AP Phyic Centipetal Acceleation All of ou motion tudie thu fa hae dealt with taight-line tuff. We haen t dealt with thing changing diection duing thei tael. Thi type of motion i called angula motion. A pecial cae of angula motion i cicula motion (when thing tael aound in cicle). We know that elocity and acceleation ae both ecto. Recall that ecto hae both magnitude and diection. Change eithe one and you hae changed the motion. We know fom the fit law that object in motion tay in motion unle an outide foce act to change that motion. We e tudied all about foce that change the magnitude of the elocity. Now we ll deal with foce that change the diection of a motion. Cicula Motion: Object that hae cicula motion ae moing in a cicula path aound a cental point which i called the axi o cente of pin. If the axi i within the body, we ay it i otating. If the axi i outide the body, then the object i eoling. The eath otate aound it axi, which caue day and night. The eath eole aound the un, which caue the fou eaon duing the yea (the time it take the eath to eole aound the un one time). An object undegoing cicula motion ha an angula elocity and undegoe an angula diplacement. Angula diplacement can be meaued in eeal unit: eolution (the numbe of time the thing make a cicle), degee, o adian. Thee ae the mot common unit. We will deal only with eolution. The angula elocity i the ate of change in angula diplacement. angula diplacement angula elocity time The Geek lowe cae lette omega i ued a the ymbol fo angula elocity. Hee i the equation: " #!! " # t t The unit fo angula elocity we will ue i: pm e min. A ecod i otating with an angula elocity of 45 pm. If it otate fo 35 econd, how many otation doe it make?! e # 45 ( 35 ) 1 min $ "! " t % & 6 e t min ' 60 ( 164
Linea Speed and Cicula Motion: When an object i otating all point on the object hae the ame angula elocity. But at the ame time each point i alo tacing out a cicle. If you diide the time it take to make one eolution into the cicumfeence of the cicle that i taced out you get the peed of the point. We call thi the linea o tangential peed. The linea elocity i not contant a it diection change eey intant. Calin fathe i coect about the two point on a ecod. But, unlike Calin, thi hould make pefect ene to you, the ophiticated adanced tudent of phyic. The futhe a point i fom the cente of pin, the lage will be it linea peed. Thi i tue fo the eath a well. You hae a much lage linea peed if you ae at the equato than you would hae if you wee at one of the pole. We can figue out the linea peed of a point uing the angula elocity. Speed i ditance diided by time, the time it take to make one eolution i: "! t! Let the angula diplacement be 1 eolution: t "! 1 t Thi i the peiod of the otation, the time it take to make " " one eolution. So we now hae the time. Hee i the equation fo peed: x t The ditance i the cicumfeence of the cicula path,!. We plug thi in and alo the alue we found fo the time. 165
d! t 1! " " So the linea peed i gien by:! " Foce and Cicula Motion: A ball i attached to a ting. The ball i whiled aound and aound. You can pictue thi ight? In the dawing, we ae looking down on the path of the ball fom aboe. w Phyic Kahuna Comment: If the ting beak at the point hown, what will be the path of the ball? The Phyic Kahuna will gie you thee choice: A B C In ode fo an object to undego cicula motion, a foce mut act. Pictue an object that ha ome elocity. What will happen to it if no foce act on it? Well, accoding to the fit law, it will continue to moe with a contant elocity. It will follow a taight-line path. To make it change diection a foce mut act on it. In ode to make it change diection contantly, a foce mut act on it contantly. What i the diection of the foce needed to do thi? Well, when you pin omething in a cicle, what do you hae to do? You jut pull it towad the cente a you go aound and aound. The object get acceleated towad the cente. We call thi the centipetal acceleation. The equation fo the centipetal acceleation i: ac 166
a c i the centipetal acceleation, i the linea o tangential peed, and i the adiu of the cicula path. Thi equation will be poided to you fo the AP Phyic Tet. A otating object ha a linea peed of 1.5 m/. It undegoe a centipetal acceleation of 3.6 m/. What i the adiu of the ma' cicula motion? ac ac! "! m " # 1 $ # 1.5 0.6 m $ # $ % & m 3.6 # $ % & The foce that bing about thi acceleation i called the centipetal foce. It diection i alo towad the cente of the cicula path. Centipetal mean "cente eeking". The centipetal foce change the diection of the object elocity ecto. Without it, thee would be no cicula path. The centipetal foce i meely a conenient name fo the net foce that i towad the cente. It i alway caued by omething it could be caued by the foce of gaity, the eaction foce between the contol uface of an aiplane with the ai, &tc. When you otate a ball aound you head in a cicle, the centipetal foce i upplied by the tenion in the ting. What i the ouce of the centipetal foce that caue a aceca to tael in a cicula path on the acetack? The foce i bought about by the tie puhing on the acetack. The fiction between the oad and the tie i ey impotant, o ace tie ae deigned to maximize fiction. What i the ouce of the centipetal foce equied to make the eath eole aound the un? Thi i whee the apple falling on Newton toy fit in. Befoe Newton no one could explain the obit of the planet and moon. Newton, the toy goe, wa elaxing unde an apple tee pondeing the poblem of the moon obit. He knew that thee had to be a foce acting on the moon to acceleating it towad the eath, but had no idea what wa the ouce of the foce. Then he aw an apple fall and the imple olution tuck him like the old thundebolt. Jut a the eath gaity eached out and made the apple fall, o it eached out and made the moon fall. Thu, the foce that keep the planet and moon following thei obital path i gaity. The AP Tet equation heet will not gie you the equation fo centipetal foce. It doe gie you the equation fo centipetal acceleation. It alo gie you the equation fo the econd law. Uing thee two equation you can eaily deie the fomula fo centipetal foce. Hee how to do it: F ma o plug in the alue of the centipetal acceleation: ac 167
! " m F ma m# $ FC % & That all thee i to it. 1. kg tone i attached to a 1.3 m line and wung in a cicle. If it ha a linea peed of 13 m/, what i the centipetal foce? F C! m " 1. kg 13 m # $ % & 1.3 m 160 N A ca i taeling at a contant peed and make a tun with a adiu of 50.0 m. It peed i15.0 m/. Find the minimum coefficient of fiction needed to keep the ca taeling along the path. Let look at the FBD: n The fictional foce mut equal the centipetal foce. m The centipetal foce i gien by: FC We know that thi mut equal the fictional foce. We alo know that the fictional foce i: f µ N F c mg f Aume the oad i flat, o n mg Set the two equal to each othe and ole fo the coefficient of fiction: µ mg m µ g! m " 1 µ # 15.0 0.459 $ % &! m " # 9.8 50.0 m $ % & A child twil a yo yo. If angle of the cod with the etical i 30.0, find a c. 0 Look at the foce in the y diection: 168
mg F y 0 T co! " mg 0 T co! The hoizontal component of T i the centipetal foce. F T in! Plug into equation fo T: C mg F C in! FC mg tan! We know that: FC mac co! T 0 o ma mg tan! a g tan! C C m o ac 9.8 tan 30.0 a C m 5.66 mg Centifugal foce: You may hae head of the centipetal foce befoe you tudied phyic. It i poible. Mot people don t ue the tem howee. Intead they talk about the centifugal foce. Jut what the heck i that? Okay, you e een the wod centifugal foce, now foget it! It teibly bad fom fo a phyicit uch a youelf to ue uch a naïe tem. Hee the deal. The centifugal foce i the thing that people blame fo the feeling that thing eem to be puhed away fom the cente of pin duing otation. You place a coin on a tuntable and then pin it eally fat. What happen to the coin? In you mind you pictue the coin flying taight away fom the cente of the ecod. The indiidual who had not tudied phyic would ay thi wa becaue of the centifugal foce. Hee i an impotant concept: The centifugal foce i a fictional foce. Thee i no actual foce that i puhing away fom the cente of a otating ytem. You feel thi centipetal foce when you ae a paenge in a ca that make a tun. When the ca ente the tun, you feel a if you hae been puhed into the doo, away fom the cente of the cicula path the ca i making. So you think, Hey, I m being puhed into the doo o thee mut be a foce puhing me away fom the cente of the tun. That cetainly how you feel at any ate. Remembe the poblem at the beginning of thi ditinguihed pape? About the path a ball would follow if the ting wee to beak? You had you thee of you baic choice: 169
A B C The coect choice, you e hopefully (actually, the Phyic Kahuna hould ay it i to be hoped, but that ound ey pompou, o we won t ay that) figued out that the coect path i B. Why? Well at the point in the cicle whee the ting beak, the ball ha a elocity that i tangent to the cicula path. The ting i poiding the centipetal foce pulling the ball towad the cente. The ball want to follow the tangential path becaue of the fit law, but the ting won t let it. The ting, ia the tenion it exet, pull the ball towad the cente, changing the diection of it motion and making it follow the cicula path. When the ting beak thee i no longe any foce to change it diection, o the ball tael in a taight-line path that i tangent to the cicle a in the B dawing in accodance with the fit law of motion. Thi i ot of what i going on in the ca with the paenge. The paenge want to tael in a taight-line path at a contant peed in accodance with the law, the fit law to be exact. The ca howee ha diffeent idea. It decide to go in a cicula path. It the tie puhing it towad the cente exeting a foce to make it all happen. So the ca change diection, but you, the paenge, do not. No foce i acting on you. So you go fowad in you oiginal diection until you puh into the doo, which then puhe you towad the cente and you then go in a cicle a well. The thid law ea it head you puh into the doo and the doo puhe into you. You feel like you ae being puhed into the doo, een though thee i no eal foce doing thi. It jut the eaction foce to you puhing into the doo. Thi i the o called centifugal foce, thi enation of being puhed away fom the cente. If the centifugal foce wa eal, i.e., thee wa a foce puhing you away fom the cente of the cicula path, then if the doo wee to uddenly pop open you would fly taight away fom the cente of the cicula path. But of coue that doe not happen. You would tael in a tangential path. The centipetal foce i eal, the centifugal foce i not. Centipetal Foce and Gaity: The Phyic Kahuna did a illy demontation inoling a bucket of wate that wa pinning in a etical cicle. The wate tayed in the bucket and did not fall out. So what wa the deal? Doe pinning omething in a etical cicle omehow cancel out gaity? Well, no, gaity i a foce that cannot be topped o canceled. It i alway thee, anytime you hae the appopiate mae. The wate doe fall, it fall but the bucket fall 170
with it and catche it. Thi only wok if the bucket i moing fat enough to catch the wate. If the bucket i too low, then the wate will fall out of it. The minimum linea peed fo thi i called the citical elocity. Citical elocity minimum elocity fo an object to tael in etical cicle and maintain it cicula path againt the foce of gaity. The ame thing i needed fo atellite in obit aound the eath o planet in obit aound the un. They too mut tael at the citical elocity. The citical elocity fomula i not poided on the AP Tet, but it i ey imple to figue out. You jut et the centipetal foce equal to the weight of the object that i in cicula motion. If the two foce ae equal, then the object won t be able to fall out of the bucket. m FC and F mg Set them equal to each othe: mg m g g So hee i the citical elocity g A canial ide tael in a etical cicle. If the ide ha a adiu of 4.5 m, what i the citical elocity?! m " m m g 4.5 m# 9.8 44.1 6.6 $ % & AP Tet Quetion Time: A heay ball wing at the end of a ting a hown hee, with negligible ai eitance. Point P i the lowet point eached by the ball in it motion, and point Q i one of the two highet point. (a) On the following diagam daw and label ecto that could epeent the elocity and acceleation of the ball at point P and Q. If a ecto i zeo, explicitly tate thi fact. The dahed line indicate hoizontal and etical diection. i. Point P 171
ii. Point Q (b) Afte eeal wing, the ting beak. The ma of the ting and ai eitance ae negligible. On the following diagam, ketch the path of the ball if the beak occu when the ball i at point P o point Q. In each cae, biefly decibe the motion of the ball afte the beak. i. Point P at P ii. Point Q at Q Okay, hee we go. Let anwe the quetion. (a) On the following diagam daw and label ecto that could epeent the elocity and acceleation of the ball at point P and Q. If a ecto i zeo, explicitly tate thi fact. The dahed line indicate hoizontal and etical diection. i. Point P Many foolih AP Phyic tudent fogot about the centipetal acceleation when they labeled the ball at point P. You wouln d foget omething like that would you? ii. Point Q a c at Q At point Q the ball ha come to et (fo an intant) and o it elocity i zeo. The only acceleation acting on it i the acceleation of gaity. (d) Afte eeal wing, the ting beak. The ma of the ting and ai eitance ae negligible. On the following diagam, ketch the path of the ball if the beak occu when the ball i at point P o point Q. In each cae, biefly decibe the motion of the ball afte the 17
beak. i. Point P at P At point P the ball i at it lowet point in the path. When the centipetal foce diipea (the boken ting, ight?) the ball continue to moe with the elocity it had at thi point, which i hoizontal. So it path look like the pojectile we tudied that tated out with a hoizontal elocty. ii. Point Q at Q At point Q the ball i at et o when the ting beak it imply fall taight down. 173