Gravitational Fields

• Newton's Law:

  • Gravity acts on any objects which have mass and is always attractive.
  • Law of gravitation shows magnitude of gravitational force between two masses is directly proportional to the product of the masses.
  • Also stated that the force is inversely proportional to the square of the distance between them.
• Gravitational Field Strength:

  • Two types of field - Uniform and Radial.
  • Arrows on field lines show direction force is acting.
  • Uniform exerts same force everywhere within a field.
  • Force in radial field depends on position.
  • The stronger the field, the higher the density of field lines.
  • Gravitational Field Strength (g) is the force per unit mass exerted by a gravitational field on an object.

  
• Gravitational Potential:

  • At a point it is the work done per unit mass when moving an object from infinity to a point.
  • At infinity, gravitational potential is 0.
  • Gravitational Potential is always negative, as energy is released as GPE is reduced.
• Gravitational Potential Difference:

  • The energy needed to move a unit mass between two points in a field.
  • Can be used to find the work done when moving an object in a gravitational field.
• Equipotential Surfaces:

  • These are surfaces which are created through joining points of equal potential together, therefore the potential on a equipotential surface is constant everywhere.
  • As gravitational potential difference is zero when moving along the surface, there is no work done when moving along the surface.

  
• Gravitational potential (V)is inversely proportional to the distance between the centres of the two objects (r). This can be represented on the graph below:

  
• Gravitational field strength (g) at a certain distance can be measured by drawing a tangent to the curve at that distance and calculating its gradient, then multiplying by -1:

  
• Kepler's Third Law:

  • This is that the square of the orbital period (T) is directly proportional to the cube of the radius(r): $T^2\propto r^3$
• Derivation of Kepler's Third:

  1. Orbiting objects experience a gravitational force towards centre of mass. Gravitational force acts as centripetal force: $\frac{mv^2}{r}=$$\frac{GMm}{r^2}$
  2. Rearrange equation to make v squared the subject: $v^2=$$\frac{GM}{r}$
• 1. Velocity is the rate of change of displacement, therefore v can be found in terms of radius and orbital period: $v =$$\frac{2\pi r}{T} \rightarrow v^2 =$$\frac{4\pi^2 r^2}{T^2}$
  2. Sub v squared in terms of r and T, into original equation:$\frac{4\pi^2 r^2}{T^2} =$$\frac{GM}{r}$
  3. Rearrange to make $T^2$ the subject: $T^2 =$$\frac{4\pi^2}{GM} \times r^3$
• As $\frac{4\pi^2}{GM}$is a constant, it shows that $T^2 \propto r^3$
• The total energy of an orbiting satellite is made up of its kinetic and potential energy, and is constant.

  For example, if the height of a satellite is decreased, its gravitational potential energy will decrease. However, it will travel at a higher speed meaning kinetic energy increases therefore total energy is kept constant.
• Escape Velocity:

  • The minimum velocity an object must travel at in order to escape the gravitational field at the surface of a mass.
  • The is the velocity at which the object's kinetic energy is equal to the magnitude of its gravitational potential energy.
• Escape Velocity Derivation:

  $E_k =$$E_p$

  $\frac{1}{2}mv^2 =$$\frac{GMm}{r}$As gravitational potential energy $=$$m\Delta V$

  $\rightarrow v =$$\sqrt\frac{2GM}{r}$
• Synchronous Orbit:

  • Where the orbital period of the satellite is equal to the rotational period of the object that it is orbiting.
  • A geosynchronous orbit is one orbiting Earth, where T would be 24 hours.
• Geostationary Satellites:

  • These follow specific orbits that mean they remain at the same point above the equator.
  • These are useful for sending TV and telephone signals as the plane of an aerial or transmitter does not need to be altered.
• Low-orbit satellites:

  • These have significantly lower orbits than geostationary satellites so they travel much faster and orbital periods are much smaller.
  • These are usefull for monitoring the weather, making scientific observations of unreachable areas and for military applications.
  • They must often work together for constant coverage of a region.