Chapter 8 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Gravity 重力 Pearson Education, Inc. Slide 8-1

Transcription

1 Chapter 8 Lecture Essential University Physics Richard Wolfson 2 nd Edition Gravity 重力 Slide 8-1

2 In this lecture you ll learn 簡介 Newton s law of universal gravitation 萬有引力 About motion in circular and other orbits 重力軌道 How to calculate changes in gravitational potential energy 重力位能 The concept of escape speed 脫離速度 The field concept 場的觀念 Slide 8-2

3 1609~1619, Johannes Kepler published 3 Laws of planetary motion based on previous astromical observations. Slide 8-3

4 不同的軌道 Slide 8-4

5 Law of Universal Gravitation 萬有引力定律 Introduced by Isaac Newton, the Law of Universal Gravitation states that any two masses m 1 and m 2 attract with a force F that is proportional to the product of their masses and inversely proportional to the distance r between them: F = Gm 1 m 2 r 2 Here G = N m 2 /kg 2 is the constant of universal gravitation. 萬有引力常數 Strictly speaking, this law applies only to point masses. But Newton showed that it applies to spherical masses of any size, and it is a good approximation for any objects that are small compared with their separation. 質點物體之間, 球形物體之間, 之間的距離很遠時 Slide 8-5

6 Henry Cavendish s torsion experiment to measure G in 1798 Slide 8-6

7 Orbits 各種軌道 Newton explained orbits using universal gravitation and his laws of motion: 萬有引力加上運動 (3) 定律 Bound orbits are generally elliptical. 束縛軌道的形狀是橢圓 In the special case of a circular orbit, the orbiting object falls around a gravitating mass, always accelerating toward its center with the magnitude of its acceleration remaining constant. 也有圓形軌道 Slide 8-7

8 Orbits 各種軌道 Newton explained orbits using universal gravitation and his laws of motion: 萬有引力加上運動 (3) 定律 Slide 8-8

9 Comet ISON is Coming!!! Comet C/2012 S1 (ISON) is on its way to skirt around the Sun in November discovered by Vitali Nevski and Artyom Novichonok on September 21, 2012 using a 40 -cm (16-inch) telescope in Russia International Scientific Optical Network (ISON) comets have been in a deep freeze for the past 4 billion years simulation Slide 8-9

10 Orbits 各種軌道 Newton explained orbits using universal gravitation and his laws of motion: 萬有引力加上運動 (3) 定律 Unbound orbits are hyperbolic or (borderline case) parabolic. 雙曲線, 拋物線的非束縛軌道 Slide 8-10

11 Projectile Motion and Orbits 拋體運動與軌道 澄清 :The parabolic trajectories of projectiles near Earth s surface are actually sections of elliptical orbits that intersect Earth. 地面上的拋物體的軌跡是橢圓軌道的一部分 The trajectories are parabolic only in the approximation that we can neglect Earth s curvature and the variation in gravity with distance from Earth s center. 這樣的結果是因為忽略地面的形狀與離地心的距離的變化 ( 等加速度自由落體 ) Slide 8-11

12 Circular Orbits 圓軌道 In a circular orbit, gravity provides the force of magnitude mv 2 /r needed to keep an object of mass m in its circular path about a much more massive object of mass M. Therefore, GMm 重力 = 向心力 = mv2 r ( 但 =mg) r 2 Orbital speed: Orbital period: Kepler s third law: v = GM r 2 4π r T = GM T 2 3 r 2 3 都由 r 決定 For satellites in low-earth orbit, the period is about 90 minutes. 近地衛星的週期 ~90minutes Slide 8-12

13 Gravitational Potential Energy 重力位能 Because the gravitational force changes with distance, it s necessary to integrate to calculate potential energy changes ΔU ( ) over large distances. Where in chapter 7, we have ΔU AB = This integration gives B A F cons. d r = W cons 1 r2gmm r2 2 r r2 1 1 Δ U12 = dr = GMm r dr = GMm = GMm r 2 r1 1 r r 1 r r Slide 8-13

14 Gravitational Potential Energy This result holds regardless of whether the two points are on the same radial line. 當 r 不變時 U 不變 It s convenient to take the zero of gravitational potential energy at infinity. Δ Then the gravitational potential energy becomes 取無窮遠處為零位能, 則 r 處的位能為 Ur ()= GMm r Slide 8-14

15 Energy and Orbits 能量與軌道 The total energy E = K + U, the sum of kinetic energy K and potential energy U, determines the type of orbit an object follows: 機械能 E=K+U 的值決定軌道形狀 : 1. E < 0: 2. E > 0: 3. E = 0: Slide 8-15

16 Energy and Orbits 能量與軌道 E < 0: The object is in a bound(closed), elliptical orbit. Special cases include circular orbits and the straightline paths of falling objects. E > 0: The orbit is unbound(open) and hyperbolic. E = 0: The borderline case gives a parabolic(open) orbit. Slide 8-16

17 Escape Speed 脫離速度 An object with total energy E less than zero is in a bound orbit and can t escape from the gravitating center. E<0 時無法脫離重力中心 With energy E greater than zero, the object is in an unbound orbit and can escape to infinitely far from the gravitating center. E>0 時能夠脫離, 進入一個 open orbit The minimum speed required to escape is given by, E=0 時能量恰好足夠能逃脫 0 = K +U = 1 2 mv2 GMm r Solving for v gives the escape speed: v esc = 2GM r Escape speed from Earth s surface is about 11 km/s. Slide 8-17

18 Energy in Circular Orbits 圓軌道的能量 In the special case of a circular orbit, kinetic energy and potential energy are precisely related: GMm U = 2K = mv2 r 2 r Thus in a circular orbit the total energy is E = K +U = K = 1U = GMm 2 2r This negative energy shows that the orbit is bound. The lower the orbit, the lower the total energy but the faster the orbital speed. v = GM r This means an orbiting spacecraft needs to lose energy to gain speed. Slide 8-18

19 Applications using earth s gravity Slide 8-19

20 Can be used to explain the causes of tides. Slide 8-20

21 The Gravitational Field 重力場 It s convenient to describe gravitation not in terms of action at a distance but rather in terms of a gravitational field that results from the presence of mass and that exists at all points in space. 超距力 與重力場 Slide 8-21

22 The Gravitational Field 重力場 A massive object creates a gravitational field in its vicinity, and other objects respond to the field at their immediate locations. The gravitational field can be visualized with a set of vectors giving its strength (in N/kg; equivalently, m/s 2 ) and its direction. Near Earth s surface On a larger scale Slide 8-22

23 We can also use gravitational potential U to show gravitational field. Connection is done by the relation: F x = du dx 加上等位線 Slide 8-23

24 Summary The Law of Universal Gravitation states that any two masses m 1 and m 2 attract with a force F that is proportional to the product of their distances and inversely proportional to the distance r between them: F = Gm 1 m 2 r 2 Motion under gravity includes Bound elliptical and circular orbits when the orbiting object s total energy is less than zero Open parabolic and hyperbolic orbits when the total energy is zero or greater The gravitational field describes the force of gravity in terms of a field at all points in space; an object then responds to the field in its immediate vicinity. Slide 8-24

25 Homework 11, 19, 41, 47, 53, 59 Slide 8-25