7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o evolutions pe unit time. Peiod and fequency ae ecipocals of each othe. In othe wods, 1 1 and f f Since peiod is a measue of time, its SI unit is the second, while the unit fo fequency is the ecipocal of this, o 1/second. Anothe way of witing 1/second is with the unit hetz (Hz). When an object spins in a cicle, the distance it tavels in one evolution is the cicumfeence of the cicle, 2. he time it takes fo one evolution is the peiod,. heefoe, 2 (adius) speed o v peiod 2p whee v is called the linea o tangential speed because at any given time, the velocity is tangent to the cicle as shown in the diagam. Although the velocity is constant in magnitude (speed), it is always changing diection. 83
Centipetal Acceleation and Centipetal Foce An object can move aound in a cicle with a constant speed yet still be acceleating because its diection is constantly changing. his acceleation, which is always diected in towad the cente of the cicle, is called centipetal acceleation. he magnitude of this acceleation is witten as 1linea speed2 2 centipetal acceleation o a c adius If a mass is being acceleated towad the cente of a cicle, it must be acted upon by an unbalanced foce that gives it this acceleation. his foce, called the centipetal foce, is always diected inwad towad the cente of the cicle. he magnitude of this foce is witten as centipetal foce (mass)(centipetal acceleation) v 2 o F c ma c mv 2 he units fo centipetal acceleation and centipetal foce ae m/s 2 and N, espectively. Solved Examples Example 1: Afte closing a deal with a client, Kent leans back in his swivel chai and spins aound with a fequency of 0.5 Hz. What is Kent s peiod of spin? Given: f 0.5 Hz Unknown:? Oiginal equation: 1 1 Solve: 2 s f 0.5 Hz 1 f Example 2: Cutis favoite disco ecod has a scatch 12 cm fom the cente that makes the ecod skip 45 times each minute. What is the linea speed of the scatch as it tuns? 84 Cicula Motion Solution: he ecod makes 45 evolutions evey 60. seconds, so find the peiod of the ecod fist. 60. s 45 ev 1.3 s Given: 12 cm Unknown: v? 1.3 s Oiginal equation: v 2p 2p112 cm2 Solve: v 58 cm/s 1.3 s 2p
Example 3: Missy s favoite ide at the opsfield Fai is the oto, which has a adius of 4.0 m. he ide takes 2.0 s to make one full evolution. a) What is Missy s linea speed on the oto? b) What is Missy s centipetal acceleation on the oto? (Read about Ameica s oldest agicultual fai at http://www.topsfieldfai.og) Solution: he ide takes 2.0 s to make one full evolution, so the peiod is 2.0 s. a. Given: 4.0 m Unknown: v? 2.0 s Oiginal equation: v 2p 2p14.0 m2 Solve: v 13 m/s 2.0 s b. Given: v 13 m/s Unknown: a c? 4.0 m Oiginal equation: a c v 2 113 m>s2 2 Solve: a c 42 m/s 2 4.0 m 2p v 2 Example 4: Captain Chip, the pilot of a 60,500-kg jet plane, is told that he must emain in a holding patten ove the aipot until it is his tun to land. If Captain Chip flies his plane in a cicle whose adius is 50.0 km once evey 30.0 min, what centipetal foce must the ai exet against the wings to keep the plane moving in a cicle? Solution: Fist, convet km to m and min to s. 50.0 km 5.00 10 4 m 30.0 min 1.80 10 3 s Befoe solving fo the centipetal foce, find the speed of the aiplane. Given: 1.80 10 3 s Unknown: v? 5.00 10 4 m Oiginal equation: v 2p 2p15.00 10 4 m2 Solve: v 175 m/s 1.80 10 3 s Use this speed to solve fo the centipetal foce. Given: m 60,500 kg Unknown: F c? v 175 m/s Oiginal equation: F c 5.00 10 4 m mv 2 160,500 kg21175 m>s2 2 Solve: F c 3.71 10 4 N 5.00 10 4 m 2p mv 2 Cicula Motion 85
Pactice Execises Execise 1: Maianne puts he favoite Backsteet Boys disc in he CD playe. If it spins with a fequency of 1800 evolutions pe minute, what is the peiod of spin of the compact disc? 1/f 1/(1800 min 1 ) (5.6 10 4 min)(60 s/min) 0.034 s 0.034 s Execise 2: Hamlet, a hamste, uns on his execise wheel, which tuns aound once evey 0.5 s. What is the fequency of the wheel? f 1/ 1/(0.5 s) 2 Hz 2 Hz Execise 3: A sock stuck to the inside of the clothes dye spins aound the dum once evey 2.0 s at a distance of 0.50 m fom the cente of the dum. a) What is the sock s linea speed? b) If the dum wee twice as wide but continued to tun with the same fequency, would the linea speed of a sock stuck to the inside be faste than, slowe than, o the same speed as you answe to pat a? a) v 2 / 2 (0.50 m)/(2.0 s) 1.6 m/s b) Faste (twice as much as pat a) a. b. 1.6 m/s faste Execise 4: What is the adius of an automobile tie that tuns with a fequency of 11 Hz and has a linea speed of 20.0 m/s? 1/f 1/11 Hz 0.091 s v/2 (20.0 m/s)(0.091 s)/2 0.29 m 0.29 m 86 Cicula Motion
Execise 5: ony twils a ound piece of pizza dough ovehead with a fequency of 60 evolutions pe minute. a) Find the linea speed of a stay piece of peppeoni stuck on the dough 10. cm fom the pizza s cente. b) In what diection will the peppeoni move if it flies off while the pizza is spinning? Explain the eason fo you answe. a) v 2 / 2 (0.10 m)/(1.0 s) 0.63 m/s b) It will go tangent to the pizza, because the velocity is tangent to a cicle. a. b. 0.63 m/s tangent to the pizza Execise 6: Eath tuns on its axis appoximately once evey 24 hous. he adius of Eath is 6.38 10 6 m. a) If some astonomical catastophe suddenly bought Eath to a sceeching halt (a physical impossibility as fa as we know), with what speed would Eath s inhabitants who live at the equato go flying off Eath s suface? b) Because Eath is solid, it must tun with the same fequency eveywhee on its suface. Compae you linea speed at the equato to you linea speed while standing nea one of the poles. a) v 2 / 2 (6.38 10 6 m)/(86 400 s) 464 m/s b) You speed is geate at the equato because you must tun though a lage distance in the same amount of time. a. b. 464 m/s geate at the equato Execise 7: Jessica is iding on a mey-go-ound on an oute hose that sits at a distance of 8.0 m fom the cente of the ide. Jessica s siste, Julie, is on an inne hose located 6.0 m fom the ide s cente. he mey-go-ound tuns aound once evey 40.0 s. a) Explain which gil is moving with the geate linea speed. b) What is the centipetal acceleation of Julie and he hose? a) Since v 2 / and is the same fo both gils, Jessica has the geate linea speed because she is fathe out fom the cente. Jessica: v 2 / 2 (8.0 m)/(40.0 s) 1.3 m/s Julie: v 2 / 2 (6.0 m)/(40.0 s) 0.94 m/s b) a c v 2 / (0.94 m/s) 2 /(6.0 m) 0.15 m/s 2 a. Jessica 1.3 m/s; Julie 0.94 m/s b. 0.15 m/s 2 Cicula Motion 87
Execise 8: A cement mixe of adius 2.5 m tuns with a fequency of 0.020 Hz. What is the centipetal acceleation of a small piece of died cement stuck to the inside wall of the mixe? v 2 / 2 (2.5 m)/(50. s) 0.314 m/s a c v 2 / (0.314 m/s) 2 /(2.5 m) 0.039 m/s 2 0.039 m/s 2 Execise 9: A popula tick of many physics teaches is to swing a pail of wate aound in a vetical cicle fast enough so that the wate doesn t spill out when the pail is upside down. If M. Lowell s am is 0.60 m long, what is the minimum speed with which he can swing the pail so that the wate doesn t spill out at the top of the path? v 2a 2110.0 m>s 2 c 210.60 m2 2.5 m/s 2.5 m/s Execise 10: o test thei stamina, astonauts ae subjected to many igoous physical tests befoe they fly in space. One such test involves spinning the astonauts in a device called a centifuge that subjects them to acceleations fa geate than gavity. With what linea speed would an astonaut have to spin in ode to expeience an acceleation of 3 g s at a adius of 10.0 m? (1 g 10.0 m/s 2 ) v 2a 2130.0 m>s 2 c 2110.0 m2 17.3 m/s 17.3 m/s 88 Cicula Motion
Execise 11: At the Femilab paticle acceleato in Batavia, Illinois, potons ae acceleated by electomagnets aound a cicula chambe of 1.00-km adius to speeds nea the speed of light befoe colliding with a taget to poduce enomous amounts of enegy. If a poton is taveling at 10% the speed of light, how much centipetal foce is exeted by the electomagnets? (Hint: he speed of light is 3.00 10 8 m/s, m p 1.67 10 27 kg) (Read about the Femilab paticle acceleato at http://www.fnal.gov) F c mv 2 / (1.67 10 27 kg)(3.00 10 7 m/s) 2 /(1.00 10 3 m) 1.50 10 15 N 1.50 10 15 N Execise 12: Roxanne is making a stawbey milkshake in he blende. A tiny, 0.0050-kg stawbey is apidly spun aound the inside of the containe with a speed of 14.0 m/s, held by a centipetal foce of 10.0 N. What is the adius of the blende at this location? mv 2 /F c (0.0050 kg)(14.0 m/s) 2 /(10.0 N) 0.098 m 0.098 m Cicula Motion 89