A LevelPhysicsCambridge (CIE)Revision Notes13. Gravitational FieldsGravitational Field of a Point MassGravitational Field Strength

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

Exam code: 9702

Leander Oates

Last updated

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 space equals space fraction numerator space F over denominator m end fraction

  • 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

F subscript G space equals space fraction numerator G M m over denominator r squared end fraction

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

F space equals space m g

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

F space equals space F subscript G

m g space equals space fraction numerator G M m over denominator r squared end fraction

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

up diagonal strike m g space equals space fraction numerator G M up diagonal strike m over denominator r squared end fraction

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

g space equals space fraction numerator G M over denominator r squared end fraction

  • 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)

Examiner Tips and Tricks

It is important to recognise the difference between the two gravitational field strength situations:

  • gravitational field strength at a point due to the object creating the gravitational field g space equals fraction numerator space F over denominator m end fraction

  • gravitational field strength due to a point mass placed in a the gravitational field of a bigger object is g space equals fraction numerator space G M over denominator r squared end fraction

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 space proportional to space 1 over r squared

  • Where: 

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

Worked Example

The mean density of the Moon is 3 over 5 times the mean density of the Earth. The gravitational field strength at the surface of the Moon is 1 over 6 the gravitational field strength at the surface of the Earth.

Determine the ratio of the Moon’s radius r subscript M and the Earth’s radius r subscript E.

Answer: 

Step 1: Write down the known quantities

rho subscript M space equals space 3 over 5 rho subscript E

g subscript M space equals space 1 over 6 g subscript E

  • g subscript M = gravitational field strength on the Moon, rho subscript M = mean density of the Moon

  • g subscript E = gravitational field strength on the Earth, rho subscript E = mean density of the Earth

Step 2: Write down the equations for the gravitational field strength, volume and density

  • Gravitational field strength is given by:

g space equals fraction numerator space G M over denominator r squared end fraction

  • The mass of each body can be described by the density equation:

rho space equals fraction numerator space M over denominator V end fraction

M space equals space rho V

  • The volume of each body can be approximated as a sphere

V space equals space 4 over 3 straight pi r cubed

Step 3:  Substitute M in terms of ρ and V

g space equals space fraction numerator G rho V over denominator r squared end fraction

Step 4: Substitute the volume of a sphere equation and simplify

g space equals space fraction numerator G rho open parentheses 4 straight pi r cubed close parentheses over denominator 3 r squared end fraction space equals space fraction numerator 4 straight pi G rho r over denominator 3 end fraction

Step 5: Find the ratio of the gravitational field strengths

g subscript M over g subscript E space equals space fraction numerator 4 straight pi G rho subscript M r subscript M over denominator 3 end fraction space divided by space fraction numerator 4 straight pi G rho subscript E r subscript E over denominator 3 end fraction space equals space fraction numerator rho subscript M r subscript M over denominator rho subscript E r subscript E end fraction

Step 6: Rearrange and calculate the ratio of the Moon’s radius rM and the Earth’s radius rE

r subscript M over r subscript E space equals space fraction numerator rho subscript E g subscript M over denominator rho subscript M g subscript E end fraction space equals space fraction numerator rho subscript E open parentheses 1 over 6 g subscript E close parentheses over denominator open parentheses 3 over 5 rho subscript E close parentheses g subscript E end fraction

r subscript M over r subscript E space equals space 1 over 6 space divided by space 3 over 5 space equals space 5 over 18 space equals space 0.28

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    Leander Oates

    Author: Leander Oates

    Expertise: Physics Content Creator

    Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.