5.34 Gravitational Potential for a Radial Field

Near the Earth's Surface

• The gravitational potential energy (G.P.E) is the energy an object has when lifted off the ground given by the familiar equation:

G.P.E = mgΔh

• When using this equation, the G.P.E on the surface of the Earth is taken to be zero
  • This means work is done to lift the object

• This equation is only used for objects that are near the Earth's surface
  • This is because, near Earth's surface, the gravitational field is approximated to be uniform
  • Far away from the Earth's surface, the gravitational field is radial because the Earth is a sphere

In a Radial Field

• In a radial field, G.P.E is defined as:

The energy an object possesses due to its position in a gravitational field

• The gravitational potential at a point is the gravitational potential energy per unit mass at that point
• Gravity is always attractive, so work must be done on a mass to move it away to a point infinitely far away from every other mass
  • 'Infinity' is the point at which the gravitational potential is zero
  • Therefore, since the potential energy of all masses increases as work is done on them to move them infinitely far away, the value of the potential is always negative

• Gravitational potential energy is defined as:

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

• It is represented by the symbol, V and is measured in J kg^–1

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

• Where:
  • V_grav = 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)

• 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)
• 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

Gravitational potential decreases as the satellite moves closer to the Earth

   Step 1: Write the gravitational potential equation

   Step 2: Substitute known quantities

= –1.3 × 10⁷ J kg^–1