Universal Gravitation

Summary

• Gravitational Force
  • Newton's law of universal gravitation states that any two objects with mass will experience a gravitational force that is…
    • universal (acts on all objects)
    • attractive (there is no such thing as antigravity)
    • directly proportional to the mass of each object (mass makes gravity)
    • inversely proportional to the square of the separation between their centers (inverse square rule)
  • Equations

    magnitude notation

    Fg = −  Gm1m2 r2

    Fg =	magnitude of the gravitational force between two objects (Note: A minus sign is often added to the equation to show that the force is attractive. It is usually ignored in practical calculations.)
    G =	universal gravitational constant, often shortened to gravitational constant, or referred to informally as Big G (6.67 × 10−11 Nm2/kg2)
    m1, m2 =	mass of the two objects
    r =	separation between the centers of the two objects

    vector notation

    Fg = −  Gm1m2  r̂ r2

    Fg =	gravitational force vector between two objects
    r̂ =	unit vector along the line separating the centers of the two objects (Note: The use of the minus sign here shows that the force vector points in the opposite direction of the separation vector. The force and separation have opposite sense.)

  • Since the universal gravitational constant is a "small" number, gravity is a "weak" force.
    • The only object that exerts an appreciable gravitational force on other objects in daily life is the Earth.
    • We perceive the gravitational force of the Earth because it is massive (5.97 × 10⁻²⁴ kg) and we are not normally far from it (usually less than 6,400 km from its center).
    • A typical human can easily push up on a small patch of ground with a normal force that is equal to or greater than the gravitational force of the entire Earth pulling them down.
  • The gravitational force between objects has infinite range.
    • There is no value of the separation, r, that can make F_g = 0 in Newton's law of universal gravitation.
    • Gravity can be reduced (by moving far away from massive objects), but it can never be eliminated.
  • Gravity is the dominant force in the universe when energies are low and volumes are large (solar systems, galaxies, clusters, superclusters, and larger structures).
• Field
  • A field is…
    • a physical quantity that has a value (or set of values) at every point in space and time
    • a mathematical function that returns a value (or set of values) for every point in space and time
    • a way to deal with philosophical arguments against action at a distance
  • Fields can be…
    • scalar fields associated with quantities such as…
      • temperature
      • pressure
      • density
      • electric charge
    • vector fields associated with…
      • the flow of fluids (the velocity field) such as…
        • wind
        • ocean currents
        • solar wind
      • non-contact forces (force fields) such as…
        • gravitational field
        • electric field
        • magnetic field
    • tensor fields associated with…
      • continuum mechanics quantities, namely the…
        • stress tensor
      • general relativity quantities such as the…
        • metric tensor
        • stress-energy tensor
        • Ricci curvature tensor
• Gravitational Field
  • Magnitude

    g =	Fg
    	m

    Where…

    g =	gravitational field (gravitational field strength)
    Fg =	gravitational force on test mass
    m =	mass of test mass

  • Units

    ⎡ ⎣	N	=	m	⎤ ⎦
    	kg		s2

  • Direction
    • Gravity is described by a vector field.
    • The direction of the gravitational field at any point in space is the direction of the net gravitational force on a "small" test mass.
    • The gravitational field around a spherically symmetric mass is radial and points inward.
    • Use vector addition to find the net gravitational field when more than one massive object is present.