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Gravitational Interactions

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Title: Gravitational Interactions


1
Chapter 13
  • Gravitational Interactions

2
13.1 Gravitational Fields
  • A magnetic field is a force field, because
    magnetic materials in it experience a force
  • The gravitational field around Earth is similar
  • A mass in the field region experiences a
    gravitational force
  • The field is strongest at the surface of Earth,
    and declines as the distance from Earths center
    increases

3
  • g is the acceleration due to gravity
  • G is the universal gravitational constant
  • g is the gravitational field vector
  • They are all different quantities that are
    related
  • G and g have the same magnitude and their units
    are equivalent N/kg m/s2

4
  • g GM/R2
  • g 9.8 N/kg is valid only near the planets
    surface
  • as R increases, g decreases
  • The strength of Earths gravitational field is
    the force per unit mass exerted by Earth on any
    object
  • g F
  • m
  • g 9.8 N/kg 9.8 m/s2
  • g weakens with increasing distance from Earth

5
Questions
  • If Earth had the same size but twice the mass,
    what would be the acceleration of freely falling
    objects at its surface?
  • Twice g, or nearly 20 m/s2
  • If Earth has the same mass but half the size,
    what would be the acceleration of freely falling
    objects at its surface?
  • Four times g, or nearly 40 m/s2

6
  • If Earth had twice the mass, and half the size,
    what would be the acceleration of freely falling
    objects at its surface?
  • Eight times g, or nearly 80 m/s2
  • What evidence would you look for to tell whether
    or not you were in a gravitational field?
  • The presence of a gravitational force

7
  • Compared to its strength at Earths surface, what
    is the strength of the gravitational field at a
    distance of two Earth radii from the center of
    Earth?
  • ¼ as much

8
13.2 Gravitational Field Inside a Planet
  • Gravitational field on Earth exists inside Earth
    as well as outside
  • Imagine a hole drilled completely through Earth

9
FALLING THROUGH THE CENTER OF THE EARTH
  • Start at the North Pole
  • Fall gain speed down to the center
  • Lose speed all the way to the South Pole
  • Gain speed toward the center
  • Lose speed moving away from the center

ag
ag/2
a0
ag/2
ag
10
  • As you fell faster and faster into the Earth,
    your acceleration diminishes because the pull of
    the mass above you partly cancels the pull below
  • At the Earths center the pulls cancel to zero
    and your acceleration is zero
  • Momentum carries you against a growing
    acceleration past the center to the opposite side
    where it is again, g

11
STUCK IN A CAVITY
  • In a cavity at the center of the Earth, your
    weight would be zero, because you would be pulled
    equally by gravity in all directions
  • You are pulled in every direction equally
  • The gravitational field at Earths center is zero

12
Questions
  • If you dropped a rock into a tunnel through
    Earth, what would happen?
  • It would gain speed until it reached Earths
    center, and then lose speed the rest of the way.
    Its speed at the far end of the tunnel would be
    the same as its initial speed. It would then
    fall back and repeat the motion in cyclic fashion.

13
  • As the velocity of the rock increases when
    falling into the Earth tunnel, what happens to
    the acceleration?
  • It decreases as the gravitational field
    decreases, and is zero at Earths center. The
    falling body has its maximum velocity at Earths
    center, where both the field and acceleration are
    zero

14
13.3 Weight and Weightlessness
  • Weight support force
  • We are as heavy as we feel
  • Weightlessness the absence of a support force
  • The queasy feeling you experience when in a car
    that speeds over the top of a hill

15
FOR EXAMPLE A SCALE
  • Stand on a bathroom scale on the floor
  • The gravitational force between you and Earth
    pulls you against the supporting floor and scale
  • By Newtons 3rd law, the floor and scale in turn
    push upward on you

16
NOWWEIGHT IN AN ELEVATOR
17
UP, UP, AND AWAY
  • If the elevator accelerates upward, the bathroom
    scale and floor would push harder against your
    feet
  • The scale would show an increase in your weight

18
GROUND FLOOR PLEASE
  • If the elevator accelerated downward, the scale
    would show a decrease in your weight
  • The support force of the floor would now be less

19
ELEVATOR CABLE BROKE
  • Elevator falls freely, the scale would read zero
  • According to the scale, you would be weightless
    and feel weightless
  • Your insides would no longer be supported by your
    legs and hips

20
Question
  • Why would you feel weightless in an elevator with
    a broken cable?
  • There would be no support force-the floor would
    fall as fast as you

21
13.4 Ocean Tides
  • Think of someone pulling on your coat
  • If some one pulled only on the sleeve, the coat
    could tear
  • But if every part of your coat were pulled
    equally, it and you would accelerate, but the
    coat wouldnt tear
  • It tears when one part is pulled harder than
    another because of a difference in forces acting
    on the coat

22
  • In a similar way, the spherical Earth is torn
    into an elliptical shape by differences in
    gravitational forces exerted by the moon

23
  • Ocean tides are caused by differences in the
    gravitational pull of the moon on opposite sides
    of Earth
  • The moons attraction is stronger on Earths
    oceans closer to the moon, and weaker on the
    oceans farther from the moon
  • gravitational forces weakened with increased
    distance

24
Why are there two tides a day?
  • There would only be one tide per day, if the
    Earth were nailed down in one place and held
    stationary, except for a daily rotation
  • But the Earth and moon are in orbit around each
    other
  • The ocean nearest the moon is pulled upward
    toward the moon, while the main body of Earth is
    pulled toward the moon
  • away from the ocean on the far side
  • Earth is closer to the moon than the far-side
    ocean is, so Earths waters get slightly
    elongated-at both ends creating ocean bulges

25
  • Earth makes one complete turn per day beneath
    these ocean bulges creating two sets of ocean
    tides per day
  • The part of Earth that passes beneath one of the
    bulges has a high tide
  • The tide cycle is every 24 hours and 50 minutes
  • The sun also contributes to ocean tides, but not
    as much as the moon.
  • The tilt of Earths axis is another factor

26
Questions
  • Which pulls harder on the oceans of Earth, the
    sun or the moon?
  • The sun
  • Which is most effective in raising tides?
  • The moon

27
  • The difference in pulls decreases as the cube of
    the distance between the centers of the bodies
  • Twice as far away produces 1/8 the tide
  • 3 times as far, only 1/27 the tide
  • Only relatively close distances result in
    appreciable tides, and so the nearby moon
    out-tides the enormously more massive but far
    away sun

28
  • When the sun, Earth, and the moon are lined up,
    the tides due to the sun and the moon coincide
  • There are higher high tides and lower low tides
  • These are called spring tides
  • They have nothing to do with spring
  • If the alignment is perfect, there is an eclipse

29
ECLIPSE
  • A lunar eclipse is produced when Earth is
    directly between the sun and moon
  • A solar eclipse is produced when the moon is
    directly between the sun and Earth

30
PHASES OF THE MOON
  • When the Earth is between the sun and the moon
    full moon
  • When the moon is between the sun and Earth new
    moon
  • Spring tides occur during a new moon and full
    moon
  • During half-moons, the tides due to the sun and
    the moon partly cancel each other
  • High tides are lower than average
  • Low tides are higher than average
  • These are called neap tides

31
Question
  • At the time of extra high tides, will extra low
    tides follow in the same day?
  • Yes, by the conservation of water. There is
    only so much water on Earth-extra high tides in
    one part of the world means extra low tides in
    another

32
13.5 Tides in Earth and the Atmosphere
  • There are tides within Earth, which is mostly
    molten lava
  • There is a greater probability of earthquakes and
    volcanoes when there is an eclipse of the sun or
    the moon
  • This is when Earth experiences spring
    tides-greater stresses on Earths crust

33
If the moon were closer
  • Ocean tides would be higher and so the tidal
    forces on the moons crust would be greater
  • If the moon were too close, Earths tidal forces
    would tear the moon into a billion pieces,
    forming a ring around Earth similar to those
    around Saturn

34
13.6 BLACK HOLES
  • Occurs with stars that are at least two to three
    times more massive than our sun
  • Once the flame of thermonuclear fusion is
    extinguished, gravitational collapse takes
    over-and it doesnt stop
  • Gravitation is so enormous that nothing can get
    back out-even light

35
  • A collapsed star represents condensed mass and
    therefore condensed gravity.
  • The mass of a black hole is no more than the mass
    of the star that collapsed to form it
  • Hence the gravitational field of the star and the
    black hole are the same at distances greater than
    the original radius of the star
  • It is only at closer distances that the enormous
    field occurs

36
Question
  • Consider a satellite companion to a star that
    collapses to become a black hole. How will the
    orbit of the companion satellite be affected by
    the stars transformation to a black hole?
  • Not at all. No terms in the gravitational
    equation change

37
Summary
  • Earth can be thought of as being surrounded by a
    gravitational field that interacts with objects
    and causes them to experience gravitational
    forces.
  • The gravitational field, g, is equal to the
    acceleration of a freely falling object.
  • Objects in orbit around Earth have a
    gravitational force acting on them even though
    they may appear to be weightless.

38
  • Ocean tides (and even tides within the solid
    Earth and within the atmosphere) are caused by
    differences in the gravitational pull of the moon
    (and sun) on opposite sides of Earth.
  • When a star runs out of fuel for fusion, it
    collapses under gravitational forces.
    Sufficiently massive stars collapse to form black
    holes.
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