Section 14 Forces in Circular Motion

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1 Secion 14 orces in Circular Moion Ouline 1 Unifor Circular Moion Non-unifor Circular Moion Phsics 04A Class Noes Wh do objecs do wha he do? The answer we have been invesigaing is forces If forces can eplain oion, hen he us be able o eplain circular oion 1 Unifor Circular Moion You have probabl never ried o swing a bucke of waer over our head, bu i can be done quie easil (hp://wwwouubeco/wach?v=bwivv--otya) Wh doesn he waer fall ou on our head? or ha aer, wh doesn he orbiing oon fall oward Earh? Wh do we keep orbiing he sun and no fall inward and ge burned o risp? The answer us be conained in Newon s Laws of Moion According o he irs Law, he waer will ove in a sraigh line unless a force acs on i Therefore, here us alwas be a force on an objec o keep i in circular oion According o he Second Law, he force and he acceleraion us poin in he sae direcion or objecs in unifor circular oion (consan speed) we alread learned ha here is acceleraion oward he cener called he cenripeal acceleraion, = v r Therefore, here us be a force acing oward he cener o keep an objec in circular oion In he case of he waer, i is he force of gravi cobined wih he noral force fro he bucke In he case of he oon orbiing Earh, i is he force of graviaion The poin is, objecs in circular oion can be reaed he sae wa we ve reaed oher objecs Here are soe eaples 14-1

2 Phsics 04A Class Noes Eaple 141: A 100g ball is wirled overhead on he end of a 400c sring a 100rp ind he ension in he cord Given: = 0100kg, = 0400, and f = 100rp ind: =? The onl force on he ball is he ension Choosing he -ais as shown and appling Newon s Second Law, = a = Using he cenripeal acceleraion, = v r Using he angenial speed, v = πr T = 1 πr r T = 4π r v T The period is he reciprocal of he frequenc, T = 1 f = 1 = 1in 100 rev in 100rev = 600s 100rev = 0600s, and he radius is he lengh of he sring inall, = 4π = (0100) 4π (0400) T (0600) = 439N Bu wai (weigh?) we forgo abou gravi The side view of he wirling ball is shown a he righ Noice ha he ball is oving in a plane slighl below he poin where he sring is held In his ore careful analsis, here are forces along wo direcions and onl he horizonal par of he ension causes he circular oion Appling he Second Law o each direcion separael, = a cosθ = v r = (1) rcosθ Σ = a sinθ = 0 sinθ = () Subsiuing for he speed in eq 1, = 1 πr = 4π r rcosθ T T cosθ Noice ha fro rigonoer, r = cosθ = 4π cos θ T cosθ v = 4π T = 439N The ension is he sae as before Since he coponen of he ension oward he cener of he circle is less, he ball us be oving slower In fac, he speed depends on he angle below horizonal, v = πr = π cosθ T T The faser i spins he saller he angle, as ou igh epec The angle below he horizonal can be found fro eq, sinθ = = g θ = arcsin g = arcsin (0100)(980) 439 θ =

3 Phsics 04A Class Noes Eaple 14: A car raveling a 500k/h rounds urve wih a 300 radius ind he iniu coefficien of fricion required o keep he car fro skidding Given: v = 500k/h = 139/s and r = 300 ind: µ =? The forces on he car are shown a he righ The are he weigh, noral and saic fricion The fricion is he force hauses he car o go in ircle and poins oward he cener You know his because urning corners on an ic road is dangerous Appling he Second Law o each direcion separael, = a sf = (1) Σ = a n = 0 n = () Using he definiion of he coefficien of saic fricion, n g sf r µ s sf,a sf µ s n µ s µ s n Appling he ass/weigh rule and he epression for cenripeal acceleraion, µ s / / g = v rg = (139) (300)(980) µ s 0657 Non-unifor Circular Moion If ou wirl a ball a he end of a sring slowl in a verical plane, insead of a horizonal plane, ou ll noice he speed isn consan The ball will be going slower a he op and faser a he boo Circular oion doesn alwas occur wih consan speed Nonunifor circular oion occurs in an cases Since he speed changes, he acceleraion us have oponen along he oion as well as he usual coponen oward he cener The coponen along he oion is called he angenial acceleraion This oo can be undersood using Newon s Laws While he ension force alwas poins oward he cener, he weigh has angenial coponens 14-3

4 Phsics 04A Class Noes Eaple 143: When he ball is 600 fro he verical oving downward, he ension is 500N ind (a)he cenripeal acceleraion, (b)he angenial acceleraion and (c)he angenial veloci Given: θ = 600 and = 500N ind: =?, a =?, and v =? Using he free bod diagra o appl he Second Law, = a cosθ = (1) Σ = a sin θ = a () (a)using he ass/weigh rule in eq 1 and solving for he cenripeal acceleraion, gcosθ = = + gcosθ Puing in he nubers, = (980)cos600 a = 549 c s The inus sign is because he -ais poins awa fro he cener of he circle ree Bod Diagra (b)the angenial acceleraion can be found using eq, / gsinθ = / a a = gsinθ = ( 980)sin600 a = 849 s Again he inus sign is due o he choice of aes (c)the angenial veloci is he veloci ha is relaed o he cenripeal acceleraion, = v r v = r = (549)(0400) v = 469 s In suar, Newon s Laws of Moion show us hircular oion requires a force oward he cener o creae he cenripeal acceleraion Tangenial forces change he speed of he circular oion Now, back o he quesion abou he oon saing in orbi Earh eers a graviaional force on he oon This force pulls i oward Earh If he oon were no oving, he graviaional force would cause he oon o fall direcl oward Earh However, he oon is no a res, i oving angeniall Think abou a baseball hrown horizonall, i falls oward Earh, bu doesn land direcl below he poin fro which i is hrown because of is horizonal oion If ou could hrow he ball faser, i will ravel farher before hiing he ground If ou could hrow i fas enough, i would fall oward Earh a eacl he sae rae he spherical shape of Earh causes he surface o fall awa fro he ball The ball would be in orbi So he oon is oving a jus he righ speed so ha as i falls oward Earh bu i never ges an closer This is wh here has o be a ver special relaionship beween he veloci, radius, and acceleraion for an objec in unifor circular oion You know his relaionship, = v r 14-4

5 Phsics 04A Class Noes You igh ask how i has coe o pass ha he oon has his eac righ veloci or ha aer, wh does ever plane in he solar sse have he perfec veloci o ainain is orbi around he sun? Reeber, here are soe objecs in he solar sse ha don have circular orbis, such as coes In 1994 oe collided wih Jupier (hp://wwwjplnasagov/sl9/) This is he evenual fae of all objecs ha don have circular orbis The reason ha we onl reall see planes and oons wih circular orbis, is ha he solar sse is old enough ha os of he hings wih non-circular orbis have sashed ino soehing alread Speaking of falling oward earh wihou geing an closer, his is wh asronaus feel weighless The force of gravi on Space Shule asronaus is onl abou 10% less han he force of gravi on Earh The are falling oward Earh all he ie, bu so is he shule Tha is wh he can floa around inside - everhing is falling ogeher Do an inerne search for voi coe You ll find inforaion abou an airplane he use o rain asronaus (and ake ovies) The plane goes up o high aliude and hen falls in a parabolic rajecor Since he passengers are falling in he sae wa he plane is, he feel weighless Secion Suar Our goal is o undersand wha objecs do and wh he do i Now we have applied Newon s Laws o undersand he causes of circular oion A ne force oward he cener of he circle is required for circular oion An forces angenial o he circle cause changes in he orbial speed 14-5

x y θ = 31.8 = 48.0 N. a 3.00 m/s

x y θ = 31.8 = 48.0 N. a 3.00 m/s 4.5.IDENTIY: Vecor addiion. SET UP: Use a coordinae sse where he dog A. The forces are skeched in igure 4.5. EXECUTE: + -ais is in he direcion of, A he force applied b =+ 70 N, = 0 A B B A = cos60.0 =