3.3 Centripetal Force

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Ernest Jordan
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1 3.3 Centipetal Foce Think of a time when ou wee a passenge in a ca going aound a shap cue at high speed (Figue 1). If the ca wee going fast enough, ou might feel the side of the ca doo pushing on ou side. If ou look closel at Figue 1, ou can see that the oad is banked. In this case, the bank assists the foce of fiction to help make the ca and passenges moe in a cicle moe safel fo a gien speed. A foce is also equied on objects when ou pla cack the whip when ice skating, o when ou swing a o-o on a sting aound ou head. If ou wee to let go of the cackthe-whip line, ou would no longe feel the foce and would continue in a staight line in the diection ou wee moing. The implications of the foces causing cicula motion is one eason highwa exit amps often hae banked cues like the one in Figue 1. CAREER LINK Figue 1 Passenges in a ca that is going aound a shap cue at high speed will expeience a stong foce pushing them towad the cente of the cue. Foces That Cause Centipetal Acceleation As ou leaned in Section 3.2, an object moing with unifom cicula motion has a centipetal acceleation of magnitude a c 2 Fom Newton s second law, we know that foces cause acceleations. So, fo an object moing with unifom cicula motion, we hae Inestigation Simulating Unifom Cicula Motion (page 13) You hae leaned about the pinciples elated to objects in unifom cicula motion. In this inestigation, ou will use a simulation pogam to eif the elationship between fequenc and aiables such as foce, mass, speed, adius, and peiod. SF ma c F c m 2 whee F c is the magnitude of the net foce equied to make an object of mass m tael with a constant speed in a cicle of adius. Centipetal acceleation is diected towad the cente of the cicle, so the centipetal foce must also be diected towad the cente of the cicle accoding to Newton s second law. To futhe analze the foces inoled in unifom cicula motion, suppose a peson is twiling a o-o on a sting so that the o-o moes in a cicle. To keep the situation simple, assume this demonstation is being pefomed b an astonaut in deep space, whee gaitational foces ae negligible (Figue 2). Since gaitational foces ae negligible, the onl foce on the o-o comes fom the sting. Accoding to Newton s second law and the equation F c m 2 a foce of magnitude m 2 causes this acceleation. And since the foce comes fom the tension F T in the sting, F T m Chapte 3 Unifom Cicula Motion NEL

2 Inestigation F T Figue 2 An astonaut in deep space twils a o-o on a sting. In deep space all gaitational foces ae negligible, so the onl foce on the o-o is due to the tension in the sting. When F T m 2, the o-o will moe with unifom cicula motion. If the sting beaks, the o-o will moe along a staight line, obeing Newton s fist law. Fo the o-o to tael in a cicle, the tension must hae this alue. In othe situations, the foce might be due to gait, fiction, o some othe souce. The net foce that causes centipetal acceleation is called the centipetal foce, F c. Without such a foce, the object cannot moe in unifom cicula motion. What happens if the sting in Figue 2 suddenl beaks? Afte the sting beaks (Figue 2), the foce on the o-o is zeo. Accoding to Newton s fist law, the o-o will then moe awa in a staight-line path with a constant elocit. The o-o does not moe adiall outwad, no does it emembe its cicula tajecto. The onl wa the o-o can moe in a cicle is when thee is a foce that makes it do so. Befoe the sting boke, the sting poided that foce. The following Tutoial models how to sole poblems that inole diffeent centipetal foces. Analzing Unifom Cicula Motion (page 136) An object in cicula motion at a constant speed is constantl undegoing centipetal acceleation diected towad the cente of the cicle. This inestigation will gie ou an oppotunit to obsee an object in unifom cicula motion and collect data to descibe elationships between the object, its mass, and the adius of its path. centipetal foce (F c ) the net foce that causes centipetal acceleation Tutoial 1 Soling Poblems Related to Centipetal Foce In this Tutoial, ou will sole fo diffeent aiables in situations in which an object is moing with unifom cicula motion. Sample Poblem 1: Detemining Centipetal Acceleation and Identifing the Centipetal Foce Suppose a bug is sitting on the edge of a hoizontal DVD. The bug has a mass of.0 g, and the DVD has a adius of 6.0 cm. The DVD is spinning such that the bug taels aound its cicula path thee times pe second. Calculate the centipetal acceleation of the bug and the net foce on the bug. Also, identif the foce o foces esponsible fo the centipetal foce. Gien: m.0 g kg; 6.0 cm m; t 1.0 s Requied: a c ; F c ; oigin of F c Analsis: Daw an FBD to detemine the foce o foces esponsible fo the centipetal foce. Calculate the speed of the DVD using Dd. Then use the speed in the equation fo centipetal acceleation, a c 2, and calculate the centipetal foce, SF c ma c ; cicumfeence of a cicle 2p. Solution: The FBD is shown in Figue 3. The bug s acceleation is in the hoizontal plane, so the bug s acceleation in the etical plane is zeo. Theefoe, the nomal foce,, and the foce of gait,, must cancel:. The bug is moing with unifom cicula motion, so we know a thid foce must poide the foce equied to poduce the centipetal acceleation, a c. This foce keeps the bug fom slipping elatie to the DVD, so the foce is the foce of static fiction. To make the bug moe with unifom cicula motion, the foce of static fiction must be diected towad the cente of the cicle. F g Figue 3 NEL 3.3 Centipetal Foce 121

The bug taels aound the cicle thee times in 1.0 s, so it taels a distance equal to thee times the cicumfeence each second: Dd 132 12p2 16p2 10.060 m2 1.0 s 1.13 m/s 1one exta digit caied2 a c 2 11.

3 The bug taels aound the cicle thee times in 1.0 s, so it taels a distance equal to thee times the cicumfeence each second: Dd p2 16p m2 1.0 s 1.13 m/s 1one exta digit caied2 a c m/s m a c 21.3 m/s 2 1one exta digit caied2 The foce is SF c ma c kg m/s 2 2 SF c 0.11 N Statement: The centipetal acceleation of the bug is 21 m/s 2, and the total foce on the bug is 0.11 N. The centipetal foce on the bug is static fiction. Sample Poblem 2: Calculating Speed Using Appaent Weight A olle coaste ca is at the lowest point on its cicula tack. The adius of cuatue is 22 m. The appaent weight of one of the passenges in the olle coaste ca is 3.0 times he tue weight. Detemine the speed of the olle coaste. Gien: 22 m; 3.0 Requied: Analsis: Daw an FBD fo the scenaio. The unifom cicula motion is in the etical plane in this poblem. At the lowest point on the cicula tack, the foces on the peson ae gait and the nomal foce. The nomal foce is the appaent weight. Since the olle coaste is at the low point of the tack, the nomal foce is diected towad the cente of the cicula ac defi ned b the tack (up), and gait is downwad, awa fom the cente. Appl Newton s second law to elate the nomal foce to the speed of the olle coaste. Then appl the equation fo cicula motion, F c m 2. Solution: Figue 4 shows the FBD. SF F c 2 m m m "2.0g " m m/s m/s F g Figue 4 Statement: The speed of the olle coaste is 21 m/s. Sample Poblem 3: Calculating Speed on a Banked Tun A ca making a tun on a d, banked highwa amp is expeiencing fiction (Figue ). The coeffi cient of static fiction between the ties and the oad is Detemine the maximum speed at which the ca can safel negotiate a tun of adius m with a banking angle of Figue x Gien: m S 0.60; m; u Requied: Analsis: In two sepaate diagams, daw the ecto components of the fictional foce and the ecto components of the nomal foce. The ca is moing with unifom cicula motion, so the centipetal foce on the ca is F c m 2 diected towad the cente of the cicula path. In Figue, the cente is m to the left of the ca. Use the diection towad the cente as positie fo the x-diection, and use up as positie fo the -diection. We hae two unknown quantities: the speed of the ca and the nomal foce. We can get two equations b appling Newton s second law along the etical and hoizontal diections. We can then sole fo and ; m S 122 Chapte 3 Unifom Cicula Motion NEL

4 Solution: The ecto component diagams ae shown in Figue 6. cos x sin x sin cos Figue 6 The ecto components fo the foce of static fiction The ecto components fo the nomal foce Vetical components of foce: The ca is not slipping up o down the incline, so the acceleation along is zeo. The total foce along must be zeo. Fom Figue 6 and 6, SF 0 SF 1 cos u 2 sin u 2 1 cos u 2 sin u 2 0 m S cos u 2 m S sin u cos u 2 m S sin u2 cos u 2 m S sin u Hoizontal components of foce: The total foce along the hoizontal diection poides the centipetal acceleation. Fom Figue 6 and 6, SF x sin u 1 cos u m S SF x sin u 1 m S cos u ma c sin u 1 m S cos u m 2 F N 1sin u 1 m S cos u2 Soling fo then gies 1sin u 1 m S cos u2 Å m Inset the esult fo the nomal foce: a Å cos u 2 m S sin u b a 1 sin u 1 m S cos u2 b m g a sin u 1 m S cos u Å cos u 2 m S sin u b sin cos m/s m2 a Å cos sin b 49 m/s Statement: The maximum speed at which the ca can safel negotiate a tun with a adius of m and with a banking angle of is 49 m/s. It is eall the hoizontal components of the nomal foce and the foce of fiction that contibute to the net foce and the acceleation. (Note: This is the maximum speed the ca can go but not the speed the ca should go.) Pactice 1. A model aiplane with a mass of kg pulls out of a die. The bottom of the die is a cicula ac with a adius of 2.6 m. At the bottom of the ac, the plane s speed is a constant 21.7 m/s. Detemine the magnitude of the upwad lift on the plane s wings at the bottom of the ac. K/U T/I A [ans:.9 N] 2. A cued oad with a adius of 40 m in the hoizontal plane is banked so that the cas can safel naigate the cue. Calculate the banking angle fo the oad that will allow a ca taelling at 97 km/h to make it safel aound the cue when the oad is coeed with black ice. (Assume no fiction.) K/U T/I A [ans: 9.38] 3. A 2.00 kg mass is spinning hoizontall in a cicle on a ituall fictionless suface. It completes.00 eolutions in 2.00 s. The mass is attached to a sting 4.00 m long. Calculate the magnitude of the tension in the sting. Ai esistance is negligible. K/U T/I A [ans: N] 4. A ban swallow chasing a moth is fling in a etical loop of adius 10 m. At the top of the loop, the etical foce exeted b the ai on the bid is zeo. At what speed is the swallow fling at this point? K/U T/I A [ans: 38 m/s]. The highwa amp in Sample Poblem 3 was d. Now suppose the highwa is wet o coeed in ice. Pedict how the maximum speed will change. Test ou pediction b using the alue 0.2 fo the coefficient of static fiction and detemining the maximum speed. K/U T/I A [ans: 36 m/s] NEL 3.3 Centipetal Foce 123

5 3.3 Reiew Summa An object moing with unifom cicula motion expeiences a net foce diected towad the cente of the object s cicula path. The net foce that causes unifom cicula motion is the centipetal foce, which ma compise one o moe othe foces such as gait, the nomal foce, o tension. Combine the equation fo Newton s second law with the equations fo centipetal acceleation to calculate the magnitude of the net foce: SF ma c ; F c m 2. Questions 1. The tack nea the top of a olle coaste has a cicula shape with a diamete of 24 m foming a hill. When ou ae at the top, ou feel as if ou weight is onl one-thid ou tue weight. Calculate the speed of the olle coaste as it olls oe the top of the hill. T/I A 2. A ca with a mass of kg is taelling oe the top of a hill, as shown in Figue 7. The hill s cuatue has a adius of 40.0 m, and the ca is taelling at 1 m/s. T/I C A Figue 7 Daw an FBD. Detemine the magnitude of the nomal foce between the hill and the ca at the top of the hill. (c) Detemine the speed equied to make the die feel weightless at the top of the hill. 3. A ciil enginee is designing a banked cue on a highwa. The banked cue is designed to allow the cas to moe safel in a hoizontal cicle. What will happen to the maximum speed of a ca on the cue when the following changes ae made? Explain ou easoning, consideing each change sepaatel. K/U T/I A The banking angle between the oad and the hoizontal is inceased. The coefficient of fiction between the ties and the oad is lage. (c) A heaie ca is used. 4. A ca moes in a hoizontal cicle on a test tack with a adius of m. The coefficient of static fiction between the ties and the oad is Daw an FBD, and calculate the maximum speed of the ca. T/I C A. Conside a banked cue on an exit amp fo a highwa in the middle of winte when the oad suface is coeed with e slippe ice. K/U T/I A How does the banking angle of the oad help dies make it safel aound the cue? What foce (o component of foce) is esponsible? Explain ou easoning. Explain wh dies must go much moe slowl unde these cicumstances. (c) A student claims, If banking angles help dies safel naigate cued sections of oad, wh not make the banking angles significantl lage? Identif one poblem that might occu if this suggestion wee used. Justif ou answe. 6. An ai puck with a mass of 0.26 kg is tied to a sting and moes at a constant speed in a cicle of adius 1.2 m. The othe end of the sting goes though a hole in the ai table and staight down to a suspended mass of 0.68 kg, which hangs at est (Figue 8). Calculate the speed of the ai puck. K/U T/I A m kg Figue 8 m kg 124 Chapte 3 Unifom Cicula Motion NEL

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