Physics 2211 Mechanics Lecture 18 Knight: 12'1 to 12'5 Universal Gravitation

1
Physics 2211 - MechanicsLecture 18 (Knight
12.1 to 12.5)Universal Gravitation

• Dr. John Evans

2
Heat
So far, we have this relation for energy
conservation
This relation does not include the
possibility that energy is being added to the
system as thermal energy. To include this
possibility, we add the term Q to the equation,
representing external heating of the system. The
heat Q transfers energy to and from the system
when there is a temperature difference between
the system and the external environment. To
include this effect, we now write
3
Power
Work transfers energy between the
environment and the system. In many cases we are
interested in how fast this energy transfer
occurs. In other words, we want to consider the
rate of energy transfer, which is called power,
defined as P º dEsys/dt. The SI unit for
power is the watt (W), which is defined as 1 J/s.
Another common power unit is horsepower (hp),
defined as 745.6999 W.
4
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 doctrine of
astrology asserted thatthe 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 createdan industry for those inclined to
star observation, learning, and skill in
calculationand geometry. Thus superstition
drove science.
5
Aristotle (384 BC - 322 BC)
The Greek philosopher Aristotle taught that
the Earth was at the centerof a nested set of
transparent spheres,with the fixed starson the
outersphere and theplanets (includingthe Sun
and Moon)attached to innerspheres, all
rotatingat differing rates.
6
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.
7
Nicolaus Copernicus (1473-1543)
Copernicus, in his book De
Revolutionibus(published posthumously) argued
that the Sunwas the center of the universe, and
that theEarth was one of the planets that
revolvedabout it in circular orbits. The
rationale forthe circles was in part
theological, circles beingperfect geometrical
objects. The Church banned Copernicus book
andpersecuted those who accepted his
ideas,because his assertions were in conflict
withthe foundations of 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.
8
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.

9
Galileo Galilei (1564 -1642)
In Pisa, Italy, Galileo Galilei heard rumors
fromvisiting sailors of a device invented in
Holland thatallowed one to obtain magnified
views of distantobjects. He experimented with
lenses until he re-discovered the trick, which
was placing a strongdiverging lens near the eye
while viewing a throughweaker and larger
converging lens placed furtheraway. 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, likethe Moon, that Saturn had rings, and
that four tinypoints of light can be seen around
Jupiter. These moonsof Jupiter formed a
miniature solar system, demonstratingthe
validity of the ideas of Copernicus and Kepler.
Galileo published his observations and ideas,
and hewas arrested by the Inquisition. He was
tried andconvicted of heresy and was forced to
publicly recant his views.
10
Isaac Newton (1642 - 1727)
Isaac Newton was born in 1642, the year of
Galileosdeath. He entered Trinity College of
CambridgeUniversity at the age of 19 and
graduated in 1665, atthe age of 23. Because the
Black Death was ravagingEurope at the time, he
then returned to his familysfarm estate for two
years to escape the pestilence. It was
during this period that he did his greatestwork.
He performed experiments in optics, laid
thefoundations of his theories of mechanics
andgravitation. Because he needed it for his
studies, heinvented the calculus as a new branch
of mathematics. 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. This is now called Newtons Law of
Gravity.
11
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 3.93
x 104 s 11 hr. However, the actual orbital
period of the Moon is about 27.3 days 2.36 x
106 s. How could this calculation be so badly
off? (Weaker gravity?) 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 2.72 x 10-3 m/s2 But an apple falls
at gE 9.80 m/s2. So 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)2
4.01x1014 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 gravitational
force law affects the apple and the Moon.
12
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.

13
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.
14
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
15
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
an alien physicist on Planet X would measure the
same value for G that we measure on Earth.
16
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.
17
Decrease of g with Distance
18
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.
19
Gravitational Potential Energy (1)
So far, we have used Ug mgy 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 m2 moving in the gravity
of mass m1 from some radius r to infinity. This
is the potential energy, with DU 0 at infinity
where the force goes to zero.
20
Gravitational Potential Energy (2)
A plot of the gravitational potential energy
Ug looks like this
21
Example Crashing into the Sun
Suppose the Earth were suddenly to halt its
motion in orbiting the Sun. The gravitational
force would pull it directly into the Sun. What
would be its speed as it crashed?
22
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.)
This is also the speed at which (in the
absence of atmosphere) a meteor, falling from
very far away, would strike the surface of the
Earth. It is called escape velocity.
23
The Flat-Earth Approximation
This is sometimes called the Flat Earth
Approximation. It is consistent with our
previous treatment of gravitational potential
energy in Chapter 10.
24
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.
25
Clicker Question 1
Which of these systems has the largest
absolute value of gravitational potential energy
Ug ?
26
End of Lecture 18

• Before the next lecture, read Knight, Sections
  13.1 through 13.3.
• Uncollected exam papers are available from Laura
  Clement, room C136 PAB.
• Check Tycho to make sure you have a grade for
  all parts of the exam. If not, see Laura.
• Regrade requests for Exam 2 will be accepted
  through noon on Monday, November 21.
• Homework Assignment 7 is posted on Tycho and is
  due by Midnight on Wednesday, November 23.

27
Satellite Orbits and Energies
The tangential velocity v needed for
acircular orbit depends on the
gravitationalpotential energy Ug of the
satellite at theradius of the orbit. The needed
tangentialvelocity v is independent of the mass
m ofthe satellite (provided mltltM).
28
ExampleThe Speed of the Space Shuttle
The Space Shuttle, in an orbit 300 km above
the surface of the Earth, wants to capture a
smaller satellite for repairs. What are the
speeds of the Shuttle and the satellite in this
orbit?
29
Keplers 3rd Law
Therefore, Keplers 3rd Law is a direct
consequence of Newtons Law of Gravity. In
the Log-Log plot to the right, the data for the
planets of the Solar System fall on a power-law
straight line specified by log10T 1.500
log10r - 9.264 The 2nd term can be used to
calculate the mass of the Sun.
30
The Solar System
31
Geosynchronous Orbit
In 1945 the science fiction author Arthur C.
Clarke pointed out that it was possible to put a
satellite in an orbit above the equator that had
a period of exactly one day, so that it rotated
around the Earth at the same rate that the Earth
rotated under it. Such a geosynchronous
satellite hangs above a particular point on the
equator and is now widely used for
communications. Clarke also envisioned
lowering a rope from a geosynchronous space
station and hauling objects into space without
rockets, using a space elevator. This is now
being seriously considered, using a super-strong
cable made from carbon nanotubes.
Notice this is the cube-root.
32
Example Extrasolar Planets
Astronomers, using the most advanced
telescopes, have recently began to discover
planets orbiting nearby stars, usually deduced
from a wobble in the stars position at the
orbital period of the planet. Suppose a
wobble with a 1200 day period (1.037x 108 s) is
observed, and it is assumed that the planet is
the same distance from its star that Jupiter is
from the Sun. What isthe mass of the star, in
solar masses?
33
Keplers 2nd Law
Keplers 2nd Law is a consequence of the
conservation of angular momentum.
34
Kepler vs. Newton
Are Keplers Laws really laws, in the
sense of Newtons Laws? No. Keplers Laws
are empirical rules deduced from data, and are
approximate, because they include only the
gravitational interaction between each planet and
the Sun, while ignoring the mutual gravitational
interactions between the planets. Newtons
Laws, on the other hand, are true Laws of Nature
that allow us to deduce all of the forces acting
in the Solar System, including planet-planet
interactions, and to calculate and predict orbits
to whatever precision we desire. We note,
however, that Newtons Laws are also
approximations, because they do not include the
effects of special and general relativity, e.g.,
relativistic mass-increase at high velocities and
time dilation in strong gravitational fields.
35
Orbital Energetics
The equation K -½Ug is called The Virial
Theorem. In effect, it says that for a planet
in orbit around the Sun, if you turned its
velocity by 90o, so that it pointed straight out
of the Solar System, you would have only half the
kinetic energy needed to escape the Suns gravity
well.
36
ExampleRaising a Satellite - LEO to Geo
How much work must be done in boosting a
1000 kg communication satellite from low Earth
orbit (h300 km) to geosynchronous orbit?
37
Chapter 12 Summary (1)
GENERAL PRINCIPLES
38
Chapter 12 Summary (2)
39
Chapter 12 Summary (3)