Gravitational Field Strength (Cambridge (CIE) A Level Physics): Revision Note

Deriving gravitational field strength (g)

• There are two situations where gravitational field strength is considered:

  • at a point

  • due to a mass

Gravitational field strength at a point

• The gravitational field strength at a point describes how strong or weak a gravitational field is at that point

• The gravitational field strength at a point is defined as

The force per unit mass acting on a small mass at that point

• Gravitational field strength at a point is given by the equation:

$g=\frac{F}{m}$

• Where:

  • g = gravitational field strength measured in newtons per kilogram (N/kg)

  • F = gravitational force measured in newtons (N)

  • m = mass of object in gravitational field measured in kilograms (kg)

Gravitational field strength due to a point mass

• The gravitational field strength due to a point mass within a gravitational field can be derived from the equations for

  • Newton’s law of gravitation

  • gravitational field strength at a point

• Newton’s law of gravitation states that the attractive force F_G between two masses M and m with separation r is equal to:

$F_G=\frac{GMm}{r^2}$

• Rearrange the definition of gravitational field strength at a point to make force F the subject:

$F=mg$

• Equate the gravitational force and the force due to the gravitational field strength:

$F=F_G$

$mg=\frac{GMm}{r^2}$

• Cancel out the mass, m, on each side:

$\cancel{m}g=\frac{GM\cancel{m}}{r^2}$

• The equation for gravitational field strength due to a point mass is:

$g=\frac{GM}{r^2}$

• Where:

  • g = gravitational field strength (N kg^-1)

  • G = Newton’s Gravitational Constant

  • M = mass of the body producing the gravitational field (kg)

  • r = distance between point source (mass, m) and position in field (m)

Calculating g

• Gravitational field strength g is a vector quantity

• The direction of g is always towards the centre of the body producing the field

  • This is the same direction as the gravitational field lines

• Gravitational field strength g and orbital radius r have an inverse square law relationship:

$g\propto \frac{1}{r^2}$

• Where: 

  • g decreases as r increases by a factor of 1/r²