Title: Physics 2211 Mechanics Lecture 19 Orbits and Rotation Knight: 12'6 and 13'1 to 13'3
1Physics 2211 - MechanicsLecture 19Orbits and
Rotation(Knight 12.6 and 13.1 to 13.3)
2Satellite 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).
3ExampleThe 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?
4Keplers 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.
5The Solar System
6Geosynchronous 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.
7Example 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?
8Keplers 2nd Law
Keplers 2nd Law is a consequence of the
conservation of angular momentum.
9Kepler 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.
10Orbital 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.
11ExampleRaising 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?
12Chapter 12 Summary (1)
GENERAL PRINCIPLES
13Chapter 12 Summary (2)
14Chapter 12 Summary (3)
15Rigid Body Rotation
A rigid body is an extended object whose
size, shape, and distribution of mass do not
change as the object moves and rotates. A
good approximation to a rigid body is a bicycle
wheel, although in practice it can flex and bend
as forces are applied to it. The purpose of
using a rigid body model to describe objects in
motion is to simplify the problem, so that
possible changes in the internal structure of the
object as it moves can be ignored.
16Rotation Translation
17Circular Motion
Speeding up or slowing down in a fixed
circular path
18The Angular Velocity wand Angular Acceleration a
Every point on the wheel has the same w and
a.
19The Signs of Angular Velocityand
AngularAcceleration
A ccw rotation rate w is positive.
A cw rotation rate w is negative.
An increase in ccw rotation rate ais positive.
An increase in cw rotation rate ais negative.
A decrease in ccw rotation rate ais negative.
A decrease in cw rotation rate ais positive.
20End of Lecture 19
- Before the next lecture, read Knight, Sections
13.3 through 13.6.