Gravity

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Gravity
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Astronomical Numbers
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Regarding how far apart the objects are

• Basic law of geometry

When something spreads out radial, there were be
more of it close to the source and less of it
from the source (when considering the same area).
Cross-Section View
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G a constant of nature

• G is not the same as g.
• g is the acceleration of things falling on earth.
• G tells you the intrinsic strength of gravity
  (with the effects of the distance and amount of
  mass extracted)
• How much bang for your buck
• How much force do you get from an explosive?
• It depends on how far away the explosives are (r)
• how much explosives there are (m)
• and whether you are using gun powder or
  nitroglycerine (G)

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Note about r

• r is the distance from the center of object 1 to
  the center of object 2.
• What is the distance between you and the earth
  right now?
• What is the distance between a satellite and the
  earth if it is orbiting 10,000 meters above the
  earth?
• r does not have to be a radius. It is simply the
  distance between the two masses (You and your
  text book)

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Note about the scope of the equation

• M implies we are working with a point mass.
• Newton believed that a sphere of any size could
  be treated as being equivalent to having a point
  mass at the CM.
• Delayed publishing his book until he had invented
  calculus to prove this hunch.

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Note about M

• The previously mentioned equivalence is only
  valid if you are outside the mass.
• Within the mass, there is some gravity pulling
  down and some gravity pulling upward.

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Inside a massive sphere

• Consider only the mass at a radius r from the
  center, where you are at a distance r.
• Small amount of mass above you, but close by.
• Large amount of mass below you, but far away.

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Inside a massive sphere

• Two effects always cancel out.
• Net force due to gravity inside a hollow sphere
  is zero at all points inside the sphere.
• But this sphere is not hollow!
• No problem, it is made of a hollow sphere (SF0)
  plus a sphere that you are at the surface of and
  FGMm/r2 applies for that part.

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The Hole Through the Center of the Earth Problem.

• How long would it take you to fall through the
  diameter of the earth?
• Write a differential equation.
• Start with Newtons 2nd law in differential form.
• How does the force change?

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Gravity Outside the Earth
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Projectiles

• Projectiles follow parabolic paths.

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But the surface of the earth follows a circular
path.
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How did the projectiles path change from
parabolic to circular?

• Parabola is from gravity always pulling in
  negative y direction.
• Circle is from gravity always pulling toward
  center.
• Any conic section can be a solution to the
  differential equation.
• Line
• Falling straight down
• Parabola
• Thrown object.
• Ellipse
• Circle
• Hyperbola
• orbit that never comes back

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Orbit!
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How fast?

• Objects go in a circle as a result of a
  centripetal force.

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Exercise

• Calculate V as a function of r using the
  centripetal force equation and Law of Universal
  Gravitation.

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For an orbit
Velocity required to orbit (in a circle) at a
distance r
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Simple rule explains a lot

• For each distance from the earth, there is one
  specific speed that the satellite needs to go.
• Mass of the satellite doesnt matter. It could
  be a TV satellite, or the moon and theyd need
  the same velocity.
• Can you think of anything that behaves similarly?
• At a close distance, you must go faster.
• At a far distance, you must go slower.

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Simple rule explains a lot

• What happens if it goes slower than that special
  speed?
• Curves in too much.
• What happens if it goes faster?
• Curves in not enough.

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Starts too slow.
Think about conservation of energy, what has
happened to its speed?
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Goes too fast
Think about conservation of energy, what has
happened to its speed? Now its going too fast.
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Too slow, too fast, ellipse
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Escape Velocity

• Can only use GPE mgh if force of gravity is
  constant.
• Works for changes in h that are small compared to
  r2
• Calculate the work (Fd) done against gravity
  (FGMm/r2) to raise a mass M from the surface of
  the earth to a distance of infinity.
• If WFd and d is going to infinity, then why
  isnt the work infinity?
• What initial velocity would the mass need to
  reach a distance of infinity?

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Escape Velocity of Earth

• escape velocity on earth
• this is if there were no air
• 11.2 km/s
• 7 mi/sec

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Keplers Laws

• Orbits are ellipses with the sun at one focus.
• A line joining any planet to the sun sweeks out
  equal areas in equal times.
• T2Cr3
• C is a conversion factor for the units. If T is
  in years and r is in astronomical units then C1
• 1 Astronomical Unit the radius of Earths orbit.

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Keplers 2nd Law

• Multiplying vectors gives the area of the
  parallelogram.

r
s
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Keplers 2nd Law
Sv(dt)
r
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Keplers 2nd Law
Sv(dt)

• The force is radial
• So the torque is zero
• So L is constant
• So dA/dt has the same value at all places

r
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Keplers 3rd Law

• You already know that for a circular orbit
• And you know that
• V D/T ? T D/V
• Where D is the circumference and T is the period
  (Time to circle once)

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Keplers 3rd Law - Modifications

• The derived equation works for circular orbits
  where the central object can be considered
  immobile (its mass is much larger than its
  satellites mass)
• Defining TE1 year and rE1 AU forces the
  constant to be 1 and gives answers for all other
  planets in multiples of these units.
• If both objects orbit a common center
• M?(M1M2)
• For elliptical orbits
• r?(rminrmax)/2

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Questions

• Is it possible for a planet to ever have a
  velocity that is not perpendicular to the force
  of gravity?
• Find the speed and distance of a geosyncronous
  orbit around earth given the mass of the earth
  and an orbital period of 1 day.
• Write an equation for the total mechanical energy
  of a satellite in circular orbit.
• Rocket goes straight up and straight down with no
  air resistance. How far away from the launch pad
  does it land?

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The End
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Escape Velocity

• The speed of light is C 2.99x108 m/s.
• Suppose the mass of the earth were compressed to
  the size of a baseball (made into a black hole).
  At what starting distance (r) would be moving at
  less than the escape velocity?
• Why are we using an idea that applies to objects
  falling and gravity, and applying it to light?
• Objects cant go faster than C, so this would be
  the closest an object could come without getting
  stuck forever.
• What about light itself? Does it get stuck
  forever? How does gravity exert a force on light
  if light has no mass?

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Projectiles

• Projectiles follow parabolic paths.

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Projectiles
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Projectiles
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Projectiles
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But the surface of the earth follows a circular
path.
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Tides65 from the moon, 35 from the sun.

• Tides from the sun are easier to understand so
  well ignore the moon for now.
• High tide happens once at midnight, once at noon.
• Gravity from the sun pulls on the ocean.
• But how do you explain the night time tide?

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Tides

• Think of the earth and the oceans as separate
  objects
• The earth has an orbit. The ocean has its own
  orbit around the sun.
• Earth has just the right velocity for its orbit.
• Oceans near the sun should be going faster to
  keep a proper orbit.
• But have to go the same speed as the earth.
• Going too slow means they fall toward the sun a
  little.
• Oceans on the far side should be going slower to
  keep a proper orbit.
• But have to go the same speed as the earth.
• Going too fast means they curve to little, which
  to us means outward

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Free Fall

• Misconception there is no gravity in space.
• Reality There is gravity in space, but as long
  as the people and their container are falling
  with each other, so the people float.

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Falling Airplanes
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Black Holes

• What would happen if the sun turned into a black
  hole?

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Sun MO
r 1.5x1011 m
Earth
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• Fg is exactly the same.
• No effect on the earth

r 1.5x1011 m
Black Hole MO
Earth
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• If compressing the mass doesnt change the
  gravity, whats so special about a black hole?

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Want more Fg so make r smaller
Earth
Astronaut
r 6.38x106 m
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r is smaller, but
Fg ½ normal
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Fg 0
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Attempt 2
r 6.38x106 m
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Fg unchanged
r 6.38x106 m
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Fg 4x normal
r 3x106 m
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Fg 100x normal
r 0.6x106 m
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How Black Holes Work

• Black holes are special because they are small.
• Small, but still have a huge mass.