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F21cos-F21sin(13341010)cos(60)-(13341010)sin(6...

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F21cos-F21sin(13341010)cos(60)-(13341010)sin(6...

    Notes for Ch&Mat.E CHAPTER 13

    Gravitation

NEWTON’S LAW OF GRAVITATION

     F ? m F ? m 121212

    F ? mm 12

    1 Also, F ? 2r

    Newton’s law of universal gravitation:

    ;Gmm12ˆ Fr12212r

    1122where the Gravitational constant G6.67×10 N)m/kg.

    F means the force acted on m by m 1212

    The negative sign indicates attraction force between the two objects. Thus in

    a general form no subscript is used.

    Gmm12ˆ r2r

Spherical Mass Distribution

     Although in general, Newtons law of gravitation applies only to point particles, there is an exception. According to the “point mass theorem” a spherically symmetric mass distribution (such as a uniform sphere or shell) attracts a point particle as if all its mass were concentrated as the center.

     Thus, we may take r to be the distance from the point particle to the center.

    1This result is a consequence of the nature of the force law. Another 2r

    consequence of the inverse square nature of the force law is that a point particle inside a uniform spherical shell experiences no net force.

    -1-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E Principle of superposition

    If there are several particles present, the net force on a given particle, say m, is 1the vector sum of the individual forces due to each of the other masses.

    ;;;;

    ?……? FFFF12131N1

    ;;;;

     FFFF1213141

    The gravitational force between m and 1

    m is not affected by the presence of a 2

    body placed between them.

    -2-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E

    Example 13-1

    m2 kg 2

    ;

     ? F2

     L2m

     m4 kg m3 kg 13

    11Gmm6.6710(4)(2)?1012F1.334×10 (N) 2122r2

    ;10ˆˆˆFcosFsin(1.334×10)cos(60?)iiFj212121

    10ˆ(1.334×10)sin(60?) j

    ;1110ˆˆ6.67×101.155×10 Fij21

    11Gmm6.6710(2)(3)?1023F1.0005×10 (N) 2322r2

    ;1010ˆˆˆˆFcosFsin(1.0005×10)cos(60?)(1.0005×10)sin(60?) Fijij232323

    ;1111 ˆˆ5.0025×108.665×10Fij23

    ;;;

     FFF22123

    11101111ˆˆˆˆ(6.67×101.155×10)(5.0025×108.665×10) ijij

    ;1110ˆˆ(1.668×10)(2.021×10) Fij2

    -3-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E GRAVITATIONAL AND INERTIAL MASS

    Newtons Second law:

    Fma ? mm inertial mass I

    Newtons Law of gravitation:

    GmM ? mm Gravitational mass FG2r

    Gravitational force can be expressed as

    mGFmg ? Fma ? ag GImI

     Basically, mm GI

    Attempt to distinguish m and m made by Newton GI

    For a simple pendulum of length L,

    mLIT2

    mgG

     When different bobs of materials were used,

    LT2

    g

    i.e, m m GI

    Principle of equivalence

    No experiment can distinguish the effect of a gravitational force from that of an

    inertial force in an accelerated frame. When a ball is released, it accelerates toward the

    floor of the cabin. The astronaut can not tell whether the acceleration relative to the rocket was

    leftcaused by the force of gravity or,

    righta consequence of the acceleration of the

    rocket in gravity-free space.

    -4-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E THE GRAVITATIONAL FIELD STRENGTH

    Consider a stationary particle of mass M exerts a force on a particle of mass m,

    which moves from a position to another, as shown in the figure.

The extent of mass m affected by the mass M is called force field.

    Note the force between the two particles is

    GMmˆ r2r

    ; and m g

    GM;ˆ Magnitude of force field /m = rg2r

    ˆ1. Origin of is at M. r

    ;2. The quantity , measured in N/kg, is called gravitational field g

    strength, the gravitational force per unit mass.

    3. The gravitational force on any particle of mass m, i.e., weight, is

    given by

    ;m gW

    -5-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E

    Acceleration due to Gravity

    Consider a particle of m at some latitude on the earth,

    ;Gravitational strength: g

    ;The gravitational force m is directed to the g

    center and serves two function:

    1. It causes the particle to fall with acceleration

    ;. g

    ;2. It produces the centripetal acceleration . ac

    ;;From m Fa

    ;;;mm() gagc

    ;;Vectors and are parallel only at the poles and at the equator. gg

    ;At the pole, 0, ggapp c

    2624?(6.4?10)4R22At the equator, a0.034 (m/s)3.4 (cm/s) c22T(86400)

    gga gg3.4 eecee

    gggg3.4 pepe

     Measurement shows gg5.2 pe

     gg1.8 ? earth is not a perfect sphere. pe

    GMTheoretically, g, R6357 km and R6378 km pe2R

    gg6.41.8 pe

    The difference is due to:

    (1) Equation used here is only applicable to uniform spherical distribution.

    (2) The density of the earth in radial direction is a non-uniform distribution.

    -6-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E KEPLER’S LAWS OF PLANETARY MOTION

    Law 1.

    The planets move around the sun in elliptical orbits with the sun at one

    focus.

    major axis: AP2a

     P: Perihelion A: aphelion

Law 2.

    The line joining the sun to a planet sweeps out equal areas in equal times.

    Area SABArea SCD

    Law 3

    The square of the period of a planet is proportional to the cube of its mean

    distance from the sun.

    mean distancesemimajor axisa

    23Ta

     : constant for a fixed star system.

    -7-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E

    Energy in an Elliptical Orbit

    At perihelion and aphelion, The conservation of angular momentum

    mrvmrv ? rvrvAAppAAPP

    The conservation of mechanical energy

    GMmGMm1122mm vvpA22rrAP

    1122 ? 2GM() vvpArrpA

    2rr112AA (1)2GM() vvvPAA2rrrrpPAP

    22rrrrpA2PA()2GM() vA2rrrAPP

    (r;r)(rr)(rr)2PAPAPA2GM vArrPA

    r2P()2GM() vrrAPArA

     Again, 2a rrAP

    GMr2P vAarA

    GMr2Av Similarly, parP

    -8-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E

    Bound and Unbound Trajectories

    A cannonball is fired from the peak of a very tall tower with speed v

Escape velocity: v esc

    GMm12 Total energy Emv 22r

     Conservative of mechanical energy;EE (EE) ifat the surfacer?~

     and E0 f

    GMm2GM12m0 ? v vescesc2r2r

    Velocity at orbital of radius r: v c

    2GMGMvAt orbital of radius r mmg and g ? v crrr

     Velocity Phenomena Energy

     0 Very small The projectile strikes the earth.

     The orbit is elliptical, with the peak as the apogee vv0 c (遠地點).

     vv0 The orbit is circular. c

     vvv0 The orbit is elliptical, with the peak as the perigee. cesc

     vv0 The object is unbounded, and the path is a parabola. esc

     vv0 The object is unbound, and the path is a hyperbola. esc

    -9-

    CHAPTER 13 GRAVITATION

    Notes for Ch&Mat.E

    CONTINUOUS DISTRIBUTIONS OF MASS

    The gravitational field strength g due to an extended object

    Gdm dg 2r

    GTotal gravitational field g dm?2r

    -10-

    CHAPTER 13 GRAVITATION

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