ACCELERATION OF GRAVITY

• gravitational force on a body on Earth's surface:

  

  where m = the body's (gravitational) mass
  M = the Earth's (gravitational) mass
  R = the Earth's radius (distance from Earth's center)

• acceleration of body due to gravitational force:
  Newton's 2nd law: acceleration = force/(inertial mass)

  

  where = inertial mass of body

  

  it is customary to use the symbol "g" for

  

• so we find the acceleration of a body under the influence of the Earth's gravity:

  

• the fact that all bodies fall the same way means that
  inertial mass = gravitational mass
• finally, we have:
  at the surface of the Earth, all bodies are subject to the same acceleration g due to Earth's gravitational force:

  

• the weight w of a body at the surface of the Earth (= the gravitational force exerted on it by Earth) is

  

GRAVITATION IN THE UNIVERSE

• gravitational force is weak, but has infinite range, and is not compensated by any repulsive "antigravity"
  in spite of its weakness, plays dominant rôle in universe;
• governs motion of planets, stars, galaxies,....
• instrumental in birth and death of stars
  

• star formation:
  • density fluctuation in interstellar gas/dust cloud can lead to run-away accumulation of matter due to gravitational attraction --
  • "gravitational collapse"
    = falling together of matter due to gravitational attraction;
  • formation of "protostar" = huge ball of gas
    (mainly hydrogen, some helium, traces of heavier stuff)
  • further contraction of protostar
    increase of temperature and pressure in its center
  • when temperature and pressure high enough,
    "nuclear fusion" process starts
  • "radiation pressure" due to nuclear fusion stops gravitational collapse;

• stable midlife star (e.g. Sun):
  (also called "main sequence star")
  dynamical equilibrium between gravitational attraction and radiation pressure from nuclear fusion of hydrogen into helium;
  star generates energy life becomes possible.
  

• death of stars:
  • when hydrogen is all used up:
    nothing to balance gravitational attraction collapse; further fate depends on mass of the star:
  • • "light" star:
      (like our sun), temperature in center not high enough to allow fusion of helium into heavier nuclei
      end as "white dwarf"
    • heavy star:
      fusion of helium into successively heavier elements possible, fusion stops when all is fused into iron;
      then gravitational collapse, stopped by "neutronization";
      abrupt stop of collapse supernova explosion;

  • "supernova remnants:"
    neutron stars, pulsars, black holes

MOMENTUM

• momentum
  is a measure of motion
  = "magnitude of motion", "impetus", "impulse"

  

• rate of change of momentum = force:

  

• if no force acts, then rate of change of momentum = 0
  momentum of an object does not change if no force acts on it.
• for a "system of objects":
  total momentum does not change if there is no outside (net) force
  -"conservation of momentum"
• examples:
  • throwing ball from a boat
  • recoil from a gun
  • jet and rocket propulsion
  • collision:
    total momentum before collision = total momentum after collision;

ANGULAR MOMENTUM

• extended bodies, or systems of bodies, can have "translational" or "rotational" motion
• rotational motion = movement around a straight line, the ``axis of rotation''

• quantities relevant to describe rotational motion:
  • "angular velocity" (= rate of change of angle)
    number of turns per unit time
  • "moment of inertia" I
    describes how mass is distributed with respect to rotation axis

  • 

  • angular momentum

    

    (note that formulae above only correct in simplest case, i.e. rotation of symmetric body around its symmetry axis)

  • "torque" = force lever arm

    

    lever arm = distance between center (axis) of rotation and line of action of the force;

  • torque = 0 if no force, or if force acts on center of rotation
    (e.g. pushing at center of wheel does not make wheel rotate, but pushing at wheel's rim does)

• conservation of angular momentum
  the total angular momentum of a rigid body or a system of bodies is conserved (does not change) if no outside torque acts on the system
• note: angular momentum is a vector - direction of axis of rotation makes a difference
  "angular momentum is conserved" means both magnitude and direction of rotation is constant
• if torque acts, then angular momentum changes: change in rotational speed, or direction of rotation, or both;
  change of axis of rotation is perpendicular to torque (precession)
• examples:
  • pirouette of ice skater
  • diver doing somersault
  • motion of planets around Sun
  • riding a bike
  • gyroscope
  • Earth's axis of rotation