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13.1 Newton's Law of Universal Gravitation

  • All masses attract one another with a gravitational force proportional to their masses and inversely proportional to the square of the distance between them.
  • Spherically symmetrical masses can be treated as if all their mass were located at the center.
  • Nonsymmetrical objects can be treated as if their mass were concentrated at their center of mass, provided their distance from other masses is large compared to their size.

13.2 Gravitation Near Earth's Surface

  • The weight of an object is the gravitational attraction between Earth and the object.
  • The gravitational field is represented as lines that indicate the direction of the gravitational force; the line spacing indicates the strength of the field.
  • Apparent weight differs from actual weight due to the acceleration of the object.

13.3 Gravitational Potential Energy and Total Energy

  • The acceleration due to gravity changes as we move away from Earth, and the expression for gravitational potential energy must reflect this change.
  • The total energy of a system is the sum of kinetic and gravitational potential energy, and this total energy is conserved in orbital motion.
  • Objects must have a minimum velocity, the escape velocity, to leave a planet and not return.
  • Objects with total energy less than zero are bound; those with zero or greater are unbounded.

13.4 Satellite Orbits and Energy

  • Orbital velocities are determined by the mass of the body being orbited and the distance from the center of that body, and not by the mass of a much smaller orbiting object.
  • The period of the orbit is likewise independent of the orbiting object’s mass.
  • Bodies of comparable masses orbit about their common center of mass and their velocities and periods should be determined from Newton’s second law and law of gravitation.

13.5 Kepler's Laws of Planetary Motion

  • All orbital motion follows the path of a conic section. Bound or closed orbits are either a circle or an ellipse; unbounded or open orbits are either a parabola or a hyperbola.
  • The areal velocity of any orbit is constant, a reflection of the conservation of angular momentum.
  • The square of the period of an elliptical orbit is proportional to the cube of the semi-major axis of that orbit.

13.6 Tidal Forces

  • Earth’s tides are caused by the difference in gravitational forces from the Moon and the Sun on the different sides of Earth.
  • Spring or neap (high) tides occur when Earth, the Moon, and the Sun are aligned, and neap or (low) tides occur when they form a right triangle.
  • Tidal forces can create internal heating, changes in orbital motion, and even destruction of orbiting bodies.

13.7 Einstein's Theory of Gravity

  • According to the theory of general relativity, gravity is the result of distortions in space-time created by mass and energy.
  • The principle of equivalence states that that both mass and acceleration distort space-time and are indistinguishable in comparable circumstances.
  • Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass.
  • Evidence for the existence of black holes is still circumstantial, but the amount of that evidence is overwhelming.
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