( ) 3600 s 1k h 1000 v 126 k h (b) Tacy was diving thee ties as fast as the speed of 126 k/h at which the noal foce povides the needed centipetal foce. Paul had to ely on fiction fo a lage pat of the needed centipetal foce. Validate An angle of banking of 18 is vey lage copaed to the banking on noal highway cuves. You would expect that it was designed fo speeds uch highe than the highway speed liit. A speed of 126 k/h is highe than highway speed liits. PRACTICE PROBLEMS 20. An enginee designed a tun on a oad so that a 1225 kg ca would need 4825 N of centipetal foce when tavelling aound the cuve at 72.5 k/h. What is the adius of cuvatue of the oad? 21. A ca exits a highway on a ap that is banked at 15 to the hoizontal. The exit ap has a adius of cuvatue of 65. If the conditions ae exteely icy and the dive cannot depend on any fiction to help ake the tun, at what speed should the dive tavel so that the ca will not skid off the ap? 22. An icy cuve with a adius of cuvatue of 175 is banked at 12. At what speed ust a ca tavel to ensue that it does not leave the oad? 23. An enginee ust design a highway cuve with a adius of cuvatue of 155 that can accoodate cas tavelling at 85 k/h. At what angle should the cuve be banked? 566 MHR Unit 5 Foce, Motion, Wok, and Enegy

17 You have studied just a few exaples of cicula otion that you obseve o expeience nealy evey day. Although you aely think about it, you have been expeiencing seveal fos of cicula otion evey inute of you life. Siply existing on Eath s suface places you in unifo cicula otion as Eath otates. In addition, Eath is evolving aound the Sun. In the next chapte, you will apply any of the concepts you have just leaned about foce and otion to the otion of planets, oons, and stas, as well as to atificial satellites Section Review 1. K/U Define unifo cicula otion and descibe the type of acceleation that is associated with it. 2. K/U Study the diaga in Figue 11.6 on page 552. Explain what appoxiation was ade in the deivation that equies you to iagine what occus as the angle becoes salle and salle. 3. C What ae the benefits of using the concept of centipetal acceleation athe than woking on a taditional Catesian coodinate syste? 4. K/U Explain how centipetal foce diffes fo coon foces, such as the foces of fiction and gavity. 5. K/U If you wee swinging a ball on a sting aound in a cicle in a vetical plane, at what point in the path would the sting be the ost likely to beak? Explain why. In what diection would the ball fly when the sting boke? 6. C Explain why gavity does not affect cicula otion in a hoizontal plane, and why it does affect a siila otion in a vetical plane. 7. C Descibe thee exaples in which diffeent foces ae contibuting the centipetal foce that is causing an object to follow a cicula path. 8. MC When aiplane pilots ake vey shap tuns, they ae subjected to vey lage g foces. Based on you knowledge of centipetal foce, explain why this occus. 9. C A centifugal foce, if it existed, would be diected adially outwad fo the cente of a cicle duing cicula otion. Explain why it feels as though you ae being thown outwad when you ae iding on an auseent pak ide that causes you to spin in a cicle. 10. K/U On a highway, why ae shap tuns banked oe steeply than gentle tuns? Use vecto diagas to claify you answe. 11. I Iagine that you ae in a ca on a ajo highway. When going aound a cuve, the ca stats to slide sideways down the banking of the cuve. Descibe conditions that could cause this to happen. UNIT PROJECT PREP Pats of you catapult launch echanis will ove in pat of a cicle. The payload, once launched, will be a pojectile. How will you launch echanis apply enough centipetal foce to the payload to ove it in a cicle, while still allowing the payload to be eleased? How will you ensue that the payload is launched at the optiu angle fo axiu ange? What data will you need to gathe fo a launch to poduce the ost coplete possible analysis of the payload s actual path and flight paaetes? Chapte 11 Pojectiles and Cicula Motion MHR 567

18 CAREERS IN PHYSICS Physics Goes to the Fai! Thee s no backing down. You ve paid fo you ticket, and you e in you seat with the estaint ba in place. You heat is pounding as you look at the tack in font of you. You e alost convinced you have nothing to woy about, but in the back of you ind, a woy flickes. Is this olle coaste safe? Well, est easy. You ll be back on the faigound in no tie, thanks to the physics of olle coaste design and the vigilance of Canada s povincial public safety inspectos. Rolle coastes have not always been safe. Ealy tack designs eployed cicula loops. When coaste cas enteed these loops at high speeds, they encounteed excessive noal foces, putting ides at isk of whiplash and boken bones. When designes tied to coect the poble by deceasing the speed at which the cas enteed the loops, the cas becoe pojectiles, unable to ake it though the loop without falling off the tack. Designes solved these pobles with the clothoid loop. Clothoid loops ae shaped like tea dops and have a constantly changing adius, whee the adius at the botto of the loop is lage than the adius at the top. The lage adius at the botto allows the cas to ente the loop at high speeds. As the cas clib the loop, they ae affected by gavity, but ae still able to ake it though the loop and aintain contact with the tack because of the salle adius at the top. Once designes ae confident they have ioned out all the kinks in auseents ides, it is the safety inspecto s tun to ake sue the ides ae safe fo the public. Alfed Bya is the Chief Public Safety Inspecto with the New Bunswick govenent, and has woked fo the past twenty-two yeas as an inspecto, testing and inspecting elevatos, and auseent ides at paks like Cystal Palace in Moncton. Bya studied constuction electicity at the New Bunswick Counity College in Saint John, and then becae a tained elevato echanic befoe joining the govenent as a safety inspecto. Befoe he could inspect auseent ides, Bya had to visit thee paks to gain onsite expeience. Safety inspectos ust be tained in echanics, especially the echanics of hydaulics. In ode to ake sue ides pass Canada s safety standads, Bya inspects ide tacks to ake sue they ae not won, cats to ake sue they ae attached and intact, and hydaulic hoses to ake sue they ae in good woking ode. Standads also equie that nondestuctive testing be done on all auseent ides. Nondestuctive testing is testing that does not destoy the pat o ateial being tested. An ipotant test done on auseent ides is agnetic paticle inspection, which uses agnetic fields and sall agnetic paticles, such as ion filings, to detect flaws in feoagnetic coponents. Accoding to Bya, safety inspectos not only have to be good with pats, they have to be good with people too. Being able to inteact diploatically with auseent pak ownes and opeatos akes his job a lot easie. As fo any equieent that inspectos be ide fanatics, Bya says thee isn t one. Bya will only ide the Feis wheel. He says, It s all the thill I need! Going Futhe 1. Most olle coaste tacks end with a seies of paabolic hills that culinate with a shap, steep dop. When ides descend the shap dop they biefly undego fee fall. What is fee fall? What kinds of foces ae involved in fee fall? 568 MHR Unit 5 Foce, Motion, Wok, and Enegy