Chapter 13 Gravity - Austin Community · PDF fileChapter 13 Gravity. MFMcGraw-PHY 2425...

Click here to load reader

  • date post

    20-May-2018
  • Category

    Documents

  • view

    225
  • download

    4

Embed Size (px)

Transcript of Chapter 13 Gravity - Austin Community · PDF fileChapter 13 Gravity. MFMcGraw-PHY 2425...

  • Chapter 13

    Gravity

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 2

    Gravity

    Keplers Laws

    Newtons Law of Gravity

    Gravitational Potential Energy

    The Gravitational Field

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 3

    Characteristics of Gravity

    Acts along a line connecting the center of mass of the two

    bodies.

    It is a central force which implies that it is a conservative

    force.

    It is only attractive. There are no repulsive gravitational

    forces, i.e. there is no anti-gravity.

    Its magnitude is proportional to the two masses

    g 1 2F m m

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 4

    Characteristics of Gravity

    1 2g 2

    12

    m mF

    r

    1 2g 2

    12

    m mF = G

    r

    g 2

    1F

    r

    The strength of the gravitational force decreases with

    distance

    Therefore

    The value of G was determined from the data of

    Henry Cavendish: G = 6.67x10-11 Nm2/kg2

    The gravitational force is

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 5

    Characteristics of Gravity

    Gravity is the weakest of all the forces. It takes large

    masses to produce a gravitational force that is

    noticeable.

    Gravity acts most on the grand astronomical scale

    where large masses are found.

    However, it is the exact balance of the positive and

    negative charges that effectively cancels the long range

    electric fields that allows gravity to be observed even at

    these large astronomical distances.

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 6

    Characteristics of Gravity

    Gravity is a vector quantity

    1 212 122

    12

    m mF = G r

    r

    12 21F = - F

    12 21r = - r

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 7

    Early Astronomers

    Tycho Brahe (1546-1601)

    He gathered observational data over a twenty year

    period of measuring planet positions.

    Johannes Kepler (1571-1630)

    Kepler worked with Brahe

    He inherited Brahes data when he died.

    Kepler deduced his three laws from this data

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 8

    Keplers Laws

    1. Law of Orbits - ellipse

    2. Law of Areas - equal area in equal times

    3. Law of Periods -

    22 3

    s

    4T = r

    GM

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 9

    Keplers 1st Law

    Perigee Apogee

    (Closest approach) (Farthest approach)

    Elliptical

    shaped orbit

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 10

    Construction of an Ellipse

    r1 +r2 = 2a

    a = semi major axis

    b = semi minor axis

    F are the focal points

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 11

    Keplers 2nd Law

    Equal areas are swept out in equal times

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 12

    Derivation of Keplers 2nd Law

    12dA = r vdt

    r mvdA = dt

    2m

    dA L=

    dt 2m

    F is a central force so L

    is conserved and

    therefore

    dA= constant

    dt

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 13

    rava = rpvp

    Conservation of Angular Momentum

    L = mrvsin = constant, therefore rvsin is constant with r,

    v and all changing but the product of the three remaining

    constant in value

    Evaluate the

    expression at = 900

    where sin = 1, at

    apogee and perigee

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 14

    Keplers LawsVariation of periods with mean orbital radius

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 15

    Keplers 3rd Law

    22 3

    s

    4T = r

    GM

    The Period Equation

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 16

    Keplers 3rd Law

    22 3

    s

    4T = r

    GM

    Astronomical Unit = Mean radius of earth orbit around sun

    1 AU = 1.50 x 10-11 m = 93.0 x 106 mi

    For orbits around the Sun 2 3

    T (Years) = R (AU)

    For other applications replace Ms in the equation above

    with the mass of the body that the object is orbiting

    around.

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 17

    The Determination of G

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 18

    The Determination of G

    According to legend, i.e. one text book author copying the

    historical aspects of physics from the authors that came before

    him, Henry Cavendish made the first accurate measurement of

    G in 1745.

    It turns out that the legend is incorrect. In fact, Cavendish

    wasnt even interested in measuring the value of G. Henry

    Cavendish wanted to measure the density of the earth.

    75 years later another researcher was able to squeeze a value of

    G out of Cavendishs old data.

    Moral of the story - take good notes while youre alive and you

    might still be making discoveries after youre dead.

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 19

    The Cavendish Experiment(minus the environmental shielding)

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 20

    The Cavendish Experiment

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 21

    Actual Cavendish Experiment Data

    Each tiny square is 1 minute.

    Scr

    e en

    Def

    lect

    ion

    - 1

    .5m

    aw

    ay

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 22

    Gravitational Potential Energy

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 23

    Bound State - Unbound State

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 24

    EGM mU(r) = -r

    Escape Speed of a Projectile

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 25

    Gravitational Field

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 26

    Preview of Electric Field Calc.-EPII

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 27

    g-Field of a Thin Spherical Shell

    2

    GMg = - r; r > R

    r

    g = 0; r < R

    g outside the mass distribution

    is the same as if all the mass

    was concentrated at the CM.

    Requires uniform mass

    distribution or a mass

    distribution that only varies

    with r.

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 28

    Gravitational Field Inside a Hollow Sphere

    gF = 02

    1 1 1

    2

    2 2 2

    1 2

    2 2

    1 2

    1

    2

    1

    m A r= =

    m A r

    m m=

    r r

    Gmg =

    r

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 29

    g-Field Distribution - Solid Sphere

    3 34 43 3

    3

    3

    3

    2 2 3 3

    M' M=

    r R

    rM' = M

    R

    GM' GM r GMg(r) = - = - = - r

    r r R R

    for r R

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 30

    Homework Problems

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 31

    Gravitational force on m and G field when m is gone

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 32

    Force on mass m at distances: 3a, 1.9a and 0.9 a

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 33

    What must their speed be if they are to orbit their common

    center under their mutual gravitational attraction?

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 34

    Extra Slides

  • Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 35