Physics 121C Mechanics Lecture 19 Universal Gravitation November 19, 2004

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Physics 121C - MechanicsLecture 19Universal
GravitationNovember 19, 2004

• John G. Cramer
• Professor of Physics
• B451 PAB
• cramer_at_phys.washington.edu

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Announcements

• Regrade requests for Exam 2 will be accepted
  through noon on Monday, November 22.
• Homework Assignment 6 has been posted on Tycho
  and is due on Wednesday, November 24.

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Lecture Schedule (Part 3)
You are here!
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The Pre-History of Gravitation
The ancients observed that the stars were
fixed, while the planets moved against the
background of fixed stars. They were very
interested in the stars because the movements of
the stars were correlated with the seasons,
growing cycles, etc. The discipline of astrology
also asserted that the movements of the planets
influenced the lives and destinies of humans, and
that future events could be predicted by studying
and codifying planetary movements. This created
an industry for those inclined to learning and
skilled in calculation and geometry.
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Aristotle (384 BC - 322 BC)
The Greek philosopher Aristotle taught that
the Earth was at the center of a nested set of
transparent spheres, with the fixed stars onthe
outer sphere andthe planets (includingthe Sun
and Moon)attached to innerspheres, all
rotatingat differing rates.
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Claudius Ptolemy (85-165)
Claudius Ptolemy (2nd century AD) noted that
some of the planets showed retrograde motion,
appearing to reverse direction as they moved
against the stars, in seeming contradiction to
Aristotles model of celestial spheres.
Ptolemy explained this by attaching the planets
to sub-spheres that rotated on the main spheres,
so that planetary motion was described by nested
epicycles.
Ptolemys cosmology became the Standard
Model of the universe for about 1,400 years.
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Nicolaus Copernicus (1473-1543)
Copernicus, in his book De
Revolutionibus(published posthumously) argued
that the Sun wasthe center of the universe, and
that the Earth wasone of the planets revolved
about it in circular orbits.The rationale for
the circles was in part theological, circles
being perfect geometrical objects. The
Church banned Copernicus book andpersecuted
those who accepted his ideas, becausehis
assertions were in conflict with the
foundationsof medieval theology. From 1570 to
1600, the Danish astronomer Tycho Brahe compiled
a set of extremely accurate (pre-telescopic)
astronomical observations. Tychos observations
revealed that there were problems with
Copernicus assertion that the planet followed
circular orbits.
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Johannes Kepler (1571-1630)

• Johannes Kepler inherited Tychos
  observations and tried to makesense of them,
  using algebra, trigonometry, and geometry. After
  a decadeof work, he was forced to conclude that
  planetary orbits were betterdescribed by
  ellipses than by circles, and that the planets
  travel in theseorbits with avarying speed. He
  deduced three laws of planetary motion
• All planets move in in elliptical orbits, with
  the Sun as a focus ofthe ellipse.
• A line drawn between Sun and planet sweeps out
  equal areas inequal times.
• The square of a planets orbit period is
  proportional to the cubeof the length of its
  semi-major axis.

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Galileo Galilei (1564 -1642)
In Pisa, Italy, Galileo Galilei hear rumors
fromvisiting sailors of a device invented in
Holland thatallowed one to obtain a magnified
view of distantobjects. He experimented with
lenses until he re-discovered the trick, which
was placing a strongdiverging lens near the eye
while viewing a weakerand larger converging lens
places further away.He discovered (or
re-discovered) the telescope. He used this
invention to view the stars and planets.He
discovered that the planet Venus has phases, like
theMoon, that Saturn had rings, and that four
tiny points oflight can be seen around Jupiter.
These moons of Jupiter formed a miniature solar
system, demonstrating the validity of the ideas
of Copernicus and Kepler. Galileo published
his observations and ideas, and he was arrested
by the Inquisition. He was tried and convicted
of heresy and was forced to publicly recant his
views.
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Isaac Newton (1642 - 1727)
Isaac Newton was born in 1642, the year of
Galileosdeath. He entered Trinity College of
Cambridge Universityat the age of 19 and
graduated in 1665, at the age of 23.Because the
Black Death was ravaging Europe at the time,he
returned to his family farm for two years to
escape thepestilence. It was during this
period that he did his greatest work.He
performed experiments in optics, laid the
foundationsof theories of mechanics and
gravitation, and, because heneeded it, invented
calculus as a new branch ofmathematics.
Newton, following an idea suggested by Robert
Hooke, hypothesized that the force of gravity
acting on the planets is inversely proportional
to their distances from the Sun.
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The Appleand the Moon
The radius of the Moons orbit is
RM3.84x108 m. If T 2pr/g½ and g9.80 m/s2,
then the Moons orbital period should be TM
2pRM/g½ 2p(3.84x108 m)/(9.80 m/s2)½
3.93x104 s 11 hr. However, the actual
orbital period of the Moon is about 27.3 days
2.36x106 s. How could this calculation be so
badly off? Lets use the Moons orbital
period and calculate gM, the acceleration due to
Earths gravity at the orbit of the Moon.gM
RM(2p/T)2 (3.84x108 m)2p/(2.36x106 s)2
2.72x10-3 m/s2 But an apple falls at gE9.80
m/s2. Lets try something. Well calculate the
product gR2 for an apple at the Earths surface
and for the Moon in orbit
gMRM2(2.72x10-3 m/s2)(3.84x108 m)24.01x1014
m3/s2 gERE2 (9.80 m/s2)(6.37x106 m)2
3.98x1014 m3/s2 These products are
essentially equal, because gravity falls off
1/R2. The same gravity affects the apple and the
Moon.
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Newtons Law of Gravity

• Newton proposed that every object in the universe
  attracts every other object with a force that has
  the following properties
• The force is inversely proportional to the
  distance between the objects.
• The force is directly proportional to the product
  of the masses of the two objects.

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Gravitational Force and Weight
With Newtons Law of Gravity, we can
calculate the gravitational force produced by the
Earth and acting on some mass on the Earths
surface. (To do this, we assume that the Earths
gravity is that same as it would be if all of the
Earths mass were concentrated at its center.)
Gravity is a very weak force, much weaker
than the other three forces of nature (the
strong, electromagnetic, and weak interactions).
However, it is a long-range force and it is
cumulative. It always adds, never subtracts,
because there is no (known) negative mass in the
universe.
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The Principle of Equivalence

• Mass appears in two roles in physics
• Inertial mass, which resists acceleration
• Gravitational mass, which produces gravitational
  attraction.

The Principle of Equivalence states that these
masses are always equal, and that the apparent
force in an accelerated reference frame is
indistinguishable from gravity.
The Principle of Equivalence
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Little g and Big G
On other planets the acceleration due to
gravity (gX) will be different, because it
depends on the mass and radius of each planet.
However, the law of gravity is universal, so a
physicist on Planet X would measure the same
value for G that we measure on Earth.
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Rotation and Little g
Notice that we calculated a value for g that
was slightly larger than 9.80 m/s2. This is
because the Earth is rotating, and part of the
force of gravitational attraction acts to provide
centripetal acceleration, keeping objects moving
in a circular path as the Earth rotates. The
centripetal acceleration is about 0.03 m/s2,
accounting for the difference.
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Decrease of g with Distance
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Weighing the Earth
Newtons gravitational constant G must be
measured in the laboratory. Henry Cavendish made
the first accurate measurement of this quantity,
using a Cavendish balance. The forces between
masses are measure using their action in twisting
a thin fiber. G is calculated from the measured
force.
A measurement of G is essentially a
measurement of the mass of the Earth.
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Gravitational Potential Energy (1)
So far, we have used Ugmgy for the
gravitational potential energy, where y is the
height above the surface of the Earth. Now we
would like to do better, using the law of gravity.
We consider a mass m2moving in the gravity
of mass m1 from some radius r to infinity. This
is the potential energy, with DU0 at infinity
where the force goes to zero.
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Gravitational Potential Energy (2)
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Example Crashing into the Sun
Suppose the Earth were suddenly to halt its
motion in orbit the Sun. The gravitational force
would pull it directly into the Sun. What would
be its speed as it crashed?
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Example Escape Speed
A 1000 kg rocket is fired straight away from
the surface of the Earth. What speed does it
need to escape from the gravitational pull of
the Earth and never return? (Assume a
non-rotating Earth.)
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The Flat-Earth Approximation
This is sometimes called the Flat Earth
Approximation. It is consistent with our
previous treatment of gravitational potential
energy.
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ExampleThe Speed of a Projectile

• A projectile is launched straight up from
  the Earths surface.
• With what speed should it be launched if it is
  to have a speed of 500 m/s at a height of 400 km?
• By what percentage would your answer be in error
  if you use the Flat-Earth approximation?

This is too big by 2.5.
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Clicker Question 1
Which of these systems has the largest
absolute value of gravitational potential energy
Ug ?
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End of Lecture 19

• Before the next lecture, read Knight, Sections
  12.6 through 13.2.
• Homework Assignment 6 has been posted on Tycho
  and is due on Wednesday, November 24.
• Regrade requests for Exam 2 will be accepted
  through noon on Monday, November 22.