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Newtons Law of Universal Gravitation

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Speed needed for a rocket to go from the planet surface to infinity... The 'gravitational drag' of the moon create the ocean tides on earth. ... – PowerPoint PPT presentation

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Title: Newtons Law of Universal Gravitation


1
Gravitation
  • Newtons Law of Universal Gravitation
  • Every mass in the universe attracts every other
    mass in the universe according to
  • F GMm
  • r2
  • where
  • G 6.67 x 10-11 Nm2kg-2

2
Gravitation
  • Newtons Law of Universal Gravitation
  • The law deals only with point masses. (A sphere
    can be considered a point mass located at the
    center of the sphere.)
  • When dealing with two objects, the gravitational
    force acting on each is equal and opposite.

3
Gravitation
  • Newtons Law of Universal Gravitation
  • Gravitational forces are always attractive.
    (Neither the mass nor the distance between the
    objects can be negative.)
  • Although gravitational force acts between all
    objects, the value of G shows the force will be
    significant only for very massive objects.

4
Gravitation
  • Newtons Law of Universal Gravitation
  • Note The law applies only to two masses at a
    time, but the gravitational force vectors may be
    added to the force vectors coming from a third
    mass in the system. This may be done iteratively
    for additional masses.
  • The resulting mathematical complexity limits
    solutions for 3 or more masses to approximate
    solutions, (the three body problem).

5
Gravitation
  • Gravitational Field Strength (g)
  • g F
  • m
  • Units are N?kg-1
  • Gravitational
  • Field Lines ?
  • (sphere and point)

6
Gravitation
  • Gravitational Field Strength
  • Since F GMm then F GM
  • r2 m r2
  • Since F mg then F g
  • m

7
Gravitation
  • Gravitational Field Strength
  • and therefore g GM
  • r2
  • is the gravitational field at the surface of a
    sphere of radius r. Use vectors to get the
    resulting field of two objects on a third.

8
Gravitation
  • Gravitational Field Strength
  • g GM
  • r2
  • G 6.67 x 10-11 Nm2kg-2
  • Mearth 5.97 1024 kilograms
  • rearth 6.4 x 106 kilometers

9
Gravitation
  • LaGrangian Points (Earth Moon)
  • L1 - Between the Earth and Moon
  • L2 - On the other side of the Moon
  • L3 - Opposite the Moon
  • L4 - 60 ahead of the Moon
  • L5 - 60 behind the Moon

10
Gravitation
  • LaGrangian Points (Earth Moon)
  • L1, L2 and L3 are quasistable, station-keeping is
    needed to keep an object on location at these
    points.
  • L4 and L5 are stable. Trash will collect at L4
    and L5. (Think Jupiter and asteroids.)

11
Gravitation
  • Gravitational Potential Energy
  • When a mass moves between two different heights
    (h) near the earths surface
  • difference in energy mgh PE
  • works only if (h) is not very large

12
Gravitation
  • Gravitational Potential Energy
  • The everyday solution for mgh treats the PE at
    the surface of the earth as 0.
  • The true solution for gravitational PE is 0 for
    PE taken at infinity. Therefore, an object at
    infinity loses potential energy as it moves
    towards a mass M.

13
Gravitation
  • Gravitational Potential Energy
  • Therefore the PE is negative for any object at
    point P, where P is not infinity.
  • This means that the work done in moving from
    infinity to point P is also negative.
  • Gravitational PE is a scalar.

14
Gravitation
  • Gravitational Potential Energy
  • By math not covered in IB...
  • PEG - GMm
  • r

15
Gravitation
  • Gravitational Potential Energy
  • STOP!
  • PROVE
  • ?PE mgh

16
Gravitation
  • Gravitational Potential Energy
  • Therefore, for r h, the change in PE
  • ?PE - GMm (- GMm)
  • r r h

17
Gravitation
  • Gravitational Potential Energy
  • ?PE - GMm GMm
  • r r h
  • Since F GMm then Fr GMm
  • r2 r

18
Gravitation
  • Gravitational Potential Energy
  • and F GMm then F(r h) GMm
  • (rh)2 (r h)
  • ?PE - Fr F(r h)
  • ?PE - Fr Fr Fh

19
Gravitation
  • Gravitational Potential Energy
  • ?PE - Fr Fr Fh Fh
  • Since F mg
  • ?PE Fh mgh

20
Gravitation
  • Gravitational Potential
  • A scalar that measures the energy per unit test
    mass.
  • VG W - GM
  • m RP
  • where W Work m Test Mass RP Planet
    Radius

21
Gravitation
  • Escape Speed
  • Speed needed for a rocket to go from the planet
    surface to infinity
  • VG - GM since VG W
  • RP m
  • and the energy difference between the surface and
    infinity is

22
Gravitation
  • Escape Speed
  • Speed needed for a rocket to go from the planet
    surface to infinity
  • VG (m) 0(m) - (- GM)(m) since W VG (m)
  • RP
  • W GM(m) and therefore W GMm
  • RP RP

23
Gravitation
  • Escape Speed
  • By the Work-Kinetic Energy Theorem
  • W ?KE ½mv2 GMm
  • RP
  • therefore v2 2GM and vescape
    (2GM/r)½ RP

24
Gravitation
  • Keplers Third Law
  • When one object is in orbit around another, the
    gravitational and centripetal forces must be
    exactly balanced.
  • FG GMm and FC mv2
  • r2 r
  • and therefore GMm mv2
  • r2 r

25
Gravitation
  • Keplers Third Law
  • Therefore the orbital speed is
  • vorbital (GM/r)½
  • Now rearranging this equation, we note that
  • GM v2r

26
Gravitation
  • Keplers Third Law
  • We can use geometry to figure the speed
  • v 2pr and v2 4p2r2
  • T T2
  • Substituting into the previous equation
  • GM 4p2r3 and therefore GM r3
  • T2 4p2 T2

27
Gravitation
  • Keplers Third Law
  • Then since G, p, and 4 are all constant, then any
    object rotating around M will be subject to the
    following constraint, usually known as Keplers
    Third Law
  • r3 constant
  • T2

28
Gravitation
  • Total Energy
  • Total energy is the sum of potential and kinetic
    energies (or TE PE KE)
  • PEG - GMm and KE ½mv2
  • r
  • therefore TE - GMm ½mv2
  • r

29
Gravitation
  • Total Energy
  • Using the equation for orbital velocity
  • vorbit (GM/r)½ and vo2 GM/r
  • therefore TE - GMm ½mGM
  • r r

30
Gravitation
  • Total Energy
  • Finally
  • TE - ½GMm
  • r

31
Gravitation
  • Gravitational Interactions
  • In essence, a small object in orbit around a
    large object is falling around the large
    object.
  • Satellites, the shuttle and the International
    Space Station are examples.

32
Gravitation
  • Gravitational Interactions
  • Two large objects rotating around each other
    actually rotate around a common center of gravity
    between them.
  • The earth and moon, or the sun and Jupiter, or
    double stars are examples.

33
Gravitation
  • Gravitational Interactions
  • The gravitational drag of the moon create the
    ocean tides on earth.
  • Tides in high latitudes can be very significant.
  • The moon creates 254,000,000,000 watts of tidal
    energy daily.

34
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35
Gravitation
  • Gravitational Interactions
  • In order of increasing gravitational attraction
    are
  • Sun
  • Blue Giant Star
  • Neutron Star
  • Black Hole
  • (of Sun Mass diameter 1 inch)

36
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40
Gravitation
  • Satellite Motion
  • We usually think of the orbits of satellites and
    the planets as circular.
  • In actuality, most orbits are elliptical, with
    the larger body at the focus of the ellipse.
  • The earth is a satellite of the sun with an
    elliptical orbit around the sun.

41
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42
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43
Gravitation
  • Keplers Three Laws
  • 1st Any object bound by a force that varies as
    1/r2 travels in an ellipse.
  • 2nd A line drawn from the sun to any planet
    will sweep out equal areas in equal times.
  • 3rd The ratio r3 /T2 is a constant for any
    planet.

44
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45
Gravitation
  • END OF GRAVITATION

46
GRAVITATION
  • Notes
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