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Gravitational Field Strength (CIE A Level Physics)

Revision Note

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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 due to 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 of a gravitational force on an object

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

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

  • Where:
    • = gravitational field strength measured in newtons per kilogram (N/kg)
    • = gravitational force measured in newtons (N)
    • = 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
    • combining the equations for Newton’s law of gravitation
    • the definition of a gravitational field 

F subscript G space equals fraction numerator space 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 gravitational field strength:

F space equals space F subscript G

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

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

up diagonal strike m g space equals fraction numerator space 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 fraction numerator space 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 Tip

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 creating the gravitational 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 fraction numerator space 1 over denominator r squared end fraction

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

Worked example

The mean density of the moon is ⅗ times the mean density of the Earth. The gravitational field strength is ⅙ on the Moon than that on Earth.

Determine the ratio of the Moon’s radius rM and the Earth’s radius rE.

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

  • gM = gravitational field strength on the Moon, ρM = mean density of the Moon
  • gE = gravitational field strength on the Earth, ρE = mean density of the Earth

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

V space equals space 4 over 3 straight pi r cubed

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

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

M space equals space rho V

Step 4: Write the gravitational field strength equation

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

Step 5:  Substitute M in terms of ρ and V

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

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

g space equals fraction numerator italic space G rho italic 4 pi r to the power of italic 3 over denominator italic 3 r to the power of italic 2 end fraction space equals fraction numerator italic space G rho italic 4 pi r over denominator italic 3 end fraction

Step 7: Find the ratio of the gravitational field strengths

g subscript M over g subscript E space equals fraction numerator space G rho subscript M 4 straight pi r subscript M over denominator 3 end fraction space divided by space fraction numerator G rho subscript E 4 straight pi 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 8: 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 9 rho subscript E close parentheses g subscript E end fraction

r subscript M over r subscript E space equals space 5 over 3 space cross times space 1 over 6 space equals space 5 over 18 space equals space 0.28 space open parentheses 2 space straight s. straight f. close parentheses

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Leander

Author: Leander

Expertise: Physics

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.