13.2.1 Gravitational Potential

• 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

• The G.P.E on the surface of the Earth is taken to be 0
  • This means work is done to lift the object

• However, outside the Earth’s surface, G.P.E can be defined as:

The energy an object possess 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
• Therefore, the gravitational potential is defined as:

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

• The equation for gravitational potential ɸ is defined by the mass M and distance r:

• Where:
  • ɸ = 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)

• The gravitational potential is negative near an isolated mass, such as a planet, because the potential when r is at infinity is defined as 0
• Gravitational forces are always attractive so as r decreases, positive work is done by the mass when moving from infinity to that point
  • When a mass is closer to a planet, its gravitational potential becomes smaller (more negative)
  • As a mass moves away from a planet, its gravitational potential becomes larger (less negative) until it reaches 0 at infinity

• This means when the distance (r) becomes very large, the gravitational force tends rapidly towards 0 at a point further away from a planet

Gravitational potential increases and decreases depending on whether the object is travelling towards or against the field lines from infinity

Step 1:            Write the gravitational potential equation

Step 2:            Determine the value of r

r is the distance from the centre of the planet

Radius of the planet = planet diameter ÷ 2 = 7600 ÷ 2  = 3800 km

r = 3800 + 400 = 4200 km = 4.2 × 10⁶ m

Step 3:            Substitute in values