5.9.2 Calculating Gravitational Potential

Calculating Gravitational Potential

• Gravitational potential V_g can be calculated at a distance r from a point mass M using the equation: 

• Where:
  • V_g = gravitational potential (J kg^–1)
  • G = Newton’s gravitational constant (N m² kg^–2)
  • M = mass of the body causing the gravitational field (kg)
  • r = distance from the centre of mass of M to the point in the field (m)

• This means that the gravitational potential is negative on the surface of a mass (such as a planet), and increases with distance from that mass (becomes less negative toward zero)
• 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 producing the gravitational field and the distance the point is from that mass

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Changes 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 between the two points
  • This is represented by the symbol ΔV

• ΔV can therefore be expressed as the difference between the 'final' gravitational V_f potential and the 'initial' gravitational potential V_i  

ΔV = V_f – V_i

• Therefore, the change in potential between two points a distance r₁ and r₂ from some mass M is given by:

• This simplifies to:

 

• Where:
  • ΔV = change in potential (J kg^–1)
  • G = Newton’s gravitational constant (N m² kg^–2)
  • M = mass causing the gravitational field (kg)
  • r₁ = initial distance from mass M (m)
  • r₂ = final distance from mass M (m)

   Step 1: Write the gravitational potential equation

   Step 2: Substitute known quantities

= – 1.3 × 10⁷ J kg^–1