Gravitational Fields

• The formula for escape velocity is sqrt(2GM/r)
• Gravity is an attractive force between two masses, with the magnitude of the force being proportional to their product and inversely proportional to the square of the distance between them.
• Which of the following graphs correctly shows the relationship between the gravitational force, F, between two masses and the distance, r, between them?

  A B C D

  
• Newton’s law of gravitation states that: ‘Any two point masses attract each other with a gravitational force that is directly proportional to the product of their masses and inversely proportional to the square of their separation.’
• Inverse square law: If the distance is doubled the , lines spread out over four times its surface area, so their concentration is reduced by a quarter.
• So the gravitational field for a spherical mass of radius 𝑅 can be represented as a radial field. The arrows always point towards the centre of a mass since gravitational force is always attractive.
• The field close to the surface of a spherical mass is approximately uniform. Hence the lines are parallel, equidistant and perpendicular to the surface.
• Gravitational field strength is a vector quantity in the direction of the gravitational force.
• Kepler’s 3rd law: Kepler showed that, for a planet orbiting the Sun, the relationship between the orbital time period 𝑻 and the orbital radius 𝒓 is given by: 𝑻^𝟐 ∝ 𝒓^3
• A satellite following a synchronous orbit has a time period equal to the rotational period of the planet being orbited.

  A satellite following a synchronous orbit around the Earth is described as having a geosynchronous orbit.

  A communication satellite requires a special type of geosynchronous orbit, e.g. a geostationary orbit.
• A satellite with a geostationary orbit around a planet will remain vertically above the same point on the equator of that planet at all times.
• The gravitational potential 𝑽 at a point is the work done ∆�� per unit mass 𝒎 (or the change in potential energy ∆𝑬𝒑 per unit mass 𝒎) to move a mass 𝒎 from infinity to that point.
• work done ∆𝑊 in moving a 𝑚 from a from infinity to a point at a distance 𝑟 from the centre of the planet of mass 𝑀 is ∆𝑾 =- GMm/r
• So the gravitational potential 𝑉 at a distance 𝑟 from the centre of a planet of mass 𝑀: V= - GM/r Hence 𝑉 is inversely proportional to 𝑟. 𝑉 is a scalar quantity.
• An object’s escape velocity is the minimum velocity required at the surface of a planet in order to escape the gravitational pull of the planet using its own kinetic energy (i.e. no further propulsion).
• The total energy of a satellite in orbit is the sum of its gravitational potential energy and its kinetic energy: 𝑬total= - GMm/2r
• For a satellite in orbit: Ek= +GMm/2r
• For a satellite in orbit: Ep= - GMm/r
• What is Newton’s law of gravitational force?

  Any two point masses that attract each other with a gravitaional force that is directly proportional to the product of their masses and inversely proportional to the square of their separation
• What are the units for gravitational field strength?


  Nkg^-1
• What is meant by the inverse square law?

  If the distance is doubled the , lines spread out over four times its surface area, so their concentration is reduced by a quarter.
• What is Kepler’s 3rd law?


  Kepler showed that, for a planet orbiting the Sun, the relationship between the orbital time period 𝑻 and the orbital radius 𝒓 is given by: T^2∝ r^3
• What is the equation that relates orbital time period T and orbital radius r?

  $T^2=$$(4π^2​/GM)r^3$
• What is the equation for orbital speed v?

  $v^2=$$GM/r$
• What is meant by a ‘Geostationary orbit’ and how is this achieved?

  A satellite with a geostationary orbit around a planet will remain vertically above the same point on the equator of that planet at all times.
• What is a satellite in a ‘Geostationary orbit’ used for?

  Telecommunications transmissions. Satellite television broadcasting.
• What are the advantages of a ‘Geostationary orbit’?

  Receiving dishes can have a fixed position pointing to the same spot in the sky and can maintain continuous contact with the satellite.
• What are the disadvantages of a ‘Geostationary orbit’?

  Each satellite communicates with a restricted area of the Earth’s surface.
• What is the definition of gravitational potential?

  The gravitational potential 𝑽 at a point is the work done ∆𝑾 per unit mass 𝒎 to move a mass m from infinity to that point.
• What is the equation for gravitational potential?


  $V=$$-(GM/r)$
• What are the units for gravitational potential?


  Jkg^-1
• • 𝑽 = 𝟎 at infinity.

  • 𝑉 = ∆𝑤/𝑚 : Work is done per unit mass as a body moves from infinity towards a point in a field.

  • Work is done by the field, so potential values become more negative as ‘r’ decreases.
• What is the area under a graph of g against r?

  ∆𝑉 the gravitational potential
• What is an equipotential?

  An equipotential in a gravitational field is a surface (or line in a two-dimensional context) on which the gravitational potential is constant. This means that at every point on an equipotential surface, a mass would have the same potential energy due to the gravitational field.
• Moving a mass perpendicularly across a field line does not alter its potential energy.
• If a mass is moved in the direction of the field lines the mass’ potential energy is reduced.
• A greater density of field lines represents a stronger field.
• An object that has escaped a gravitational field has zero potential energy.
• The total energy of a satellite orbiting the Earth has a negative value.
• The gravitational potential gradient at a point has the same numerical value as the gravitational field strength at that point.