Gravitational Fields

• Gravity is an attractive force that acts on all objects with mass
• G (universal gravitational constant) = 6.67 x 10 ^-11 m^3kg^-1s^-2
• A gravitational field is a region where masses experience a gravitational force
• Grav. field lines show the path a small test mass will follow if placed in the field
• What can field lines tell you about a field?

  The direction and strength of the field
• A test mass is a hypothetical object with negligible mass
• field lines

  A) Uniform

  
• field lines

  A) radial

  
• Gravitational field strength (g) is the force per unit mass
• grav. field strength

  A) M

  B) r

  C) F

  D) m

  
• Field strength is constant in a uniform field, but varies in a radial field
• Force is often written as negative as the displacement is often defined as the positive direction
• Field strength is a vector quantity
• As field strength is a vector, there is a point between two point masses, where their field strength sums to zero - > the field strengths are equal and opposite.

  • This point will always be closer to the LIGHTER mass
• Graph of field strength and distance between two masses

  A) 9.81

  
• Gravitational potential at a point is the work done per unit mass to move a test mass from infinity to that point
• potential is defined as 0 at infinity. Therefore work is done on the object to move it from a point to infinity.

  • This means that the value of potential is negative at any point in the universe
• For a radial field

  A) M

  B) r

  
• Gravitational potential difference is the energy needed to move a unit mass between two points. TF WD = mass x potential difference

  
• Field strength = - potential gradient

  for a graph of g against r
• g = -V / r
• Gravitational potential difference is the difference in the gravitational potentials of two points in a gravitational field
• Gravitational potential is a scalar quantity
• Unit of potential: Jkg^-1
• Total gravitational potential between two massive objects: sum of the two (negative) potentials, as it is scalar
• To find the field strength at a point from a potential graph, draw a tangent to the graph and workout the gradient.

  A) -g

  B) radius

  
• A potential v distance graph will have an asymptote at the objects radius and will be situated in the 4th quadrant

  
• Area under field strength v distance graph = grav. potential difference

  
• Gravitational potential energy = -GMm/r Unit: J
• We can use E = mgh for situations where g can be considered constant e.g. close to a planets surface where field can be considered uniform
• Gravitational potential energy is negative everywhere in the universe as it is equal to V x m , where potential is defined as 0 at infinity
• Equipotential surfaces are surfaces over which the potential is the same
• No work is done moving between points on an equipotential surface
• Field lines and equipotentials are mutually perpendicular
• RADIAL

  Equipotentials get further apart as gravity can't be considered constant at this scale.

  g gets smaller as you get further away so lines get further apart

  
• UNIFORM

  evenly space as at surface, gravity can be considered constant, so it requires the same amount of work to move between equipotentials as E = mgh

  
• When should gravitational potential reach zero?

  never
• The escape velocity is the minimum velocity needed to escape a gravitational well
• potential vs distance graph is in the fourth quadrant

  • never reaches zero

  
• ESCAPE VELOCITY DERIVATION

  At infinity:

  Ep = 0, Ek = 0, Etot. = 0

  At surface:

  Ep = -GMm/r , Ek = 1/2 mv^2

  1. 1/2mv^2 = GMm/r
  2. Vescape= root(2GM/r)
  3. = root(2xmod(potential))