Gravitational Potential Gradient (DP IB Physics: HL): Revision Note

Gravitational Potential Gradient

• 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 

$g=-\frac{∆V_g}{∆r}$

• 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 gravitational potential and distance graphs follow a -1/r relation

• 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 $-\frac{1}{r}$ relation

  • The gradient of the graph at any particular point is the value of g at that point, $g=-V_g\times -\frac{1}{r}=\frac{V_g}{r}$

  • 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:

  $V_g=-\frac{GM}{r}$

  where G and M are constant