Universal Gravitation

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Conceptual PhysicsChapter 12

• Universal Gravitation

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What is gravity?

• In the early part of the twentieth century,
  Albert Einstein made the amazing discovery that
  gravity arises from the warping of space and
  time.

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What is gravity?

• Every mass in the universe reaches out to attract
  every other one, and every mass feels the
  attraction from every other one.

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12.2 The Falling Moon

• Newton realized that if the moon did not fall, it
  would move off in a straight line and leave its
  orbit.

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The Falling Moon

• Thus the moon falls in the sense that it falls
  beneath the straight line it would follow if no
  force acted on it.

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The Falling Moon

• He hypothesized that the moon was simply a
  projectile circling Earth under the attraction of
  gravity.

Earth and Moon
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Tangential velocity

• The moon has a component of velocity parallel to
  Earths surface called tangential velocity.

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Tangential velocity

• Tangential velocity is the sideways velocity
  the component of velocity perpendicular to the
  pull of gravity.

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Relative distance from Earths center

• The moon was already known to be 60 times farther
  from the center of Earth than an apple at Earths
  surface.

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Free-fall distance at Earths surface

• The apple will fall nearly 5 m in its first
  second of fall or more precisely, 4.9 m.

d ½gt2
d ½ (9.8 m/s2)(1 s)2
d 4.9 m
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Gravity becomes weaker with distance

• The influence of Earths gravity on the moon
  should be diluted 1/60 of 1/60, or 1/(60)2.

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Free-fall distance of moon

• So in one second the moon should fall 1/(60)2 of
  4.9 m, which 1.4 millimeters.

d 1/(60)2 ? 4.9 m d 0.0014 m (1.4 mm)
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Geometric orbital distance

• Using geometry, Newton calculated how far the
  circle of the moons orbit lies below the
  straight-line distance the moon otherwise would
  travel in one second.
• His value turned out to be about the 1.4-mm
  distance accepted today.

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Universal gravitation

• Newton generalized his moon finding to all
  objects, and stated that all objects in the
  universe attract each other.

Horsehead nebula
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12.3 The Falling Earth

• The sun occupies the center, and the Earth and
  the planets orbit the sun in the same way that
  the moon orbits Earth.

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12.4 Newtons Law of Universal Gravitation

• Newton did not discover gravity.
• What Newton discovered was that gravity is
  universal.

Jupiter-Io-Europa
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Newtons law of universal gravitation

• Every object attracts every other object with a
  force that for any two objects is directly
  proportional to the mass of each object.
• The greater the masses, the greater the force of
  attraction between them.

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Distance decreases gravity

• Newton deduced that the force decreases as the
  square of the distance between the centers of
  masses of the object increases.
• The farther away the objects are from each other,
  the less the force of attraction between them.

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Law expressed by symbols

• The law can be expressed symbolically as

where m1 is the mass of one object, m2 is the
mass of the other, and d is the distance between
their centers of mass.
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Meaning of the law

• The greater the masses m1 and m2, the greater the
  force of attraction between them.
• The greater the distance d between the objects,
  the weaker the force of attraction.

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The Universal Gravitational Constant, G

• The proportionality form of the law of universal
  gravitation can be expressed as an exact equation
  when the constant of proportionality G, called
  the universal gravitational constant, is
  introduced.

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Force of gravity equation

• Then the equation is

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In words,

• The force of gravity between two objects is found
  by multiplying their masses, dividing by the
  square of the distance between their centers, and
  then multiplying this result by the constant G.

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The value of G

• In scientific notation,
• G 6.67 ? 10-11 N?m2/kg2
• In decimal notation,
• G 0.0000000000667 N?m2/kg2

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Measuring G

• G was first measured 150 years after Newtons
  discovery of universal gravitation by an English
  physicist, Henry Cavendish.

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Measuring G

• Cavendish accomplished this by measuring the tiny
  force between lead masses with an extremely
  sensitive torsion balance.

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Gravity is a very weak force

• The value of G tells us that the force of gravity
  is a very weak force.
• It is the weakest of the presently known four
  fundamental forces.

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Your weight depends on

• In addition to your mass, your weight also
  depends on your distance from the center of
  Earth.
• You weigh less on top of a mountain than in a
  valley.

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Known facts

• The force that Earth exerts on a mass of 1
  kilogram at its surface is 9.8 newtons.
• The distance between the 1-kilogram mass and the
  center of mass of Earth is Earths radius, 6.4 ?
  106 meters.

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Finding the mass of the earth

• Therefore, where m1 is the mass of Earth,

from which the mass of Earth m1 6 ? 1024
kilogram.
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12.5 Gravity and Distance The Inverse- Square
Law

• When the quantity varies as the inverse square of
  its distance from its source, it follows an
  inverse-square law.

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Gravity and Distance The Inverse- Square Law

• This law applies to the weakening of gravity with
  distance.

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Interpreting the graph

• The greater the distance from Earths center, the
  less an object will weigh.

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Interpreting the graph

• But no matter how far the distance, Earths
  gravity does not drop to zero.

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Conclusion

• The gravitational influence of every object,
  however small or far, is exerted through all
  space.