13.1.5 Gravitational Field Strength

• The gravitational field strength at a point describes how strong or weak a gravitational field is at that point
• The gravitational field strength due to a point mass can be derived from combining the equations for Newton’s law of gravitation and gravitational field strength
  • For calculations involving gravitational forces, a spherical mass can be treated as a point mass at the centre of the sphere
  • The gravitational field strength unit is N kg^-1

Gravitational Field Strength Formula:

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

• The gravitational field strength at a point is defined as the force F per unit mass m

• Substituting the force F with the gravitational force F_G leads to:

• Cancelling mass m, the equation becomes:

• 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 from the mass where you are calculating the field strength (m)

• Gravitational field strength, g, is a vector quantity
• The direction of g is always towards the centre of the body creating the gravitational field
  • This is the same direction as the gravitational field lines

What is the Gravitational Field Strength on Earth?

• On the Earth’s surface, g has a constant value of 9.81 N kg^-1
• However outside the Earth’s surface, g is not constant
  • g decreases as r increases by a factor of 1/r²
  • This is an inverse square law relationship with distance

• When g is plotted against the distance from the centre of a planet, r has two parts:
  • When r < R, the radius of the planet, g is directly proportional to r
  • When r > R, g is inversely proportional to r² (this is an ‘L’ shaped curve and shows that g decreases rapidly with increasing distance r)

Graph showing how gravitational field strength varies at greater distance from the Earth’s surface

• Sometimes, g is referred to as the ‘acceleration due to gravity’ with units of m s^-2
• Any object that falls freely in a uniform gravitational field on Earth has an acceleration of 9.81 m s^-2

Step 1:            Write down the known quantities

g_M = gravitational field strength on the Moon, ρ_M = mean density of the Moon

g_E = gravitational field strength on the Earth, ρ_E = mean density of the Earth

Step 2:            The volumes of the Earth and Moon are equal to the volume of a sphere

Step 3:            Write the density equation and rearrange for mass M

M = ρV

Step 4:            Write the gravitational field strength equation

Step 5:            Substitute M in terms of ρ and V

Step 6:            Substitute the volume of a sphere equation for V, and simplify

Step 7:            Find the ratio of the gravitational field strengths

Step 8:            Rearrange and calculate the ratio of the Moon’s radius r_M and the Earth’s radius r_E