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制作 张昆实 Yangtze University

Bilingual Mechanics. Chapter 7 Gravitation. 制作 张昆实 Yangtze University. Chapter 7 Gravitation. 7-1 What Is Physics? 7-2 Newton's Law of Gravitation 7-3 Gravitation and the Principle of Superposition 7-4 Gravitation Near Earth's Surface

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制作 张昆实 Yangtze University

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  1. BilingualMechanics Chapter 7 Gravitation 制作 张昆实 Yangtze University

  2. Chapter 7 Gravitation 7-1 What Is Physics? 7-2 Newton's Law of Gravitation 7-3 Gravitation and the Principle of Superposition 7-4 Gravitation Near Earth's Surface 7-5 Gravitation Inside Earth 7-6 Gravitational Potential Energy 7-7 Planets and Satellites: Kepler's Laws 7-8 Satellites: Orbits and Energy

  3. 7-1 What Is Physics Have you ever imaged how vast is the universe? The sun is one of millions of stars that form the Milky Way Galaxy. We are near the edge of the disk of the galaxy, about 26000 light-years from its center. Milky Way galaxy

  4. 7-1 What Is Physics Andromeda galaxy The universe is made up of many galaxies, each one containing millions of stars. One of the galaxies is the Andromeda galaxy. The great galaxy M31 in the Constellation Andromeda is more than 100000 light-years across.

  5. 7-1 What Is Physics ★The most distantgalaxies are known to be over 10 billionlight years away ! ★What forcebinds together these progressively larger structures, from star togalaxy to supercluster ? ★It is the gravitational force that not only holds you on Earthbut also reaches out across intergalactic space.

  6. 7-1 What Is Physics The great steps of China toward the space Lauching Shenzhou five (神州五号) Space ship

  7. 7-1 What Is Physics China CE-1 project Exploring the Moon Orbit around the Moon: 2007-11-5 Lauching: 2007-10-24 shifting: 2007-11-1 Moon’s orbit

  8. A report on exploring deep space & CE- project by academician Ou Yang Ziyuan in Yangtze University

  9. Academician Ou Yang Ziyuanpresent Yangtze Universitywiththe all-around picture of the Moon taking by CE-1

  10. Chinese astronauts Jing Haipeng(L), Zhai Zhigang(C) and Liu Boming wave hands during a press conference in Jiuquan Satellite Launch Center (JSLC) in Northwest China's Gansu Province, September 24, 2008. The Shenzhou VII spaceship will blast off Thursday evening from the JSLC to send the three astronauts into space for China's third manned space mission.

  11. China´s manned spacecraft Shenzhou-7 blasts off

  12. Chinese taikonauts report they feel "physically sound"

  13. Astronauts assemble EVA suit for spacewalk

  14. Chinese astronaut Zhai Zhigangis ready for spacewalk

  15. Chinese astronautZhai Zhigangistraveling indeep space

  16. Congratulations to the successful launching of Shenzhou-7 ! The fundamental principles of space flight is Mechanics ! Physics is the cradle of modern science and technology !

  17. Translating this into an equation (7-1) ( Nowton’s law of gravitation ) 7-2 Newton's Law of Gravitation Nowton published the law ofgravitation In 1687. It may be stated as follows: Every particlein the universeattractsevery other particlewitha forcethat isdirectely proportionalto the product of the masses of the particles andinversely proportinal to the square of thedistance between them.

  18. (7-2) Particle 2 attracts particle 1 with Particle 1 attracts particle 2 with and are equal in magnitude but opposite in direction. Fig.14-2 These forces are notchanged even if there are bodies lie between them (7-1) ( Nowton’s law of gravitation ) 7-2 Newton's Law of Gravitation is the gravitational constant with a value of

  19. What about an apple and Earth? Shell theorem: A uniform spherical shell of matter attracts a particle that is outside the shell as ifall the shell’s mass were concentrated at its center. 7-2 Newton's Law of Gravitation Nowton’s law of gravitation applies strictly to particles; also appliesto real objects as long as their sizes are small compared to the distance between them (Earth and Moon).

  20. Given a group of nparticles, there are gravitational forces between any pair of particles. 1 1 Finding the net force acting on particle 1 from the others extented body First, compute the gravitational force that acts on particle 1 due to each of the other particles, in turn. Then, add these forces vectorialy. (7-4) For particle- (7-6) (7-5) extented body 7-3 Gravitation and the Principle of Superposition 3 the Principle of Superposition 5 2 i n 4 a group of nparticles

  21. A particle (m) locates outside Earth a distance r from Earth’s center. The magnitude of the gravitational force from Earth (M) acting on it equals (7-7) If the particle is releaced, it will fall towards the center of Earth with the gravitatonal acceleration : Gravitation Near Earth's Surface (7-8) the gravitatonal acceleration (7-9) 7-4Gravitation Near Earth's Surface

  22. 7-1 Gravitation Near Earth's Surface (7-9) 7-4Gravitation Near Earth's Surface the gravitatonal acceleration

  23. m We have assumed that Earth is an inertial frame (negnecting its actual rotation). This allowed us to assume the free-fall acceleration is the same as the gravitationalacceleration However differs from (7-9) Weight differs from (7-7) Because: 7-4Gravitation Near Earth's Surface (1)Earth is not uniform, (2)Earth is not a perfect sphere, (3)Earth rotates.

  24. Crust Oute core Mantle Inner core Thus, varies from region to regionover the surface. 7-4Gravitation Near Earth's Surface (1)Earth is not uniform Thedensity of Earth varies radially: Inner core 12-14 (103 kg/m3) Outer core 10-12 (103 kg/m3) Mantle 3-5.5 (103 kg/m3) and the density of the crust (outer section) of Earth varies from region to region over Earth’s surface.

  25. This is one reason the free-fall acceleration increasesas one proceeds, at sea level, from the equator toward either pole. 7-4Gravitation Near Earth's Surface (2)Earth is not a perfect sphere Earth is approximately an ellipsoid, flattened at the poles and bulging at the equattor. Its equatorial radius is greater than its polar radius by 21km. equator Thus, a point at the poles is closer to the dense core of Earth than is a point on the equator.

  26. How Earth’s rotationcauses to differ from ? Put a crate of mass on a scaleat the equator and analyze it. Free-body diagram Normal force (outward in direction ) Gravitational force (inward in direction ) Centripital acceleration (inward in direction ) Newton’s secend law for the axis (7-10) 7-4Gravitation Near Earth's Surface (3)Earth is rotating. An object located on Earth’ssurface anywhere (except at two poles) must rotate in a circleabout the Earth’s rotation axis and thus have a centripital acceleration ( requiring a centripital net force ) directed toward the center of the ciecle. equator

  27. Reading on the scale (7-11) = magnitude of gravitation force mass times centripetal acceleration mearsure weight - (7-12) Relation between and - = Newton’s secend law for the axis gravitation acceleration centripetal acceleration Free-fall acceleration (7-10) 7-4Gravitation Near Earth's Surface (3)Earth is rotating.

  28. 7-5Gravitation Inside Earth Newton’s shell theorem can also be applied to a particle located Inside a uniform shell: A uniform spherical shell of matter exerts nonet gravitationalforce on a particle located insideit. • If a particle were to move into Earth, the • gavitational Force would change : • It would tend to increase because the • particle would be moving closer to the • center of Earth. • (2) It would tend to decrease because the • thickening shell of material lying outside • the particle’s radial position would not • exert any net force on the particle.

  29. ( particle on Earth’s surface ) (4-55) (4-55) However, we now choose a referance configuration with equal to zero as the seperation distance is large enough to be approximated as infinite. gravitational potential energy At finite 7-6 Gravitational Potential Energy The gravitational potential energy of a particle-Earth system (P101)

  30. (gravitational potential energy) (7-17) For any finite value of , the value of is negative. The gravitational potential energy is a property of the system of the two particles rather than of either particle along However, for Earth and a apple, We often speak of “potential energyof the apple”, because when a apple moves in the vicinity of Earth, (apple) 7-6 Gravitational Potential Energy

  31. gravitational potential energy (7-17) For a system of three particles, the gravitational potential energy of the system isthe sum of the gravitational potential energies of all three pairs of particles. ( calculating as if the other particle were not there ) (7-18) 7-6 Gravitational Potential Energy

  32. Findthegravitational potential energyof a ball at point P, at radial distance Rfrom Earth’s center. The work done on the ball by the gravita- tional forceas the ball travels from point P to a great (infinite) distance from Earth is (7-19) (7-20) 7-6 Gravitational Potential Energy Proof of (7-17): Differential displacement (7-21)

  33. (7-21) FromEq. 4-47 7-6 Gravitational Potential Energy Differential displacement (7-17)

  34. The work done along each circular arc is zero, because at every point. Earth 7-6 Gravitational Potential Energy Path Independence Moving a ball from A to G along a path: consisting of three radial lengths and three circular arcs (cented on Earth). The work doneby the gravitational forceon the ball as it moves along ABCDEFG: the gravitational force is a conservative force, the work done by it on a particle is independent ofthe actual path taken between points A and G.

  35. (4-47) (7-22) Since the work done by a conservative forceis independent ofthe actual path taken. The change in gravitational potential energy is also independent ofthe actual path taken. Earth 7-6 Gravitational Potential Energy From Eq. 4-47:

  36. We derived the potential energy function from the force function . radially (7-23) inward 7-6 Gravitational Potential Energy potential energy and force Now let’s go the other way: derive the force functionfrom thepotential energy function This is Newton’s law of gravitation (7-1) . ( Derivation is the inverse operation of integration )

  37. Escape Speed:The minimum initial speed that will cause a projectile to move up forever is called the (Earth) escape speed. Consider a projectile () leaving the surface of a planet with escape speed Its kinetic energy Its potential energy Its kinetic energy Its potential energy From the principle of conservation of energy Escape Speed: (7-24) 7-6 Gravitational Potential Energy When the projectile reches infinity, it stops.

  38. Escape Speed (7-24) From Earth: The escape speed does not depend on the direction in which a projectile is fired from a planet. However, attaining that speedis easier if the projectile is fired in the directionthe launch siteis moving as the planet rotates about its axis. 7-6 Gravitational Potential Energy eastward For example, rockets are launched eastward at XiChang to take the advantage of the eastward speed of 1500km/h due to Earth’s rotation.

  39. The motion of the planets have been a puzzle since the dawn of history. 1 THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus. is the semimajor axis of the orbit is the eccentricity of the orbit is the distance from the center of the ellipse to either focus the eccentricityof Earth’s orbit is only 0.0167 7-7Planets and Satellites: Kepler's Laws Johannes Kepler(1571-1630) worked out the empiricallaws that governed these motions based on the data from the observations by Tycho Brahe(1546-1601).

  40. 7-7 Planets and Satellites: Kepler's Laws 2 THE LAW OF AREAS: A line that connects a planet to the Sun sweeps outequal areas in the plane of the planet’s orbit in equal times; that is, the rate dA/dt at which it sweeps out area A is constant. This second law tell us that the planet will move most slowly when it is farthest from the Sun and most rapidly when it is nearest to the Sun.

  41. (7-26) (7-27) constant constant 7-7 Planets and Satellites: Kepler's Laws Proof of Kepler’s second law is totally equivalent to the law of conservation of angular momentum. The area of the wedge The instantaneousrate at which area is been sweept out is The magnitude of the angular momen-tum of the planet about the Sun is

  42. (7-29) From Eq. 11-20 (7-30) 7-7 Planets and Satellites: Kepler's Laws 3 THE LAW OF PERIODS: The square of theperiod of any planet is proportional tothe cube of the semimajor axis of its orbit. Applying Newton’s second law to the orbiting planet : The quantity in parentheses is a constant that depends only the mass M of the central body about which the planet orbits.

  43. (7-30) 7-7 Planets and Satellites: Kepler's Laws 3 THE LAW OF PERIODS: （水星） （金星） （地球） （火星） （木星） （土星） （天王星） （海王星） （冥王星）

  44. The potential energy of the system (or the satellite) is (7-34) ( Circular orbit ) 7-8 Satellites: Orbits and Energy As a satelliteorbits Earth on its elliptical path, its speed and the distance from the center of Earthfluctuate with fixed periods. However, the mechanical energy E of the satellite remains constant. To find the kinetic energy of the satellite, use Newton’s second law Compare U and K (7-32) (7-33)

  45. (7-35) ( circular orbit ) Compare E and K ( circular orbit ) (7-36) For a satellite in an elliptical orbit of semimajor axis ( elliptical orbit ) (7-37) 7-8 Satellites: Orbits and Energy The total mechanical energy E of the satellite is

  46. 7-15 7-16 7-8 Satellites: Orbits and Energy

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