Gravitational Potential (CIE A Level Physics)

• The gravitational potential energy is the energy an object has when lifted off the ground given by the familiar equation:

$E_p = mg∆h$

• The gravitational potential energy 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 of a Meteor

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

• The gravitational potential energy (G.P.E) at point in a gravitational field is defined as:

 The work done in bringing a mass from infinity to that point

• The equation for G.P.E of two point masses m and M at a distance r is:

• The change in G.P.E is given by:

ΔG.P.E = mgΔh

• Where:
  • m = mass of the object (kg)
  • ɸ = gravitational potential at that point (J kg^-1)
  • Δh = change in height (m)

• Recall that at infinity, ɸ = 0 and therefore G.P.E = 0
• It is more useful to find the change in G.P.E e.g. a satellite lifted into space from the Earth’s surface
• The change in G.P.E from for an object of mass m at a distance r₁ from the centre of mass M, to a distance of r₂ further away is:

• The change in potential Δɸ is the same, without the mass of the object m:

Change in Gravitational Potential Energy of a Satellite

Gravitational potential energy increases as a satellite leaves the surface of the Moon

Maths tip

• Multiplying two negative numbers equals a positive number, for example: