Gravitational Field Strength | CIE A Level Physics Revision Notes 2025

Deriving Gravitational Field Strength (g)

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

$F_{G} = \frac{G M m}{r^{2}}$

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

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

Rearranging to make for F the subject:

$F = m g$

Equating the two forces:

$F = F_{G}$

$m g = \frac{G M m}{r^{2}}$

Cancelling out the mass, m, on each side:

$m g = \frac{G M m}{r^{2}}$

The equation becomes:

$g = \frac{G M}{r^{2}}$

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 creating the gravitational field
- This is the same direction as the gravitational field lines
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 Gravitational Field Strength at Distance from Earth's Centre

g v R graph on Earth (1), downloadable AS & A Level Physics revision notes

The value of g increases in direct proportion to distance from the centre of the Earth until the Earth's surface, then it decreases with distance at an increasing rate

Distances from Centre of Earth in Terms of R

R increases with distance from the centre of the Earth

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

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 r_M and the Earth’s radius r_E.

Answer:

Step 1: Write down the known quantities

$ρ_{M} = \frac{3}{5} ρ_{E}$

$g_{M} = \frac{1}{6} g_{E}$

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

$V = \frac{4}{3} π r^{3}$

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

$ρ = \frac{M}{V}$

$M = ρ V$

Step 4: Write the gravitational field strength equation

$g = \frac{G M}{r^{2}}$

Step 5: Substitute M in terms of ρ and V

$g = \frac{G ρ V}{r^{2}}$

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

$g = \frac{G ρ 4 π r^{3}}{3 r^{2}} = \frac{G ρ 4 π r}{3}$

Step 7: Find the ratio of the gravitational field strengths

$\frac{g_{M}}{g_{E}} = \frac{G ρ_{M} 4 π r_{M}}{3} \div \frac{G ρ_{E} 4 π r_{E}}{3} = \frac{ρ_{M} r_{M}}{ρ_{E} r_{E}}$

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

_{$\frac{r_{M}}{r_{E}} = \frac{ρ_{E} g_{M}}{ρ_{M} g_{E}} = \frac{ρ_{E} (\frac{1}{6} g_{E})}{(\frac{3}{9} ρ_{E}) g_{E}}$}

_{$\frac{r_{M}}{r_{E}} = \frac{5}{3} \times \frac{1}{6} = \frac{5}{18} = 0.28 (2 s . f .)$}

Gravitational Field Strength (CIE A Level Physics)

Revision Note

Deriving Gravitational Field Strength (g)

Calculating g

Graph Showing Gravitational Field Strength at Distance from Earth's Centre

Distances from Centre of Earth in Terms of R

Worked example

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Author: Leander