Universal Gravitation Lecturer: Professor Stephen T. Thornton

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Universal Gravitation Lecturer Professor
Stephen T. Thornton
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Reading Quiz
The International Space Station is at an altitude
of 200 km above the surface of the Earth. What
is the net force on an astronaut at rest inside
the Space Station?

• Equal to her weight on Earth.
• A little less than her weight on Earth.
• Less than half her weight on Earth.
• Zero (she is weightless).
• Somewhat larger than her weight on Earth.

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Answer B

• The astronaut is falling around the Earth. The
  gravitational force is keeping her from going in
  a straight line. The acceleration of gravity is
  a little less than g at 200 km above the Earth.

200 km
6380 km
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Last Time

• Non-uniform circular motion
• Drag
• Terminal velocity
• Fundamental forces

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Today

• History of gravitation
• Newtons law of universal gravitation
• Keplers laws

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History of Gravitation

• Greeks used the geocentric frame in which the
  Earth was at the center.
• Ptolemy, 2nd A.D., prepared a detailed
  formulation of heavenly body motion.
• epicycles
• retrograde motion

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? Nicolaus Copernicus (1473-1543 ) introduced
heliocentric frame with the Sun at the center
of the solar system.? Catholic church thought
this was heresy.? Danish astronomer Tycho Brahe
(1546- 1601) made huge number of observations.
Johannes Kepler continued his work and did
analysis.
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Galileo (1564-1642) is said to have invented the
telescope. He made many observations.Issac
Newton (1642-1727) was one of the smartest
persons to ever live.He invented calculus so he
could solve the problem of how the moon rotates
around Earth.Newton looked at Keplers results
and figured everything out in a manner of days!
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Newtons Law of Universal Gravitation
If the force of gravity is being exerted on
objects on Earth, what is the origin of that
force?
Newtons realization was that the force must come
from the Earth. He further realized that this
force must be what keeps the Moon in its orbit.
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conic sections
Newton showed that planetary motion and other
similar motion had to be of conic sections.
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Newtons Law of Universal Gravitation

• The force of gravity between any two point
  objects is attractive and of magnitude

G is the universal gravitational constant
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Gravitational Force Between Point Masses

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Important points about gravity

1. Gravitational force is a vector and always
   attractive.
2. Gravity is difficult to measure, except for large
   bodies.
3. Gravity has an exact 1/r2 dependence.
4. For several masses, just add forces. Called
   superposition.

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Gravitational Force Between a Point Mass and
a Sphere (uniform mass density)
Use symmetry. For this case, force acts at
center.
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Gravitational Force Between
the Earth and the Moon
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Conceptual Quiz
A) the Earth pulls harder on the Moon B) the
Moon pulls harder on the Earth C) they pull on
each other equally D) there is no force between
the Earth and the Moon E) it depends upon
where the Moon is in its orbit at that time

• Which is stronger, Earths pull on the Moon, or
  the Moons pull on Earth?

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Conceptual Quiz
A) the Earth pulls harder on the Moon B) the
Moon pulls harder on the Earth C) they pull on
each other equally D) there is no force between
the Earth and the Moon E) it depends upon where
the Moon is in its orbit at that time

• Which is stronger, Earths pull on the Moon, or
  the Moons pull on Earth?

By Newtons 3rd Law, the forces are equal and
opposite.
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Conceptual Quiz
A) one quarter B) one half C) the same D)
two times E) four times

• If the distance to the Moon were doubled, then
  the force of attraction between Earth and the
  Moon would be

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Conceptual Quiz
A) one quarter B) one half C) the same D)
two times E) four times

• If the distance to the Moon were doubled, then
  the force of attraction between Earth and the
  Moon would be

The gravitational force depends inversely on the
distance squared. So if you increase the
distance by a factor of 2, the force will
decrease by a factor of 4.
Follow-up What distance would increase the
force by a factor of 2?
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Force of gravity on a mass m on the surface of
the Earth is mg.Lets use Newtons universal law.
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Keplers three laws follow naturally from
Newtons Law of Universal Gravitation.First law
occurs because of 1/r2. Orbits must be ellipses.
We will derive Keplers laws later after we
study angular momentum.
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Keplers 1st and 2nd laws
http//physics.bu.edu/duffy/semester1/semester
1.html
2nd law Radius vector sweeps out equal areas in
same time interval.
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Keplers 3rd law follows directly from the form
of the gravitational force law.
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Keplers Third Law and Some Near Misses
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Free Fall

• Look at http//galileoandeinstein.physics.virginia
  .edu/more_stuff/Applets/newt/newtmtn.html

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Gravitational Attraction. Two objects attract
each other gravitationally with a force of
when they are 0.25 m apart. Their
total mass is 4.00 kg. Find their individual
masses.
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Suns Mass Determination. Determine the mass
of the Sun using the known value for the period
of the Earth and its distance from the Sun.
Hint The force on the Earth due to the Sun is
related to the centripetal acceleration of the
Earth. Compare your answer to that given in
your textbook.
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Conceptual QuizThe gravitational constant
G is  A) equal to g at the surface of Earth.
B) different on the Moon than on Earth.
C) obtained by measuring the speed of
falling objects having different masses.
D) all of the above. E) none of the above
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Answer E

• None of them determine G.

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Newtons Law of Universal Gravitation
Using calculus, we can show Particle outside a
thin spherical shell gravitational force is the
same as if all mass were at center of
shell. Particle inside a thin spherical shell
gravitational force is zero. See next slide. Can
model a sphere as a series of thin shells
outside any spherically symmetric mass,
gravitational force acts as though all mass is at
center of sphere.
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