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Fundamentals of Physics School of Physical Science and Technology

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1 Fundamentals of Physics School of Physical Science and Technology
Mechanics (Bilingual Teaching) 张昆实 School of Physical Science and Technology Yangtze University

2 Chapter 14 Gravitation 14-1 The World and the Gravitational Force
14-2 Newton's Law of Gravitation 14-3 Gravitation and the Principle of Superposition 14-4 Gravitation Near Earth's Surface 14-5 Gravitation Inside Earth 14-6 Gravitational Potential Energy 14-7 Planets and Satellites: Kepler's Laws 14-8 Satellites: Orbits and Energy

3 14-1 The World and the Gravitational Force
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 14-1 The World and the Gravitational Force
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 light-years across. Andromeda galaxy

5 14-1 The World and the Gravitational Force
The most distant galaxies are known to be over 10 billion light years away ! What force binds together these progressively larger structures, from star to galaxy to supercluster ? It is the gravitational force that not only holds you on Earth but also reaches out across intergalactic space.

6 14-1 The World and the Gravitational Force
The great steps of China toward the space Yang Liwei and ShenZhou 5

7 14-1 The World and the Gravitational Force
China CE-1 project Exploring the Moon Orbit around the Moon : Lauching: shifting: Moon orbit

8 14-2 Newton's Law of Gravitation
Nowton published the law of gravitation In It may be stated as follows: Every particle in the universe attracts every other particle with a force that is directely proportional to the product of the masses of the particles and inversely proportinal to the square of the distance between them. ( Nowton’s law of gravitation ) (14-1) Translating this into an equation

9 14-2 Newton's Law of Gravitation
( Nowton’s law of gravitation ) (14-1) is the gravitational constant with a value of (14-2) Fig.14-2 Particle 2 attracts particle 1 with Particle 1 attracts particle 2 and are equal in magnitude but opposite in direction. These forces are not changed even if there are bodies lie between them

10 14-2 Newton's Law of Gravitation
( Nowton’s law of gravitation ) (14-1) is the gravitational constant with a value of (14-2) Fig.14-2 Particle 2 attracts particle 1 with Particle 1 attracts particle 2 and are equal in magnitude but opposite in direction. These forces are not changed even if there are bodies lie between them

11 14-2 Newton's Law of Gravitation
Nowton’s law of gravitation applies strictly to particles; also applies to real objects as long as their sizes are small compared to the distance between them (Earth and Moon). What about an apple and Earth? Shell theorem: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

12 14-2 Newton's Law of Gravitation
Nowton’s law of gravitation applies strictly to particles; also applies to real objects as long as their sizes are small compared to the distance between them (Earth and Moon). What about an apple and Earth? Shell theorem: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

13 14-3 Gravitation and the Principle of Superposition
Given a group of n particles, there are gravitational forces between any pair of particles. 1 1 extented body 3 5 Finding the net force acting on particle 1 from the others 2 i n First, compute the gravitational force that acts on particle 1 due to each of the other particles, in turn. 4 a group of n particles Then, add these forces vectorialy. (14-3) (14-4) (14-5) extented body For particle- the Principle of Superposition

14 14-4 Gravitation Near Earth's Surface
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 (14-8) If the particle is releaced, it will fall towards the center of Earth with the gravitatonal acceleration : Gravitation Near Earth's Surface (14-9) the gravitatonal acceleration (14-10)

15 14-4 Gravitation Near Earth's Surface
the gravitatonal acceleration (14-10) Gravitation Near Earth's Surface

16 14-4 Gravitation Near Earth's Surface
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 gravitational acceleration m However differs from (14-10) Weight differs from (14-8) Because: (1) Earth is not uniform, (2) Earth is not a perfect sphere, (3) Earth rotates.

17 14-4 Gravitation Near Earth's Surface
Crust Inner core Oute core Mantle (1) Earth is not uniform The density of Earth varies radially: Inner core (103 kg/m3) Outer core (103 kg/m3) Mantle (103 kg/m3) and the density of the crust (outer section) of Earth varies from region to region over Earth’s surface. Thus, varies from region to region over the surface.

18 14-4 Gravitation 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. This is one reason the free-fall acceleration increases as one proceeds, at sea level, from the equator toward either pole.

19 14-4 Gravitation Near Earth's Surface
(3) Earth is rotating. An object located on Earth’s surface anywhere (except at two poles) must rotate in a circle about the Earth’s rotation axis and thus have a centripital acceleration ( requiring a centripital net force ) directed toward the center of the ciecle. equator How Earth’s rotation causes to differ from ? Put a crate of mass on a scale at 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 (14-11)

20 14-4 Gravitation Near Earth's Surface
(3) Earth is rotating. Newton’s secend law for the axis (14-11) Reading on the scale (14-12) mearsure weight = magnitude of gravitation force mass times centripetal acceleration - (14-13) Relation between and = Free-fall acceleration gravitation centripetal -

21 Gravitation Inside Earth
14-4 Newton’s shell theorem can also be applied to a particle located Inside a uniform shell: A uniform spherical shell of matter exerts no net gravitational force on a particle located inside it. 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.

22 14-6 Gravitational Potential Energy
In section 8-3 (P144) the gravitational potential energy of a particle-Earth system is studied. Usually we chose the reference configuration ( the particle was on Earth’s surface) as having a gravitational potential energy of zero. (8-9) Here, considerthe gravitational potential energy of two particles, of masses and , seperated by a distance 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

23 14-6 Gravitational Potential Energy
(14-20) 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 energy of the apple”, because when a apple moves in the vicinity of Earth, (apple)

24 14-6 Gravitational Potential Energy
(14-20) For a system of three particles, the gravitational potential energy of the system is the sum of the gravitational potential energies of all three pairs of particles. ( calculating as if the other particle were not there ) (14-21)

25 14-6 Gravitational Potential Energy
Find the gravitational potential energy of a ball at point P, at radial distance R from Earth’s center. Proof of (14-20): Differential displacement The work done on the ball by the gravita- tional force as the ball travels from point P to a great (infinite) distance from Earth is (14-22) (14-23) (14-24)

26 14-6 Gravitational Potential Energy
Differential displacement (14-24) From Eq. 8-1 (14-20)

27 14-6 Gravitational Potential Energy
Path Independence Earth Moving a ball from A to G along a path : consisting of three radial lengths and three circular arcs (cented on Earth). The work done by the gravitational force on the ball as it moves along ABCDEFG: The work done along each circular arc is zero, because at every point. the gravitational force is a conservative force, the work done by it on a particle is independent of the actual path taken between points A and G.

28 14-6 Gravitational Potential Energy
Earth From Eq. 8-1: (8-1) (14-25) Since the work done by a conservative force is independent of the actual path taken. The change in gravitational potential energy is also independent of the actual path taken.

29 14-6 Gravitational Potential Energy
We derived the potential energy function from the force function potential energy and force Now let’s go the other way: derive the force function from the potential energy function (14-26) radially inward This is Newton’s law of gravitation (14-1) . ( Derivation is the inverse operation of integration )

30 14-6 Gravitational Potential Energy
Escape Speed The 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 potential energy Its kinetic energy When the projectile reches infinity, it stops. Its kinetic energy Its potential energy From the principle of conservation of energy (14-27) Escape Speed:

31 14-6 Gravitational Potential Energy
Escape Speed (14-27) From Earth: The escape speed does not depend on the direction in which a projectile is fired from a planet. eastward However, attaining that speed is easier if the projectile is fired in the direction the launch site is moving as the planet rotates about its axis . For example, rockets are launched eastward at XiChang to take the advantage of the eastward speed of 1500km/h due to Earth’s rotation.

32 14-1 The World and the Gravitational Force
China CE-1 project Exploring the Moon Orbit around the Moon : launched eastward Lauching: shifting: Moon orbit

33 14-7 Planets and Satellites: Kepler's Laws
The motion of the planets have been a puzzle since the dawn of history. Johannes Kepler ( ) worked out the empirical laws that governed these motions based on the data from the observations by Tycho Brahe ( ). 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 eccentricity of Earth’s orbit is only

34 14-7 Planets and Satellites: Kepler's Laws
2 THE LAW OF AREAS: A line that connects a planet to the Sun sweeps out equal 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.

35 14-7 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 instantaneous rate at which area is been sweept out is (14-29) The magnitude of the angular momen- tum of the planet about the Sun is (14-30) constant constant

36 14-7 Planets and Satellites: Kepler's Laws
3 THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. Applying Newton’s second law to the orbiting planet : (14-32) From Eq (14-33) The quantity in parentheses is a constant that depends only the mass M of the central body about which the planet orbits.

37 14-7 Planets and Satellites: Kepler's Laws
(14-33) 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)

38 14-8 Satellites: Orbits and Energy
As a satellite orbits Earth on its elliptical path, its speed and the distance from the center of Earth fluctuate with fixed periods. However, the mechanical energy E of the satellite remains constant. The potential energy of the system (or the satellite) is To find the kinetic energy of the satellite, use Newton’s second law Compare U and K (14-41) (14-43) ( Circular orbit ) (14-42)

39 14-8 Satellites: Orbits and Energy
The total mechanical energy E of the satellite is (14-44) ( circular orbit ) Compare E and K (14-45) ( circular orbit ) For a satellite in an elliptical orbit of semimajor axis (14-46) ( elliptical orbit )

40 14-8 Satellites: Orbits and Energy


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