Knowledge-14 GF

• Gravity is a universal attractive force that acts between all matter.
• A point is one that behaves as if all its mass concentrated at its centre , like a sphere of uniform density.
• Newtons law of universal gravitation for two point masses is:

  • gravitational force F = Gmn / r^2
  • Where G = gravitational force constant , 6.67x10^-11
• the gravitational force between two point masses is:

  • always attractive
  • proportional to the product of their masses
  • an inverse square law F proportional to 1/r^2
• A force field is a region of space in which an object experiences a non-contact force.
• Gravitational fields:

  • exist around all objects with mass
  • Any mass will experience an attractive force if placed in the gravitational field of another mass.
  • as both masses have a gravitational field the other mass will experience an attractive force of same size in opposite directions.
• • the arrows on field lines show the direction in which the force acts
  • the separation of field lines indicates the strength of the field , the closer together the stronger the field.
• Radial fields:

  • where field lines point towards or away from the object causing the field , the separation between them increases with distance , showing that field strength decreases with time.
• Uniform fields:

  • where the field strength is the same at every point , indicating that field lines are parallel to each other and equally spaced.
• For a planet of radius R , the gravitational field strengths at its surface is :

  • g (surface) = GM / r^2
• The gravitational potential at a point in a field is defined as the work done per unit mass to move a small object from infinity to that point.
• The gravitational potential at a point in a radial field caused by mass M is :

  • V = -GM / r
• The gravitational potential energy of an object of mass m at any point in a gravitational field can be calculated using :

  • E = Vm = - GMm / r
• Work must be done against gravitational attraction to move masses apart , so an object gain E as it moves toward infinity but since the maximum value E can have is zero all the points in the field must have negative value for gravitational potential.
• Gravitational potential difference is the difference between the gravitational potential at two points in a gravitational field.
• The work done in moving a mass m between two points in a gravitational field is :

  • work done = mass x gravitational potential difference
  • the change in gravitational potential can be found from the area under a g against r graph.
• An equipotential surface is one where the gravitational potential is the same at all points.
• No work is done when moving along an equipotential difference between any two points on the surface is zero.
• Equipotential surfaces are perpendicular to field lines
• The potential gradient at a point in a field is the change of potential per metre at that point:

  • potential gradient = change in V / change in r
• Gravitational field strength is related to potential gradient by:

  • g = - change in V / change in r
• the gradient of the graph of gravitational potential against distance is -change in V / change in r.
• The value of g at any point in a gravitational field can be found by finding the gradient of V against r on a graph at that point.
• The escape velocity of an astronomical body is the minimum speed an object must be given to escape the body's gravitational field when launched from the surface.
• To derive escape velocity:

  • 1/2 mv^2 = GMm / R
  • 1/2 v^2 = GM/R
  • V^2 = 2GM/R
  • V = root (2GM/R)
• The time period T of a planet or satellite in a circular orbit is related to the radius of the orbit r by:

  • T^2 is proportional to r^3
• For any planets orbiting the same star:

  • T^2 / r^3 = constant
• For any two planets orbiting the same star:

  • T1^2 / r1^3 = T2^2 / r2^3
• for an object of mass m in circular orbit of radius r around an object mass M , gravitational attraction provides the centripetal force:

  • GMm / r^2 = mv^2 / r
• To find time period of an circular object:

  • T^2 = 4 pie^2 r^2 / GM
• Geostationary orbits:

  A satellite with a stationary orbit has a time period equal to that rotational period of the object it is orbiting.
• A geostationary satellite is an example of a satellite with a synchronous orbit that:

  • remains above the same point on the earths equator
  • orbits in the equatorial plane
  • has a time period of 24 hours
  • moves in the same direction as earths rotation
• low orbiting satellites are those that orbit less than 2000 km above the earth. Satellites with a polar orbit move in a plane that is perpendicular to the equatorial plane.
• A satellite in a circular orbit remains at the same height above the object it is orbiting and moves with a constant speed , so its kinetic and gravitational energies are constant.
• A satellites total energy is:

  • E total = Ek + Ep = GMm / 2r