Physics 55 Friday, September 23, 2005

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Physics 55Friday, September 23, 2005

1. Keplers empirical laws of planetary motion
2. Newtons laws of motion and related concepts such
   as mass, acceleration, and forces.

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Need to Understand Some Physics of Gravity and
Light

• To make further progress in understanding
  astronomy, you need to know some basic physics
  concepts
• Newtons laws mass, force, and universal law of
  gravitation.
• Fundamental conservation laws of energy,
  momentum, angular momentum.
• What we can learn from light surface
  temperature, speed toward or away, rotation rate,
  presence of atmosphere, atomic composition,
  presence and strength of electric or magnetic
  fields,

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Calculating Periods

• Planets motion in sky results from combination
  of true motion and Earths motion. Planet orbits
  Sun each sidereal period.
• What we see recurs every synodic period (relative
  configuration of Earth, Sun, planet).
• For an inferior planet, a synodic period has
  elapsed when the planet has lapped Earth.
• For a superior planet when Earth has lapped it.
  
• If sidereal period is P days, planet moves 360/P
  degrees a day. Earth moves 360/E degrees a day
  where E365 is sidereal period of Earth.
• Planet laps Earth (completing a synodic period of
  S days) when S(360/P) S(360/E) 360. This is
  same as 1/P 1/E 1/S.
• For a superior planet find similarly 1/P 1/E
  1/S.

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Galileos Smoking Scope

• Smoking gun evidence for Copernician model
  required new technology.
• Galileo (1610) turns new telescope up and finds
  phases of Mercury.
• The correlation between phase and position in sky
  agrees with heliocentric model, not with
  Ptolemaic model.

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Cultural Issues

• Galileo also discovers that Jupiter is itself
  accompanied by moons that orbit the planet, much
  like our Moon orbits Earth. Nature repeats on
  different scales.
• Motion of Jupiters moons studied closely, forms
  first Nautical clock for longitude measurement.
• Despite all this, Galileo tried for heresy and
  sentenced to house arrest.

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More on Galileo

• Galileo made many other discoveries of
  importance. With his telescope, he discovered
  mountains on the moon, size and shape of planets,
  nature of Milky Way, moving spots on Sun, among
  others.
• He also studied mechanics, properties of motion
  in general. Formulated principle of inertia
  object tends to remain in its state of motion
  unless disturbed externally. We know we need to
  work to move things. Galileos insight reminds us
  we also work to stop or turn them.
• Galileo almost got mechanics right. What stopped
  him was the fact that the mathematics needed to
  formulate the theory was not known. To make
  progress, Newton had to invent Calculus.

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Brahe and Kepler

• First steps to deeper insight were careful
  observations by Brahe (1580) of planetary motion
  to great precision.
• Using these, Kepler (1609) finds three laws of
  planetary motion.

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Keplers Laws

• 1. Orbit of a planet is an ellipse with Sun at
  one focus.
• Ellipse is shape of all points such that sum of
  their distances from two points (foci) is
  constant.
• Eccentricity (e) measures how far the foci are
  relative to size. e0 is a circle.
• Other focus is nothing (not even same for all
  planets).
• Typically e small, .017 for Earth, .2 for Mercury.

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• 2. Line connecting planet to Sun sweeps out
  equal areas in equal time intervals as planet
  orbits.
• Planet moves faster at perihelion, slower at
  aphelion.
• This causes slight change in rate of Suns motion
  discussed earlier.

• This effect much more dramatic for comets which
  follow highly eccentric orbits.

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• 3. Square of sidereal period is proportional to
  cube of semimajor axis.
• This relates the orbital motions of different
  planets orbiting same Sun.
• Write this as P2 a3. This is valid if P is
  measured in years and a in AU.

• Recall 1AU 1.496 108 km 93 million miles is
  average Earth-Sun distance.

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• Keplers laws are amazing progress. They give
  planetary motion with unprecedented accuracy.
  Whats more, they are universal they apply to
  any orbital system, from an atom through Saturns
  moons to Galaxy clusters.
• In physics such universality means there are
  fundamental laws at work here.
• These were found by Newton (1670) who at first
  was not thinking at all about Astronomy.

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PRS Question

• An asteroid with an orbital period of 8 years
  lies at
• an average distance from the Sun equal to
• 2 AU
• 4 AU
• 8 AU
• 16 AU
• Need to know the asteroids mass.

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PRS Question

• The period of revolution p of a point on a
  spinning
• CD is related to its distance r from the center
  of the
• CD (its axis of rotation) by the expression
• p2 c r3 for some constant c.
• p2 c r for some constant c.
• p c r for some constant c.
• p does not depend on r.

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PRS Question Prediction
If steel ball is being swung around in circle on
a rope and if the rope breaks at point P, which
path does ball follow next? (For PRS, A1, B2,
etc.)
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PRS Question Prediction
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PRS Question Prediction
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Newtons Laws of Motion

• 1. An object upon which no forces act will move
  in a straight line with constant velocity.
  (inertia)

• Familiar when velocity is zero object at rest
  will stay there.

• Need to remember in our world two forces (at
  least) always get in the way gravity pulls us
  down friction slows all motion. To see Newton 1
  need to minimize these or imagine them removed.

• Velocity is a vector has direction as well as
  magnitude. So constant velocity means no change
  in direction or speed.

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• 2. When a force acts on an object, it will
  change its velocity. The acceleration will be
  proportional to the force (and pointed in the
  same direction). The proportionality constant is
  called mass.
• F ma

• Acceleration a is rate of change of velocity v.
  So measured in (m/sec)/sec or m/sec2.
• Like v it is a vector and has direction. Note
  that changing direction of v requires
  acceleration, just as does changing magnitude of
  v.

• m is mass. Measured in kg total amount of
  stuff.

• F is force. Measured in kg (m/sec2) N(ewton)

• Acceleration of gravity here is g 9.8 m/sec2,
  so force of gravity on 1kg. is 9.8 N.

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My Van

• My van can go 0 to 60 mph in 12 sec.
• This is an acceleration of
• a (60 mi/hr)/12 sec
• 5 (mi/hr)/sec
• (5 1609 m/3600 sec)/sec
• 8045 m/3600 sec2
• 2.34 m/sec2
• Its mass is 800 kg. Force required is
• F ma 800 kg2.34 m/sec2
• 1788 kgm/sec2 1788 N

a (1) 5 mph/sec (2) 5 m/sec2 (3) 5
miles/sec2
F (1) 1743 m/sec2 (2) 1788 N (3)
1967 N
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Uniform Circular Motion

• Planet moves at uniform speed v around circle of
  radius R. Period is P2pR/v
• Is velocity constant?

• NO. Direction changes.

• Guess acceleration
• Points inwards
• Grows with larger v (m/sec).
• Smaller with larger R (m).
• Measured in m/sec2.

a (1) v/ R2 (2) v2 R (3) v2 / R

• a v2/R.
• So F ma m v2/R

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Numbers for Earth

• As Earth spins, we move at
• Vspin (2pR)/P 4107 m/24 hr
• 463 m/sec 1036 mph
• As Earth orbits, we move at
• Vorbit (2pR)/P
• 6.28 1.5 1011 m/36586400 sec.
• 29871 m/sec 100,595 mph

• a v2/R
• 4632/6.38106
• 0.034 m/sec2

a (1) 9.8 m/sec2 (2) 0.034 m/sec2
(3) 0.45 m/sec2

• F ma
• 100kg. 0.034m/sec2
• 3.4 N

• a v2/R
• 298712/1.51011
• 0.0059 m/sec2

• F ma
• 100kg. 0.0059m/sec2
• 0.59 N

• When rocks in space hit Earth the relative
  velocities are about 100,000 mph. That is why
  they burn in atmosphere as meteors!

• Lets compute the forces required to keep in
  these circular motions a person of mass m100 kg.
  He weighs 9.8 m/sec2 100 kg 980 N

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• 3. When one object applies a force to another,
  the latter applies a force to the former, equal
  in magnitude and opposite in direction. (action
  and reaction).
• This explains how we walk. I push Earth back, it
  pushes me forward!

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Deductions by Newton

1. Elliptical orbit suggested to Newton an
   inverse-square law for gravity.
2. Keplers first law was almost but not exactly
   correct ellipses are the true shape of an orbit
   only for two isolated masses. Can deduce
   position, mass of unknown planets from tiny
   deviation of known planet from ellipse.
3. Keplers second law holds for any central force,
   is really a statement about conservation of
   angular momentum.
4. Keplers third law can be generalized to a more
   useful form that allows one to deduce the mass of
   the less massive object in orbit.
5. Two masses orbit around their center of mass,
   which is at a focus of their elliptical orbits.
   Important for binary stars, Pluto and its moon
   Charon.
6. Other conic sections such as parabolas and
   hyperbolas can describe unbounded orbits of one
   mass moving near a second mass.
7. Escape velocity, implication for black holes.
8. Tidal stresses, origins of tides, Roche limit,
   black holes.

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Brief Review at Whiteboard of Acceleration, Mass,
Forces, Gravity
Logic Objects like balls and planets often have
nonuniform motion called acceleration.
Acceleration has physics units speed over time or
m/s2. Experiments and thinking suggested to
Newton that acceleration can only arise from
something called a force, which acts on a body to
change its speed or direction. Many kinds of
forces gravitational, electrical, magnetic,
friction. Acceleration a is related to force F
by a positive quantity called the mass m of the
object
a (1/m)F. The bigger the
mass, the smaller the acceleration for a given
force. Note mass is measured in units of
kilograms kg force is measured in units of mass
x acceleration kg m/s2 called a newton and
abbreviated as N. By brilliant mathematical and
scientific thinking, Newton discovered a formula
for the gravitational force F of one mass on
another mass. Newton also realized that the
formula is universal and applies to any two
masses in space apples, Moons, planets, stars.
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Simplest Motion Uniform Motion
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Nonuniform Motion Acceleration
Speed is not constant or direction of motion is
not constant (but speed can be constant in case
of circular motion).
A
Where are speeds large in this picture if
stroboscope samples at equal times? Where are
speeds small in this picture?
B
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Demo Ball in Circular Motion
Ball would go in straight line (B) if a force
didnt act on the ball, which here is the string
pulling the ball toward the person at the center
of the circle.
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PRS Question
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Newtons Great InsightNonuniform Motion Caused
by Forces
Something from one object like Sun somehow
influences motion of other object like Earth.
That something is still not understood in any
fundamental sense but Newton discovered could be
described by an astonishingly simple and precise
mathematical rule now known as the universal law
of gravitation. The gravitational force becomes
weaker with distance but has an effect no matter
how far one object is from other object.. Total
force on object is sum of forces from all other
objects so depends on relative positions of all
the other objects. Mathematics can be hard,
computers have helped to obtain insight.
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Orbit Falling Around the Earth
Marvelous insight and calculation by Newton Moon
falls around Earth exactly as apple falls to the
ground, gravity is quantitatively universal.
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How to FlyThe Hitchhikers Guide to the Galaxy
There is an art, it says, or rather, a knack to
flying.The knack lies in learning how to throw
yourself at the ground and miss. Pick a nice day,
it suggests, and try it.The first part is easy.
All it requires is simply the ability to throw
yourself forward with all your weight, and
willingness not to mind that it's going to hurt.
That is, it's going to hurt if you fail to miss
the ground. Most people fail to miss the ground,
and if they are really trying properly, the
likelihood is that they will fail to miss it
fairly hard.