Gravitation Applications Lecturer: Professor Stephen T. Thornton

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Gravitation Applications Lecturer Professor
Stephen T. Thornton
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Reading Quiz
A) They are so far from Earth that Earths
gravity doesnt act any more. B) Gravitys force
pulling them inward is cancelled by the
centripetal force pushing them outward. C) While
gravity is trying to pull them inward, they are
trying to continue on a straight-line path. D)
Their weight is reduced in space so the force of
gravity is much weaker.

• Astronauts float in the space shuttle because

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Reading Quiz
A) They are so far from Earth that Earths
gravity doesnt act any more. B) Gravitys force
pulling them inward is cancelled by the
centripetal force pushing them outward. C) While
gravity is trying to pull them inward, they are
trying to continue on a straight-line path. D)
Their weight is reduced in space so the force of
gravity is much weaker.

• Astronauts float in the space shuttle because

Astronauts in the space shuttle float because
they are in free fall around Earth, just like a
satellite or the Moon. Again, it is gravity
that provides the centripetal force that keeps
them in circular motion.
Follow-up How weak is the value of g at an
altitude of 300 km?
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Last Time

• History of gravitation
• Newtons law of universal gravitation
• Keplers laws
• Free floating in space

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Today

• Orbital maneuvers
• Ocean tides
• Geophysical applications
• Free floating in space
• Satellites and weightlessness
• Principle of Equivalence
• Black holes

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Conceptual Quiz
A) Its in Earths gravitational field B) The net
force on it is zero C) It is beyond the main pull
of Earths gravity D) Its being pulled by the
Sun as well as by Earth E) none of the above is
precise enough

• The Moon does not crash into Earth because

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Conceptual Quiz
A) It is attracted to Earth B) The net force on
it is zero C) It is beyond the main pull of
Earths gravity D) Its being pulled by the Sun
as well as by Earth E) None of the above is
precise enough

• The Moon does not crash into Earth because

The Moon does not crash into Earth because of
its high speed. If it stopped moving, it would,
of course, fall directly into Earth. With its
high speed, the Moon would fly off into space if
it werent for gravity providing the centripetal
force.
Follow-up What happens to a satellite orbiting
Earth as it slows?
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The Global Positioning System
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Orbital Maneuvers
Move to higher orbit
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Orbital Maneuvers
Move to lower orbit
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Ocean Tides Arrows denote force due to the
moon relative to the force at the center of
Earth. Newton finally correctly explained tides!
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Gravitation Force Due To Ring. A mass M is ring
shaped with radius r. A small mass m is placed at
a distance x along the rings axis as shown in
the figure. Show that the gravitational force on
the mass m due to the ring is directed inward
along the axis and has magnitude Hint
Think of the ring as made upof many small point
masses dMsum over the forces due to each dMand
use symmetry.
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Vector Form of Newtons Universal Gravitation
If there are many particles, the total force is
the vector sum of the individual forces
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We can relate the gravitational constant to the
local acceleration of gravity. We know that on
the surface of the Earth Solving for g gives
g can be measured to 1 part in 109 so that
mineral and oil deposits can be detected using
sensitive gravitometers.
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The acceleration due to gravity varies over the
Earths surface due to altitude, local geology,
and the shape of the Earth, which is not quite
spherical.
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Geosynchronous satellite. A
geosynchronous satellite stays above the same
point on the Earth, which is possible only if it
is above a point on the equator. Such satellites
are used for TV and radio transmission, for
weather forecasting, and as communication relays.
They must have an orbit of precisely 24 hours.
In order to do that, they must be about 22,000
miles above the Earth and have a precise speed.
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Lagrange point The mathematician
Joseph-Louis Lagrange discovered five special
points in the vicinity of the Earths orbit about
the Sun where a small satellite (mass m) can
orbit the Sun with the same period T as Earths
( 1 year).
One of these Lagrange Points, called L1, lies
between the Earth and Sun on the line connecting
them. Several satellites are being placed in
Lagrange points. We probably will not be able to
service them like we have done with the Hubble.
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Satellites and Weightlessness
Objects in orbit are said to experience
weightlessness. They do have a gravitational
force acting on them, though! The satellite and
all its contents are in free fall, so there is no
normal force. This is what leads to the
experience of weightlessness.
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More properly, this effect is called apparent
weightlessness, because the gravitational force
still exists. It can be experienced on Earth as
well, but only briefly
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Gravitational Field
The gravitational field is the gravitational
force per unit mass
The gravitational field due to a single mass M is
given by
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Principle of Equivalence
Inertial mass the mass that appears in Newtons
second law. Gravitational mass the mass that
appears in the universal law of
gravitation. Principle of equivalence inertial
mass and gravitational mass are the same.
We can do no experiment to tell the difference
between gravitational and inertial mass.
Fundamental tenet of the General Theory of
Relativity.
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One way to visualize the curvature of space (a
two-dimensional analogy)
If the gravitational field is strong enough, even
light cannot escape, and we have a black hole.
Einstein predicted in 1915 that light should be
attracted by gravity to mass.
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At rest
Light should be deflected by a massive object.
On the right side, the person can not tell
whether acceleration caused the light to bend or
whether gravity did it.
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This bending has been measured during total solar
eclipses.
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Gravitational Lensing
Massive stars can collapse under the
gravitational force. They can become black
holes, and nothing can escape even
light. Einstein showed gravity even bends light!
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Gravity Assist

• http//www.youtube.com/watch?vI3F88w3LkiI

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Milky Way Galaxy. The Sun rotates about the
center of the Milky Way Galaxy (see figure) at a
distance of about 30,000 light-years from the
center (1 ly 9.5 x 1015 m). If it takes about
200 million years to make one rotation, estimate
the mass of our Galaxy. Assume that the mass
distribution of our Galaxy is concentrated mostly
in a central uniform sphere. If all the stars had
about the mass of our Sun (2 x 1030 kg), how
many stars would there be in our Galaxy?
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Conceptual Quiz
A) B) C) D) its the same E) 2

• Two satellites A and B of the same mass are
  going around Earth in concentric orbits. The
  distance of satellite B from Earths center is
  twice that of satellite A. What is the ratio of
  the centripetal force acting on B compared to
  that acting on A?

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Conceptual Quiz
A) B) C) D) its the same E) 2

• Two satellites A and B of the same mass are
  going around Earth in concentric orbits. The
  distance of satellite B from Earths center is
  twice that of satellite A. What is the ratio of
  the centripetal force acting on B compared to
  that acting on A?

• Using the Law of Gravitation
• we find that the ratio is .

Note the 1/r2 factor
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Conceptual Quiz

• A planet of mass m is a distance d from Earth.
  Another planet of mass 2m is a distance 2d from
  Earth. Which force vector best represents the
  direction of the total gravitation force on Earth?

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Conceptual Quiz
A planet of mass m is a distance d from Earth.
Another planet of mass 2m is a distance 2d from
Earth. Which force vector best represents the
direction of the total gravitation force on Earth?
The force of gravity on the Earth due to m is
greater than the force due to 2m, which means
that the force component pointing down in the
figure is greater than the component pointing to
the right.
F2m GME(2m) / (2d)2 GMm / d 2 Fm GME m
/ d 2 GMm / d 2