The Copernican Revolution
(Gravity and Motion)

        A. Ptolemaic Universe

  • The Ptolemaic system is geocentric: The Moon, Sun, and planets orbit the Earth (in circular orbits).
  • Retrograde motion is explained by epicycles = extra circles added to motion.
  • The model is based on Aristotle's physics:
    • The heavens are made of a quintessence (fifth element) which is like nothing on Earth. Heavenly bodies are perfect and unchanging.
    • Massive bodies fall faster than light ones.

    B. Nicolaus Copernicus

  • Pointed out that a heliocentric model also fit observations.
  • In the Copernican model, the Earth and planets orbit the Sun, while the Moon orbits the Earth.
  • He also used circular planetary orbits.
  • Retrograde motion explained by the Earth ``passing'' or being passed by other planets.
  • Epicycles still needed to get precise motions of the planets.

    C. Tycho Brahe

  • Made precise unaided-eye measurements of planetary positions.
  • Observed a ``new star'' (1572) and a comet (1577) and used parallax to show that they were both beyond the Moon.
  • Parallax is measuring the distance to an object by measuring the apparent change in position angle of that object as you move.

    D. Johannes Kepler

  • Used Brahe's observations to calculate planetary orbits.
  • He found that he could make the Copernican model work, without epicycles, if planetary orbits were ellipses, not circles.
  • Developed Laws of Planetary motion:
    1. Each planet moves in an elliptical path, with the Sun at one focus.
    2. The line between the Sun and the planet sweeps out equal areas in equal times.
    3. The ratio of the cube of the semi-major axes to the square of the orbit periods is the same for all planets. T2 proportional to a3
  • The closest point between the orbit and the Sun is the periapse (or perihelion). The furthest point is the apoapse or aphelion.
  • The eccentricity of the orbit is one minus the ratio of the periapse distance to the semimajor axis. e = 1 - (p/a)
  • The eccentricity is also the ratio of the apoapse distance to the semimajor axis minus 1. e = (ap/a) - 1

    E. Galileo Galilei

  • Used the first telescopes for astronomy
  • Discovered: mountains on Moon, sunspots, phases of Venus, 4 moons of Jupiter. The last two showed fundamental flaws in Ptolemaic model of universe.
  • Showed flaws in Aristotelean physics: different sized bodies fell with the same constant acceleration.

    F. Sir Isaac Newton

  • Newton argued that all motion is governed by simple mathematical laws.

    F.1 1st Law of Motion: A body remains at rest or moves with a constant speed and direction unless acted on by a net outside force. This law explains why it's so hard to stop on ice: inertia (the tendency to keep moving) isn't opposed by any friction, so you keep going.

    F.2 2nd Law of Motion: A net outside force causes an acceleration given by: a = F/m.

  • Acceleration (a) = change in speed (faster or slower) or direction. Measured in cm/sec2 (or m/sec2).
  • Friction is a common form of acceleration.
  • Mass (m) = quantity of material in a body. Measured in grams or kilograms.
  • Force: Measured in grams centimeters/second2 = dynes,
    or kilograms meters/second2 = Newtons.
  • Weight = force of gravity on a body = mass times acceleration due to gravity.

    F.3 3rd Law of Motion For every force on one body, there is an equal and opposite force on another body.

  • In other words, when you push on something, that something exerts an equal force on you.

    F.4 Universal Gravity

  • Observation: All bodies fall with the same constant acceleration. 9.8 cm/s2.
  • Theory: Since a = F/m, the force of gravity Fg = m1g, where m1 = mass of body feeling force. So the force on a person with mass 50kg is 50kg x 9.8 m/s2 = 490 Newtons.
  • Observation: Large masses exert greater force.
  • Theory: Fg is proportional to m1 times m2, where m2 = mass of body exerting force.
  • Observation: Objects further away are accelerated much less than objects close by (as for planets orbiting the Sun).
  • Theory: Fg is proportional to 1/d2, where d = distance between centers of bodies. We can combine these observations and theories to produce a universal law of gravity, Fg = Gm1m2/d2, where G is a universal constant. G = 6.67 x 10-11 m3/kg/s2.

    F.5 Implications of Newton's Laws.

  • Everything in the universe attracts everything else.
    • eg. Consider two people, one with a mass 50kg, the other with a mass 75kg, who are 0.5m apart. What is the force between them in Newtons?
      Fg = (6.67 x 10-11 m3/kg/s2) x (50kg) x (75kg) / (0.5m)2
      = 1 x 10-6 Newtons.
  • Your weight changes depending on what planet you're on. If its mass is greater, your weight is greater, but if its radius is greater, your weight gets smaller.
  • The planets are ``falling'' around the Sun.
  • Kepler's laws of planetary motion can be derived from Newton's laws of motion and gravity. Bound orbits can be circles (eccentricity e = 0), or ellipses (0 < e < 1). Unbound orbits can be parabolae (e = 1) or hyperbolae (e > 1).
  • Weightless = Freefall. The space shuttle feels the Earth's gravity, but the astronauts inside it are "falling" with it, and hence they appear to be "weightless".

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