5. Cicula otion By Liew Sau oh Content 5.1 Angula diplaceent and angula elocity 5. Centipetal acceleation 5.3 Centipetal foce Objectie a) expe angula diplaceent in adian b) define angula elocity and peiod c) deie and ue the foula = d) explain that unifo cicula otion ha an acceleation due to the change in diection of elocity e) deie and ue the foulae fo centipetal acceleation a = / and a = Objectie f) explain that unifo cicula otion i due to the action of a eultant foce that i alway diected to the cente of the cicle g) ue the foulae fo centipetal foce F = / and F = h) ole poble inoling unifo hoizontal cicula otion fo a point a i) ole poble inoling etical cicula otion fo a point a (knowledge of tangential acceleation i not equied). 5.1 Angula diplaceent and angula elocity Unifo cicula otion Suppoe that an object execute a cicula obit of adiu with unifo tangential peed. Unifo cicula otion he intantaneou poition of the object i ot coneniently pecified in te of an angle. Unifo cicula otion Fo intance, we could decide that = 0 coepond to the object' location at, in which cae we would wite = t, whee i the angula elocity of the object.
Unifo cicula otion Fo a unifoly otating object, the angula elocity i iply the angle though which the object tun in one econd. Angula diplaceent Conide the otion of the object in the tie inteal between and. Hee, the object otate though an angle, and tace out a cicula ac of length. Angula diplaceent It i faily obiou that the ac length i diectly popotional to the angle, an angle of 360 coepond to an ac length of. Hence, an angle ut coepond to an ac length of 360 Angula diplaceent At thi tage, it i conenient to define a new angula unit known a a adian (ybol ad). Angula diplaceent An angle eaued in adian i elated to an angle eaued in degee ia the following iple foula: Angula diplaceent hu, 360 ad, 180 ad, 90 ½ ad, and 57.96 1 ad. ad 360 Angula diplaceent When i eaued in adian, 360 Angula elocity Conide the otion of the object in the hot inteal between tie t and t + t. In thi inteal, the object tun though a all angle and tace out a hot ac of length, whee =. iplifie geatly to gie =.
Angula elocity Now / t (i.e., ditance oed pe unit tie) i iply the tangential elocity, wheea / t (i.e., angle tuned though pe unit tie) i iply the angula elocity w. hu, diiding = by t, we obtain = w. Angula elocity Note, howee, that thi foula i only alid if the angula elocity w i eaued in adian pe econd. Fo now on, in thi coue, all angula elocitie ae eaued in adian pe econd by default. Angula elocity An object that otate with unifo angula elocity w tun though w adian in 1 econd. Hence, the object tun though adian (i.e., it execute a coplete cicle) in = / w econd. Angula elocity Hee, i the epetition peiod of the cicula otion. If the object execute a coplete cycle (i.e., tun though 360 ) in econd, then the nube of cycle executed pe econd i f = 1/ = w /. In othe wod, w = /. Angula elocity Hee, the epetition fequency, f, of the otion i eaued in cycle pe econd--othewie known a hetz (ybol Hz). Angula elocity A an exaple, uppoe that an object execute unifo cicula otion, adiu = 1., at a fequency of f = 50Hz (i.e., the object execute a coplete otation 50 tie a econd). he epetition peiod of thi otion i iply = 1/f = 0.0. Angula elocity Futheoe, the angula fequency of the otion i gien by w = f = 314.16 ad/ Finally, the tangential elocity of the object i = w = 1. 314.16 = 376.99 /. 5.3 Centipetal acceleation
Cicula otion Cicula otion he tenion in the ting! he peed tay contant, but the diection change R Bat wing the tenni ball aound hi head in a cicle. he ball i acceleating, what foce ake it acceleate? he acceleation in thi cae i called centipetal acceleation Centipetal acceleation, a C Centipetal acceleation R a C he acceleation point towad the cente of the cicle towad the cente of the cicle Centipetal acceleation An object executing a cicula obit of adiu with unifo tangential peed poee a elocity ecto whoe agnitude i contant, but whoe diection i continuouly changing. Centipetal acceleation It follow that the object ut be acceleating, ince acceleation (ecto) i the ate of change of elocity (ecto), and the elocity (ecto) i indeed aying in tie. Centipetal acceleation Suppoe that the object oe fo point to point between tie t and t + t, a hown in the figue aboe. Suppoe, futhe, that the object otate though adian in thi tie inteal. Centipetal acceleation he ecto, hown in the diaga, i identical to the ecto. Moeoe, the angle ubtended between ecto and i iply.
Centipetal acceleation he ecto epeent the change in ecto elocity,, between tie t and t + t. Centipetal acceleation It can be een that thi ecto i diected towad the cente of the cicle. Fo tandad tigonoety, the length of ecto i = in( /). Centipetal acceleation Howee, fo all angle in, poided that i eaued in adian. Hence,, It follow that a = / t = / t =, whee = / t i the angula elocity of the object, eaued in adian pe econd. Centipetal acceleation In uay, an object executing a cicula obit, adiu, with unifo tangential elocity, and unifo angula elocity w = /, poee an acceleation diected towad the cente of the cicle:- i.e., a centipetal acceleation:- of agnitude a = w = / = w. 5.3 Centipetal foce Centipetal acceleation centipetal acceleation a foce i needed to poduce thi centipetal acceleation CENRIEAL FORCE whee doe thi foce coe fo? a C = R Centipetal foce Suppoe that a weight, of a, i attached to the end of a cable, of length, and whiled aound uch that the weight execute a hoizontal cicle, adiu, with unifo tangential elocity. Centipetal foce A we hae jut leaned, the weight i ubject to a centipetal acceleation of agnitude /. Hence, the weight expeience a centipetal foce f = /.
Centipetal foce What poide thi foce? Well, in the peent exaple, the foce i poided by the tenion in the cable. Hence, = /. Centipetal foce Suppoe that the cable i uch that it nap whenee the tenion in it exceed a cetain citical alue ax. Centipetal foce It follow that thee i a axiu elocity with which the weight can be whiled aound: naely, ax ax Centipetal foce If exceed ax then the cable will beak. A oon a the cable nap, the weight will ceae to be ubject to a centipetal foce. Centipetal foce So it will fly off; with elocity ax along the taight-line which i tangential to the cicula obit it wa peiouly executing. uetion 1 A.0 kg a winging at the end of a 0.50 ting i taeling 3.0 /. What i the a. centipetal acceleation of the a? b. centipetal foce on the a? Anwe 1 a) a c = / a c = (3.0 /) /(0.5 ) a c = 18 / b) F c = a c = 36 J uetion A tudent wing a ball in a cicle of adiu 70 c in the etical plane. he angula elocity of the ball i 10 ad 1. a) What i the elocity of the ball? b) How long doe the ball take to coplete one eolution?
Anwe a) = = (10)(0.70) = 7.0-1 b) (0.70) 7 10 0.63 uetion (continue) A tudent wing a ball in a cicle of adiu 70 c in the etical plane. he angula elocity of the ball i 10 ad 1. he tudent eleae the ball when it i at A, which i 130 c aboe the gound, and the ball tael etically upwad. Calculate a) the axiu height, aboe the gound, the ball will each; b) the tie taken fo the ball to hit the gound afte it eleae fo A. a) = u + a 0 = (7) + (-9.8) => =.50 => ax. height =.5 + 1.30 = 3.8 b) Oeall: A => ax. height => gound: = ut + ½ at -1.30 = 7t ½ (9.8)t tie, t = 1.59 Ball on a ting op iew he tenion in the ting poide the neceay centipetal foce to keep the ball going in a cicle. ath of ball if the ting beak Exaple What i the tenion in a ting ued to twil a 0.3 kg ball at a peed of -1 in a hoizontal cicle of 1 ete adiu? Anwe: Foce = a x acceleation [ a c ] Acceleation, a c = / R = () / 1 = 4 - Foce, F = a c = 0.3 4 = 1. N Exaple If the ting i not tong enough to handle thi tenion (1. N) it will beak and the ball goe off in a taight line. Vetical Cicula Motion If a pail i whiled in a etical cicle, the peed of the pail aie along it cicula otion g O Vetical Cicula Motion he cetipetal acceleation, a = / towad O. Uing F = a, g co = / enion, = / + g co g O
Vetical Cicula Motion Vetical Cicula Motion = / + g co At lowet point, = 0 = / + g At highet point, = = / g When = / o 3 / = /; (co = 0) g O = / + g co At highet point, = = / g he pail well not dop if / g > 0 / > g > (g) 1/ g O Conical endulu An object i oing in a hoizontal cicle (foing a conical pendulu) Fo hoizontal cicula otion, a = / (towad cente) g Conical endulu Fo etical coponent, l co = g co = g / l = co-1(g/l ) y g l l x Conical endulu Fo hoizontal cicula otion, a = / (towad cente) F = a = / Fo hoizontal coponent, in = / = = (l in ) = l y g y = co l x, x = in Negotiating a flat (leel) tun he centipetal foce i poided by the fiction foce between the oad and tie. thi foce i educed if the oad i wet o icy y = co, x = in Banked un Banked tun Velodoe 31 degee bank Since the oad i banked (not hoizontal) the foce of the oad on the box i not etical. at of the foce on the box fo the oad point towad the cente of the cicle. hi poide the centipetal foce. No fiction i neceay to keep the box in the cicle. N F CEN R
he ed object will ake the tun only if thee i enough fiction on it 0 it goe taight the appaent outwad foce i called the centifugal foce it i NO A REAL foce! an object will not oe in a cicle until oething ake it! object on the dahboad taight line object natually follow Centifugal foce: Definition Foce epeent the effect of inetia that aie in connection with otation and which ae expeienced a an outwad foce away fo the cente of otation Silly Silo (Roto) wall puhing in on Bat Fiction between Bat and wall Suay: Cicula otion Angula Diplaceent & Angula Velocity Centipetal Acceleation = / t = a = / a = a = weight he inwad wall foce keep Bat in the cicle. Fiction keep hi fo falling down. Centipetal Foce F = / = = Motion in hoizontal cicle Motion in etical cicle