Chapter 13. VectorValued Functions and Motion in Space Velocity and Acceleration in Polar Coordinates


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1 13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along a cuve in the pola coodinate plane, we expess its position, velocity, and acceleation in tems of the moving unit vectos u = (cos θ)i + (sinθ)j, u θ = (sin θ)i + (cos θ)j. The vecto u points along the position vecto OP, so = u. The vecto u θ, othogonal to u, points in the diection of inceasing θ. Figue 13.30, page 757
2 13.6 Velocity and Acceleation in Pola Coodinates 2 Note. We find fom the above equations that du dθ du θ dθ = (sinθ)i + (cos θ)j = u θ = (cos θ)i (sin θ)j = u. Diffeentiating u and u θ with espect to time t (and indicating deivatives with espect to time with dots, as physicists do), the Chain Rule gives u = du dθ θ = θu θ, u θ = du θ dθ θ = θu. Note. With as a position function, we can expess velocity v = ṙ as: v = d dt [u ] = ṙu + u = ṙu + θu θ. This is illustated in the figue below. Figue 13.31, page 758
3 13.6 Velocity and Acceleation in Pola Coodinates 3 Note. We can expess acceleation a = v as a = ( u + ṙu ) + (ṙ θu θ + θu θ + θ u θ ) = ( θ 2 )u + ( θ + 2ṙ θ)u θ. Example. Page 760, numbe 4. Definition. We intoduce cylindical coodinates by extending pola coodinates with the addition of a thid axis, the zaxis, in a 3dimensional ighthand coodinate system. The vecto k is intoduced as the diection vecto of the zaxis. Note. The position vecto in cylindical coodinates becomes = u + zk. Theefoe we have velocity and acceleation as: v = ṙu + θu θ + żk a = ( θ 2 )u + ( θ + 2ṙ θ)u θ + zk. The vectos u, u θ, and k make a ighthand coodinate system whee u u θ = k, u θ k = u, k u = u θ.
4 13.6 Velocity and Acceleation in Pola Coodinates 4 Figue 13.32, page 758 Theoem. Newton s Law of Gavitation. If is the position vecto of an object of mass m and a second mass of size M is at the oigin of the coodinate system, then a (gavitational) foce is exeted on mass m of F = GmM 2. The constant G is called the univesal gavitational constant and (in tems of kilogams, Newtons, and metes) is Nm 2 kg 2.
5 13.6 Velocity and Acceleation in Pola Coodinates 5 Note. Newton s Second Law of Motion states that foce equals mass times acceleation o, in the symbols above, F = m. Combining this with Newton s Law of Gavitation, we get o m = GmM 2 = GM 2., Figue 13.33, page 758 Note. Notice that is a paallel (o, if you like, antipaallel) to, so = 0. This implies that d [ ṙ] = ṙ ṙ + = 0 + = = 0. dt So ṙ must be a constant vecto, say ṙ = C. Notice that if C = 0, then and ṙ must be (anti)paallel and the motion of mass m must be in
6 13.6 Velocity and Acceleation in Pola Coodinates 6 a line passing though mass M. This epesents the case whee mass m simply falls towads mass M and does not epesent obital motion, so we now assume C 0. Lemma. If a mass M is stationay and mass m moves accoding to Newton s Law of Gavitation, then mass m will have motion which is esticted to a plane. Poof. Since ṙ = C, o moe explicitly, (t) ṙ(t) = C whee C is a constant, then we see that the position vecto is always othogonal to vecto C. Theefoe (in standad position) lies in a plane with C as its nomal vecto, and mass m is in this plane fo all values of t. Q.E.D. Figue 13.34, page 759
7 13.6 Velocity and Acceleation in Pola Coodinates 7 Theoem. Keple s Fist Law of Planetay Motion. Suppose a mass M is located at the oigin of a coodinate system. Let mass m move unde the influence of Newton s Law of Gavitation. Then m tavels in a conic section with M at a focus of the conic. Note. Keple would think of mass M as the sun and mass m as one of the planets (each planet has an elliptical obit). We can also think of mass m as an asteoid o comet in obit about the sun (comets can have elliptic, paabolic, o hypebolic obits). It is also easonable to think of mass M as the Eath and mass m as an object such as a satellite obiting the Eath. Poof of Keple s Fist Law. The computations in this poof ae based on wok fom Celestial Mechanics by Hay Pollad (The Caus Mathematical Monogaphs, Numbe 18, Mathematical Association of Ameica, 1976). Let (t) = be the position vecto of mass m and let (t) = (t), o in shothand notation =. Then d dt = d dt [ ] = ṙ ṙ 2 whee the dots epesents deivatives with espect to time t, = 2 ṙ ṙ 3 = ( )ṙ ( ṙ) 3
8 13.6 Velocity and Acceleation in Pola Coodinates 8 since d dt [2 ] = 2ṙ by the Chain ule and d dt [2 ] = d [ ] = 2 ṙ, we dt have ṙ = ṙ, = ( ṙ) 3 since, in geneal, (u v) w = (u w)v (v w)u (see page 723, numbe 17). That is, Gm, o d dt [ ] = ( ṙ) 3 GM d dt GM d dt [ ] = C, o, multiplying both sides by 3 [ ] = C = C GM 3 GM 3. ( ) Fom Newton s Law of Gavitation and Newton s Second Law of Motion, we have = GM 2 = GM, and so ( ) becomes 3 GM d dt [ ] = C ( ) = C. ( ) Integate both sides of ( ) and add a constant vecto of integation e to get GM ( + e ) = ṙ C ( ) (emembe C is constant). Dotting both sides of ( ) with gives GM ( + e ) = (ṙ C)
9 13.6 Velocity and Acceleation in Pola Coodinates 9 o GM 2 + e = ( ṙ) C by a popety of the tiple scala poduct (see page 704), o whee C = C, o GM( + e) = C C = C 2 + e = C2 GM. ( ) As commented above, if C = 0 then we have motion along a line towads mass M at the oigin, so we assume C 0. Finally, we intepet e = e. Fist, suppose e = 0. Then = C 2 /(GM) (a constant) and so the motion is cicula about cental mass M. Recall that a cicle is a conic section of ( ) eccenticity 0. Second, suppose e 0. Fom ( ), GM + e = ṙ C whee C = ṙ. By popeties of the coss poduct, + e and ae both othogonal to C. Theefoe C = 0 and ( ) + e C = 1 ( C) + e C = 0 + e C = e C = 0. So e is othogonal to C. Since C is othogonal to the plane of motion, then e lies in the plane of motion (when put in standad position). Intoduce vecto e in the plane of motion (say the xyplane) and let α be the angle between the positive xaxis and e. Let (t) be in standad position and
10 13.6 Velocity and Acceleation in Pola Coodinates 10 epesent the head of (t) as P(, β) in pola coodinates and β. Define θ as β α: The elationship between, e, α, β, and θ. Then e = e cosθ. So equation ( ) gives + e = C2 GM o + e cosθ = C2 GM o = C2 /(GM) 1 + e cosθ. This is a conic section of eccenticity e in pola coodinates (, θ) (see page 668). Notice that is a minimum when the denominato is lagest. This occus when θ = 0 and gives 0 = C2 /(GM). We can solve fo C e to get C 2 = GM(1 + e) 0. Theefoe the motion is descibed in tems of e and 0 as = (1 + e) 0. Fo (noncicula) obits about the sun, 1 + e cos θ 0 is called the peihelion distance (if the Eath is the cental mass, 0 is the peigee distance). In conclusion, the motion of mass m is a conic section of eccenticity e and is descibed in pola coodinates (, θ) as = (1 + e) e cos θ. Q.E.D.
11 13.6 Velocity and Acceleation in Pola Coodinates 11 Theoem. Keple s Second Law of Planetay Motion. Suppose a mass M is located at the oigin of a coodinate system and that mass m move accoding to Keple s Fist Law of Planetay Motion. Then the adius vecto fom mass M to mass m sweeps out equal aeas in equal times. Figue 13.35, page 759 Note. If we know the obit of an object (that is, if we know the conic section fom Keple s Fist Law which descibes the objects position), then Keple s Second Law allows us to find the location of the object at any given time (assuming we have some initial position fom which time is measued).
12 13.6 Velocity and Acceleation in Pola Coodinates 12 Poof of Keple s Second Law. In Lemma we have seen that the vecto (t) ṙ(t) = C is a constant. If we expess the position vecto in pola coodinates, we get (t) = = ( cos θ)i + ( sin θ)j. Theefoe ṙ(t) = (ṙ cos θ θ sin θ)i + (ṙ sinθ + θ cosθ)j. We also know that C = Ck. So the equation (t) ṙ(t) = C yields (t) ṙ(t) = i j k cos θ sinθ 0 ṙ cosθ θ sinθ ṙ sinθ + θ cos θ 0 = { ( cos θ(ṙ sinθ + θ cos θ)) sinθ(ṙ cos θ θ sin θ) } k = (ṙ cos θ sin θ + 2 θ cos 2 θ ṙ sin θ cos θ + 2 θ sin 2 θ)k = ( 2 θ)k = Ck. Now in pola coodinates, aea is calculated as A = b 1 a 2 2 (θ)dθ and so the deivative of aea with espect to time is (by the Chain Rule and the Fundamental Theoem of Calculus Pat I) da = da dθ dt dθ dt = (θ) θ = 1 da 2 2 θ. Theefoe dt = θ = C whee C is constant. Hence the ate 2 of change of time is constant and the adius vecto sweeps out equal aeas in equal times. Q.E.D.
13 13.6 Velocity and Acceleation in Pola Coodinates 13 Theoem. Keple s Thid Law of Planetay Motion. Suppose a mass M is located at the oigin of a coodinate system and that mass m move accoding to Keple s Fist Law of Planetay Motion and that the obit is a cicle o ellipse. Let T be the time it takes fo mass m to compete one obit of mass M and let a be the semimajo axis of the elliptical obit (o the adius of the cicula obit). Then, T 2 a = 4π2 3 GM. Note. Keple s Thid Law allows us to find a elationship between the obital peiod T of a planet and the size of the planet s obit. Fo example, the semimajo axis of the obit of Mecuy is 0.39 AU and the obital peiod of Mecuy is 88 days. The semimajo axis of the obit of the Eath is 1 AU and the obital peiod is days. The semimajo axis of Neptune is AU and the obital peiod is 60,190 days (165 Eath yeas). Poof of Keple s Thid Law. The aea of the ellipse which descibes the obit is A = T 0 da dt dt = T C dt = 1 2 CT since da/dt = 1 C fom the poof of Keple s Second Law. The aea of an 2 ellipse with semimajo axis of length a and semimino axis of length b is
14 13.6 Velocity and Acceleation in Pola Coodinates 14 πab. Theefoe since fom page 666 e = a2 b 2 b = a 1 e 2. Theefoe 1 2 CT = πab = πa2 1 e 2 a o e 2 a 2 = a 2 b 2 o b 2 = a 2 (1 e 2 ) o C 2 T 2 4 = π 2 a 4 (1 e 2 ) o T 2 a = 4π2 (1 e 2 )a. ( ) 3 C 2 Now the maximum value of, denoted max, occus when θ = π: max = (1 + e) 0 = 0(1 + e) 1 + e cos θ θ=π 1 e. Since 2a = 0 + max, then a = 0 + 0(1+e) 1 e 2 = 0(1 e) + 0 (1 + e) 2(1 e) = 0 1 e. Figue 13.36, page 760
15 13.6 Velocity and Acceleation in Pola Coodinates 15 Substituting this value of a into ( ) give T 2 a = 4π2 (1 e 2 ) 0 3 C 2 1 e = 4π2 (1 e 2 ) 0 GM(1 + e) 0 1 e since C 2 = GM(1 + e) 0 fom the poof of Keple s Fist Law. Theefoe T 2 a = 4π2 3 GM. Q.E.D. A Histoical Note. Claudius Ptolemy (90 ce 168 ce) pesented a model of the univese which was widely accepted fo almost 1400 yeas. In his Almagest he poposed that the univese had the Eath in the cente with the planets Mecuy, Venus, Mas, Jupite, and Satun, along with the sun and moon, obiting aound the Eath once evey day in cicula obits. In addition, the stas wee located on a sphee which otated once a day. His model was quite complicated and equied a numbe of epicycles which wee additional cicles needed to explain the complicated obseved motion of the planets (in paticula, the occasional etogade movement seen in the motion of the supeio planets Mas, Jupite, and Satun). Some of these ideas wee inheited fom Ptolemy s pedecessos such as Hippachus and Apollonius of Pega (both living aound 200 bce). Supisingly, anothe ancient Geek astonome which pedates each of these, Aistachus of Samos (310 bce 230 bce), poposed that the Sun is the cente of the univese and that the Eath is a planet obiting the
16 13.6 Velocity and Acceleation in Pola Coodinates 16 sun, just like each of the othe five planets. Howeve, Ptolemy s model was much moe widely accepted and adopted by the Chistian chuch. Polish astonome Nicolaus Copenicus ( ) poposed again that the sun is the cente of the univese and that the planets move in pefect cicles aound the sun with the sun at the cente of the cicles. His ideas wee published shotly befoe his death in De evolutionibus obium coelestium (On the Revolutions of the Celestial Sphees). The Copenican system became synonymous with heliocentism. Copenicus s model was meant to simplify the complicated model of Ptolemy, yet its pedictive powe was not as stong as that of Ptolemy s model (due to the fact that Copenicus insisted on cicula obits). Johannes Keple ( ) used extemely accuate obsevational data of planetay positions (he used the data of Tycho Bahe which was entiely based on naked eye obsevations) to discove his laws of planetay motion. Afte yeas of tying, he used data, pimaily that of the position of Mas, to fit an ellipse to the data. In 1609 he published his fist two laws in Astonomia nova (A New Astonomy). Howeve, Keple s wok was not based on any paticula theoetical famewok, but only on the obsevations. It would equie anothe to actually explain the motion of the planets.
17 13.6 Velocity and Acceleation in Pola Coodinates 17 Isaac Newton ( ) invented calculus in 1665 and 1666, but failed to publish it at the time (which lead to yeas of contovesy with Gottfied Leibniz). In 1684, Edmund Halley (of the comet fame) asked Newton what type of path an object would follow unde an invesesquae law of attaction (of gavity). Newton immediately eplied the shape was an ellipse and that he had woked it out yeas befoe (but not published it). Halley was so impessed, he convinced Newton to wite up his ideas and Newton published Pincipia Mathematica in 1687 (with Halley coveing the expense of publication). This book was the invention of classical physics and is sometimes called the geatest scientific book of all time! Newton s poof of Keple s laws fom his invesesquae law of gavitation is cetainly one of the geatest accomplishments of classical physics and Newton s techniques (which we have seen in this section) uled physics until the time of Albet Einstein ( )!
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