Gravitational Potential (CIE A Level Physics)

• Gravitational potential, φ, at a point is defined as:

The work done per unit mass in bringing a small test mass from infinity to the point

• Gravitational potential always has a negative value because:
  • It is defined as having a value of zero at infinity
  • Since the gravitational force is attractive, work must be done on a mass to reach infinity

• On the surface of a mass (such as a planet), gravitational potential also has a negative value
  • The value becomes less negative, i.e. it increases, with distance from that mass

• Work has to be done against the gravitational pull of the planet to take a unit mass away from the planet
• The gravitational potential at a point depends on:
  • The mass of the object
  • The distance from the centre of mass of the object to the point

Changing gravitational potential

Gravitational potential decreases as the satellite moves closer to the Earth

• The gravitational potential, φ, in a field due to a point mass, M is given by the equation:

• Where:
  • φ = gravitational potential (J kg^-1)
  • G = Newton’s gravitational constant
  • M = mass of the body producing the gravitational field (kg)
  • r = distance from the centre of the mass to the point mass (m)

Change in gravitational potential

• Two points at different distances from a mass will have different gravitational potentials
  • This is because the gravitational potential increases with distance from a mass
• Therefore, there will be a gravitational potential difference Δφ between the two points

• Where:
  • = initial gravitational potential (J kg^–1)
  • = final gravitational potential (J kg^–1)

• The change in gravitation potential Δφ is calculated using the equation:

• Where:
  • r₁ = initial distance from the centre of mass to the point mass (m)
  • r₂ = final distance from the centre of mass to the point mass (m)

Gravitational potential energy

• In a Radial field, gravitational potential energy (GPE) describes the energy an object possesses due to its position in a gravitational field

• The gravitational potential energy of a system is defined as:

The work done to assemble the system from infinite separation of the components of the system

• The equation for GPE between two point masses m₁ and m₂ at a distance r is:

• Where:
  • G = universal gravitational constant (N m² kg^–2)
  • m₁ = larger mass producing the field (kg)
  • m₂ = mass moving within the field of m₁ (kg)
  • r = distance between the centre of m₁ and m₂ (m)

Gravitational potential energy vs distance

Gravitational potential energy increases as a satellite leaves the surface of the Moon (of mass M)

Change in gravitational potential energy

• The change in GPE when a small mass, m₂ moves towards, or away from, another larger mass, m₁ is given by:

• Where:
  • m₁ = mass that is producing the gravitational field (e.g. a planet) (kg)
  • m₂ = mass that is moving in the gravitational field (e.g. a satellite) (kg)
  • r₁ = first distance of m₂ from the centre of m₁ (m)
  • r₂ = second distance of m₂ from the centre of m₁ (m)

Work done

• The Work done against a gravitational field is equal to the change in gravitational potential energy (GPE) of an object in a gravitational field
  
  • For example, a satellite lifted into space from the Earth’s surface

• Recall that work done is equal to:
  • force applied multiplied by the displacement of the object
  • energy transferred

• Recall that Newton's Law of Gravitation relates the magnitude of the gravitational force F between two masses m₁ and m₂ separated by distance r is:

• So the work done by the gravitational force is equivalent to the change in gravitational potential energy: