Gravitational Field

• Newton's Law of Gravitation states that the gravitational force of attraction between two point masses is directly proportional to the product of the masses and inversely proportional to the square of the separation between their centres.
• $F=$$\frac{Gm_1m_2}{r^2}$ (in units)


  F = N

  G = N m^2 kg^-2

  $m_1$= kg

  $m_2$= kg

  r = m
• The gravitational constant, G, is approximately 6.67x10^-11 N m^2 kg^-2.
• Point masses have non-zero mass and no volume.
• The gravitational forces between two masses are equal and opposite and constitute an action-reaction pair. The forces always act along the line joining the two point masses. The attractive nature is written as:

  $\overrightarrow{F}=$$-\frac{Gm_1m_2}{r^2}\widehat{r}$
• A gravitational force acts on a mass placed in a gravitational field.
• A field of force is a region of space where there is a force acting on an object in that field.
• Gravitational field is invisible and represented by imaginary field lines.
• For (a):

  The gravitational field is uniform, where the field strength is the same at all points. This is represented by parallel field lines with equal spacing.

  
• For (b):

  The gravitational field is non-uniform, where the field lines originate from a single point mass. The field lines are thus drawn radially pointing towards the centre of the mass.

  
• The direction of a field at a point is along a tangent to the field line at that point.
• The density of the field lines at a point corresponds to the strength of the field at that point.
• Gravitational field strength at a point is defined as the gravitational force per unit mass exerted on a small test mass placed at that point.
• Gravitational field strength is a vector quantity and it is in the same direction as the gravitational force. Its SI unit is N kg^-1.
• SI unit of gravitational field strength can also be represented as m s^-2.
• $g=$$\frac{F}{m}$ (in units)


  g = N kg^-1 or m s^-2

  F = N

  m = kg
• $g=$$\frac{GM}{r^2}$ (in units)


  g = N kg^-1

  G = N m^2 kg^-2

  M = kg

  r = m
• The approximate constancy should be appreciated by considering the value of g at height h above the surface, where h is small compared to the radius R of the Earth of mass M.

  $g=$$\frac{GM}{(h+R)^2}\approx\frac{GM}{R^2}$

  since h << R
• The gravitational field strength near the surface of the Earth is approximately constant at 9.81 N kg^-1 or m s^-2.
• $\Delta{U}=$$mgh$ (in units)


  $\Delta{U}$= J

  m = kg

  g = m s^-2

  h = m
• A change in gravitational potential energy from a conveniently chosen lower reference level as the zero potential energy level is represented by the quantity mgh.
• The gravitational potential energy of a mass at a point is defined as the work done on the mass in moving it from infinity to that point.
• Gravitational potential energy of a mass at a point can be understood as the work done against the gravity in moving the mass from infinity to that point, where the work is done by an external force acting in the opposite direction to the gravitational attraction.
• $U=$$-\frac{GMm}{r}$ (in units)


  U = J

  G = N m^2 kg^-2

  M = kg

  m = kg

  r = m
• The gravitational potential at a point is defined as the work done per unit mass in moving a small test mass from infinity to that point.
• $\phi=$$\frac{U}{m}=$$-\frac{GM}{r}$ (in units)


  $\phi$= J kg^-1

  U = J

  m = kg

  G = N m^2 kg^-2

  M = kg

  r = m
• The unit of gravitational potential is J kg^-1 and is a scalar quantity.
• Gravitational field strength is negative of the gravitational potential gradient.

  $g =$$-\frac{d\phi}{dr}$
• Gravitational force is negative of the gravitational potential energy gradient.

  $F =$$-\frac{dU}{dr}$
• Kepler's 3rd Law: The square of the orbital period is directly proportional to the cube of the orbital radius.

  $T^2\propto{r^3}$
• Geostationary orbits made by satellites about the Earth's axis would appear stationary when observed from a point on the surface of the Earth, which provides permanent coverage of a given wide area.
• Satellites in geostationary orbits must:

  • Move from West to East
  • Have an orbital period of 24 hours
  • Orbit around the Earth's equator
• Geostationary satellites are used for telecommunications, television broadcasting, and weather monitoring.
• Polar orbital satellites are often used for Earth-mapping, Earth observation, and for reconnaissance.