7.2 gravitational fields

• Newtons law of gravitation, 1687

  • Every particle of matter in the universe attracts every other particle with a force that is:
  • Directly proportional ro the product of their masses
  • Inversely proportional to the square of their distance apart
• Force between two point masses

  $F=$$\frac{Gm_1m_2}{r^2}$

  G - the universal constant of gravitation

  $6.672\times10^{-11}Nm^2kg^{-2}$
• Gravity 

  $g=$$\frac{F}{m}$

  Magnitude of g in a radial field:

  $g=$$\frac{GM}{r^2}$

  Gravitational field - a region of space around an object of mass where another object mass experiences a force

  • Force is always attractive
• Gravitational field line - the direction of force on a test mass at that point in the field

  • The direction of movement on a test mass when placed at that point in the field

  Gravitational field strength - the force per kilogram at a point in the field

  • Property of field
  • Vector

  $Nkg^{-1}$
• Gravitational potential

  • Work done per unit mass to move a mass from infinity to a point in the field
  • Property of field
  • Scalar
  • $Jkg^{-1}$
  • Zero at infinity
  • Potentials are always negative
  • Fields are always attractive
  • Need to put energy in, to move away from an object

  $V=$$\frac{W}{m}=$$-\frac{GM}{r}$
• Gravitational potential difference

  • Two points at different distances from a mass have different gravitational potentials
  • Gravitational potential difference - the difference in potential between these points
• Equipotentials

  • Positions of constant potential in a gravitational field
  • Lines or surfaces
  • Always meet the field at 90°
  • No work is done by the field moving a mass along an equipotential
  • As work done = change in potential x mass, if there is no change in potential the work done stays constant
  • Evenly spaced in uniform fields
  • Spacing gets further apart in radial fields
• Potential gradient

  • Change in potential per meter at a point within a gravitational field
  • $Jkg^{-1}m^{-1}$
  • Change in ‘r’ considered as really small
  • Comparably negligible to the earth’s radius
  • Same value as the magnitude of the gravitational field strength
  • Negative $g=$$-\frac{\Delta V}{\Delta r}$
  • The area under the graph of g against r is the change in potential 

  
• Relationship between radius and period

  • Force on an object in circular motion
  • Force of attraction due to gravity between two objects
  • Equate the equations for force
  • Rearrange for velocity
  • Speed is inversely proportional to the square root of its orbital radius
  • v ∝ 1/r
  • Rearrange speed of an object in circular orbit to find the orbital period
  • Substitute v and rearrange
  • Square both sides 
  • $T^2=$$\frac{4\pi^2r^3}{GM}$
  • Therefore $T^2$∝ $r^3$
• Energy of an orbiting satellite

  • Has kinetic and potential energy
  • Total energy is always constant
  • In circular motion, the speed and the distance above the mass it’s orbiting are constant
  • Therefore KE and PE are constant
  • Elliptical orbit speed increases as height decreases
  • KE increases as PE decreases
  • Total energy remains constant
• Escape velocity 

  • Velocity at which an objects kinetic energy is equal to minus its gravitational potential energy
  • Total energy is zero 
  • $V=$$\sqrt{\frac{2GM}{r}}$

  Derivation: 

  • Multiply gravitational potential energy per unit mass by mass to get gravitational potential energy
  • Total energy is zero
  • $\frac{1}{2}mv^2=$$\frac{GMm}{r}$
  • Cancel out mass
  • Rearrange for velocity
• Geosynchronous  

  • have the same time period as earth
  • 24 hours
  • geostationary 
  • fixed position above Earth 
  • orbits along the equator 
  • perfect for TV broadcasting
• Geostationary orbit

  • Synchronous - orbital period is the same as rotational period
  • Geostationary - above a fixed point on the Earth
  • Must always be directly above the equator
  • Travels at the same angular speed as the earth
  • Orbit takes one day
  • Orbital radius = 42 000km, 36 000km above surface
  • TV and telephone signals - don't have to alter angle of receiver
• Low orbit 

  • Orbit between 180-2000km above the Earth
  • Cheaper to launch and require less powerful transmitters
  • Useful for communications
  • Proximity to the earth and high orbital speed means multiple are needed to maintain constant coverage
  • Close enough to see the earth in high level detail
  • Imaging satellites - spying, monitoring weather
  • Orbits lie in the plane including the north and south pole
  • Each orbit is over a new part of Earth's surface so the whole earth can be scanned