Gravitational fields

• • All objects with mass create a gravitational field around them
  • The field extends all the way to infinity, but it gets weaker as the distance from the centre of mass if the object increases becoming negligible at long distances
  • Any other object with mass placed in a gravitational field will experience an attractive force towards the centre of mass of the object creating the field
• Gravitational field strength --> The gravitational force exerted per unit mass on a small object placed at that point within the field

  g = F/m
• • Gravitational field patterns can be mapped around an object with gravitational field lines
  • These lines don't cross, and the arrows on the lines show the direction of the field, which is the direction of the force on a mass at that point in the field
  • Gravitational force is always attractive, the direction of the gravitational field is always towards the centre of mass of the object producing the field
  • A stronger field is represented by field lines that are closer together
  • The field lines around a spherical mass form a radial field
• • If the field lines are parallel & equidistant, the field is said to be a uniform gravitational field
  • In a uniform field, the gravitational field strength doesn't change
• Newton's law of gravitation states the force between 2 point masses is:

  • directly proportional to the product of the masses
  • inversely proportional to the square of their separation
  • F is directly proportional Mm/r^2
  • -G shows that gravitational force is attractive
  • F = -GMm/r^2
  • The attractive force F between objects decreases with distance in an inverse - square relationship
• Multiple objects:

  • If several objects are involved, the resultant force can be determined by vector addition
• The gravitational field strength g at a distance r from the centre of an object of mass M is:

  g = -GM/r^2
• In a radial field the gravitational field strength at a point is:

  • directly proportional to the mass of the object creating the gravitational field
  • inversely proportional to the square of the distance from the centre of mass of the object
• Kepler's 1st law: The orbit of a planet is an ellipse with the sun at one of the 2 foci
• • An ellipse is a elongated circle with 2 foci. The orbits of all the planets are elliptical
• • The orbits have a low eccentricity ( a measure if how elongated the circle is)
• Kepler's 2nd law: A line segment joining a planet & the sun sweeps out equal areas during equal intervals of time
• Kepler's 3rd law: The square of the orbit period T of a planet is directly proportional to the cube of its average distance r from the sun
• T^2 = (4 x pi^2 / GM ) x r^3

  • the ratio T^2 / T^3 is a constant and equal to 4 x pi^2 / GM
  • the gradient of a graph of T^2 against r^3 must be equal to 4 x pi^2 / GM
• • For any satellite in orbit, the gravitational force F is given by
  • F = mv^2 / r = GMm / r^2
• • Since the only force acting on a satellite is the gravitational attraction between it & the Earth, it is always falling towards the Earth
  • As it is travelling so fast, it travels such a great distance that as it falls the Earth curves away beneath it, keeping it at the same height above the surface
  • v^2 = GM / r
• Uses of satellites:

  • Communications: satellite phones ( not mobile phones), TV, some types of satellite radio
  • Military uses: reconnaissance
  • Scientific research
  • Weather & climate: predicting & monitoring the weather across the globe & monitoring long - term changes in climate
  • Global positioning
• Geostationary satellites:

  • be in orbit above the Earth's equator
  • rotate in the same direction as the Earth's rotation
  • have an orbital period of 24 hours
• Gravitational potential Vg --> Work done per unit mass to move an object to that point from infinity
• • Infinity refers to a distance so far from the object producing the gravitational field that the gravitational field strength is zero
  • Gravitational potential is a scalar quantity - it only has magnitude
  • All masses attract each other. It takes energy, external work must be done, to move objects apart
  • Gravitational potential is a maximum at infinity
  • All values if gravitational potential are negative
• Gravitational potential in a radial field:

  • The gravitational potential at any point in a radial field around a point mass depends on 2 factors
  • The distance r from the point mass producing the gravitational field to that point
  • the mass M of the point mass
• • The gravitational potential Vg is directly proportional to M and inversely proportional to r
• • Vg = -GM / r
• • All values of Vg within the region of the gravitational field will be negative and when r tends to infinity the Vg = 0
• Changes in gravitational potential:

  • Moving towards a point mass results in a decrease in gravitational potential
  • Moving away from a point mass results in an increase in gravitational potential
• Gravitational potential energy --> work done to move the mass from infinity to a point in a gravitational field
• E = mVg

  E = -GMm / r
• Escape velocity:

  • In order to escape the gravitational field of a mass like a planet, an object must be supplied with energy equal to the gain in gravitational potential energy needed to lift it out of the field
  • In order for the projectile to have just enough energy to leave the gravitational field, the loss of kinetic energy must equal the gain in gravitational potential energy
  • 1/2mv^2 = GMm / r
  • The minimum velocity v for this condition to be met is called the escape velocity
  • V^2 = 2GM / r
  • The escape velocity on a given planet is therefore the same for all objects regardless of their mass