Gravitational Potential Gradient (HL) (HL IB Physics)

• A gravitational field can be defined in terms of the variation of gravitational potential at different points in the field:

The gravitational field at a particular point is equal to the negative gradient of a potential-distance graph at that point

• The potential gradient is defined by the equipotential lines
  • These demonstrate the gravitational potential in a gravitational field and are always drawn perpendicular to the field lines

• The potential gradient in a gravitational field is defined as:

The rate of change of gravitational potential with respect to displacement in the direction of the field

• Gravitational field strength, g and the gravitational potential, V can be graphically represented against the distance from the centre of a planet, r 

• Where:
• g = gravitational field strength (N kg^-1)
• ΔV_g = change in gravitational potential (J kg^-1)
• Δr = distance from the centre of a point mass (m)

• The graph of V_g against r for a planet is:

• The key features of this graph are:
  • The values for V_g are all negative (because the graph is drawn below the horizontal r axis)
  • As r increases, V_g against r follows a relation
  • The gradient of the graph at any particular point is the value of g at that point, 
  • The graph has a shallow increase as r increases

• To calculate g, draw a tangent to the graph at that point and calculate the gradient of the tangent
• This is a graphical representation of the gravitational potential equation:

where G and M are constant