7.2.2 Calculating Gravitational Potential

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

• Where:
  • V = 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 always is negative near an isolated mass, such as a planet, because:
  • The potential when r is at infinity (∞) is defined as 0
  • Work must be done to move a mass away from a planet (V becomes less negative)

• It is also a scalar quantity, unlike the gravitational field strength which is a vector quantity

• Gravitational forces are always attractive, this means 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