Class 6 - Circular Motion and Gravitation

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Class 6 - Cicula Motion and Gavitation pdf vesion [http://www.ic.sunysb.edu/class/phy141d/phy131pdfs/phy131class6.

exploed hee [http://www.ic.sunysb.edu/class/phy141d /ciculaotion.htl].

1 Class 6 - Cicula Motion and Gavitation pdf vesion [ Fequency and peiod Fequency (evolutions pe second) [ o ] Peiod (tie fo one evolution) [ ] π Velocity v = = πf [ s 1 ] T T = 1 f f s 1 Hz s The elationship between fequency and peiod can be exploed hee [ /ciculaotion.htl]. Newton's Fist Law Evey object continues in its state of est, o of unifo velocity in a staight line, as long as no net foce acts on it. Fo Newton's fist law we can see that cicula otion can only occu when a net foce acts on an object. Releasing a ball fo a cicula path Unlike in baseball, cicketes ay not extend thei elbows duing the bowling action. (Clealy, flexing the elbow, o chucking is just not cicket ). See soe bowling basics [ (o even lean about [ /wiki/cicket] a bette spot). Neglecting effects such as spin, you can see that the path of the ball afte elease is essentially tangential to the cicula velocity. Acceleation in cicula otion

2 The agnitude of the velocity does not change, but it's diection does. The vecto fo the change of the velocity always points towads the cente of the cicle. Thee is thus an acceleation, naed the centipetal acceleation pointing to the cente of the cicle. Fo sall changes in the position. Δv = v sin( Δθ = Δl vδθ = v Δl Δθ ) vδθ a R = = = Δv li Δt 0 Δt v Δl li Δt 0 Δt v Visualizing the acceleation We can look at the diection of the acceleation of an object oving in cicula otions using the acceleoetes in a satphone o tablet.to display the acceleation vecto fo the x-y plane of you phone o tablet click hee fo ios [ o click hee fo Andoid [ /phy141d/iphone/vecto.htl] (You device ust have an acceleoete and gyoscope chip fo this to wok. Fo easons I don't begin to copehend Apple and Google define the acceleoete axes diffeently which is why thee ae two vesions of this.). Centipetal foce Newton's second law tells us we can only have a centipetal acceleation towads the cente of a cicula otion path if the su of the foces in the adial diection Σ = = F R a R v This centipetal foce ust be povided in soe way fo cicula otion to occu. Do not be isled in to thinking that such a thing as centifugal foce exists. As we saw ealie if the centipetal foce is eoved an object continues in a tangential, not pependicula path to the otion. Of couse in situations whee you ae in cicula otion, it can feel as if you ae being pushed out fo the cente, but this is in fact you applying the equal and opposite eaction foce equied by Newton's 3d Law. The foce on you is diected inwads. Centifuges NASA uses this centifuge to subject people to centipetal acceleations of up to 0G!

If a peson sits 10 fo the cente of the centifuge, how fast does the centifuge have to tun fo a peson to be subjected to foce of 0G? Centifuge solution ΣF = 0g p v = 0g v = 00g v = 44.

3 If a peson sits 10 fo the cente of the centifuge, how fast does the centifuge have to tun fo a peson to be subjected to foce of 0G? Centifuge solution ΣF = 0g p v = 0g v = 00g v = 44.3s 1 Anothe Centifuge Video [ Cicula otion on a sping Spings extend o contact as a foce is exeted on the. The foce on a sping can be consideed to be diectly popotional to it's extension. If a sping is used to povide the centipetal foce on an object, the equal and opposite eaction foce on it will stetch it out. By swinging a weight on a sping in hoizontal cicle above y head I can deonstate that the foce equied depends on

4 the velocity. Vetical cicle If we now ty to ove the ass on sping in a vetical cicle we can see it doesn't happen. The foce equied is diffeent at the top and botto because gavity altenatively assists o hindes us at the top and botto of the path and as the sping changes it's length accoding to the tension on it the esulting otion is not cicula. A ball on a sting woks, poviding the velocity is geate than v = g Looping plane A plane in a cicula loop is a siila poble. Hee we can also conside the foce F A that the plane needs to apply as it

goes though the loop. Things not attached [http://www.youtube.co/watch?v=vp0bv39nrm] to the plane will still be subject to gavity. Conical Pendulu See 5.6 of this physclips video [http://www.

5 goes though the loop. Things not attached [ to the plane will still be subject to gavity. Conical Pendulu See 5.6 of this physclips video [ fo a nice exaple. How fast does it go? v F T cos θ = g F T sin θ = v F sin θ = T v = g ( ) sin θ = cos θ g sin θ cos θ Rathe than we would like an expession in l so we sub in = l sin θ lg sin θ v = cos θ π πl sin θ v To get the peiod of otation, divide the path length = by T = πl sin θ cos θ = π l cos θ lg sin θ g Cas and tuns When a ca ounds a bend the question of whethe it slips o not is a static fiction poble. We conside the ca not to be oving in the diection pependicula to it's otion. It will eain stationay in this diection if the axiu possible static fiction foce v μ s N = μ s g > o v < g μ s As you can see, on wet o icy oads you should slow down ound bends!

Banked tuns Roads designed fo high speed taffic will often used banked tuns to incease the axiu speed fo which slipping does not occu. A well designed banked tun eans the ca should not ely on fiction.

6 Banked tuns Roads designed fo high speed taffic will often used banked tuns to incease the axiu speed fo which slipping does not occu. A well designed banked tun eans the ca should not ely on fiction. (It also akes the poble easie..) The elationship between axiu speed, adius and the bank angle can be found fo consideing the foces. Fo ideal banking v F N sin θ = F N cos θ = g sin θ g = cos θ v tan θ = v g o with fiction v F N sin θ + μf N cos θ = F N cos θ μ sin θ+μ cos θ cos θ μ sin θ = F N v g sin θ = g Newton's Law of Univesal Gavitation Newton's faous, and supposedly apple inspied, idea that the sae foce that caused objects to be bound to the Eath's suface was what ade the planets obit each othe was a huge step fowads, pioneeing the concept of foces which act a distance. Reasoning based on the Moon's obit aound the Eath 384, 000k v 10s 1 Suppose we know both the distance of the oon fo the eath ( ) and the speed of it's obit ( ). v a R = Fo ou last lectue we know that the centipetal acceleation of an object is. Fo the known velocity and acceleation it can be found that a R = 0.007s. Clealy this is uch less than the acceleation due to gavity at the Eath's suface ( )! 9.8s So if we want to follow the idea that the sae foce is esponsible fo both the centipetal acceleation of the oon and the falling of an apple it is clea that the foce due to gavity ust depend on how fa objects ae fo each othe. 6400k Suppose we also know the adius of the Eath ( ). We can see that oon is about 60 ties as fa fo the cente of 1 the Eath as the apple. The acceleation howeve is. 3600

7 Fo this Newton concluded (although Robet Hooke [ also laid clai to this idea) that the dependence of the gavitational foce should depend on the invese squae of the distance. Newton's Second Law tells us that the foce on the oon ust be popotional to it's ass M. Newton's Thid Law tells us that as well as the Eath exeting a foce on the Moon, the Moon should exet an equal and opposite foce on the Eath, so the foce needs to also be popotional to E. This cobination of consideations leads us to: F E M Gavitational Constant We can define a Gavitational Constant F = G 1 such that the foce between two asses is Although Newton could deteine the fo of the gavitational foce he could not deteine the constant in his law. (He did not know the ass of the Eath o Moon.) The gavitational constant can be deteined using a tosion balance [ /9/91/Cavendish_Tosion_Balance_Diaga.svg]. This expeient was fist pefoed by Heny Cavendish [ in the Cavendish Expeient [ who used it to easue the density of the Eath. Othe scientists late used his esults to deteine the value of G. The value of is. Vecto fo G G N kg F 1 F 1 1 = G 1 = G = G 1 = G F 1 1 is the foce on paticle 1 (ass ) due to paticle (ass ). 1 is the distance between the two paticles. 1 is a unit vecto which points fo paticle towad towads paticle 1. Gavitational Field The gavitational field due to a ass is g = GM N The noal units fo gavitational field ae. (Note these ae diensionally equivalent to s.) kg The foce on an object F = g g M due this field is

8 Looks failia? Nea the Eath's suface we can find that = 9.8s. Distibuted Mass g The fo of the gavitation law we have pesented iplicitly assues that we can appoxiate all the ass of an object as being at it's cente. Now that we have an expession fo the field we can show this to be explicitly tue fo spheical objects. This is tivial if you know how to do suface integals as Newton's Law of gavity can be expessed as Gauss' Law [ (We'll be etuning to Gauss' law in PHY 13!) Assuing that we don't know how to do suface integals we can also deive the esult using a shell odel [ We can also see fo the shell odel that a shell futhe fo the cente of the sphee than an object exets no gavitational foce on it. What does this ean fo an object that falls though the cente of a planet [ featue=elated]? Falling though a planet An object that falls though a hole in a planet expeiences a foce due to all the ass that is within a sphee such that it is close to the cente of the planet than the object. If we can assue that the density of the planet ( ρ) is unifo then the ass within a given adius is equivalent to ρ 4π 3 3 The foce on an object at a distance ρ 4π 3 3 F = G = Gρ 4π 3 and is always diected to the cente of the planet. fo the cente is theefoe Whateve gains in velocity ade as you fall to the cente, will be lost on the way back to the suface. If the object stats at est it will tun aound at the othe side and epeat the otion in the othe diection, oscillating back and fowad fo eve. (We'll coe back to this as an exaple of siple haonic otion late in the couse.) Gavity on diffeent planets g M P R P The acceleation due to gavity on the suface of any given planet of ass and adius g = GM P R P Of couse it can also be useful to expess this in tes of the density of the planet. 4π g = G ρr 3 P

9 G = N kg Planet Mass Radius Mecuy Venus Eath Mas Jupite Satun Uanus Neptune Gavity at diffeent heights above the suface Be caeful that to find the gavitational acceleation of an object at height needs to be added to the adius! g (h) = Also eebe that this foula is not valid fo Useful nubes: kg 400k kg 6050k kg 6380k kg 3390k kg 69900k kg 5800k kg 5400k kg 4600k GM E ( R E +h) M E = kg R E = 6380k G = N kg h < 0 h above the suface of the eath, the height (as we showed befoe) because of the excluded ass. phy131studio/lectues/class6.txt Last odified: 014/09/14 :18 by dawbe

CHAPTER 5: Circular Motion; Gravitation

CHAPTER 5: Circular Motion; Gravitation CHAPER 5: Cicula Motion; Gavitation Solution Guide to WebAssign Pobles 5.1 [1] (a) Find the centipetal acceleation fo Eq. 5-1.. a R v ( 1.5 s) 1.10 1.4 s (b) he net hoizontal foce is causing the centipetal