15.2 Gravitational Field Strength

A gravitational field exists in the region of space surrounding any mass. Any other mass placed in this region experiences a gravitational force.

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Definition of Gravitational Field Strength

The gravitational field strength $g$ at a point is defined as the gravitational force per unit mass acting on a small test mass placed at that point:

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

where $F_g$ is the gravitational force (in N) and $m$ is the test mass (in kg).

• $g$ is a vector quantity — it points radially inward toward the centre of the source mass.
• SI units: $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$ or equivalently $\mathrm{m}{\mathrm{s}}^{-2}$.

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Gravitational Field Lines

A gravitational field can be represented by field lines (also called lines of force):

• Lines point radially inward toward the centre of the mass (direction of force on a positive test mass).
• The closer together the lines, the stronger the field.
• Near the surface of a planet, lines are dense; they spread out with increasing distance.
• For a uniform spherical mass, the pattern is perfectly radially symmetric.

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Derivation of Gravitational Field Strength at Distance $r$

Using Newton's Law of Universal Gravitation, the force on a test mass $m$ at distance $r$ from a source mass $M$ is:

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

Substituting into the definition $g=F_g/m$:

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

where:

• $G=6.67\times 10^{-11}\mathrm{N}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-2}$ is the universal gravitational constant
• $M$ is the mass of the source body
• $r$ is the distance from the centre of the source mass

Key point: $g$ depends only on the source mass $M$ and distance $r$ — it is independent of the test mass $m$.

At Earth's surface ($r=R_E$):

$g_s=\frac{G{M}_E}{R_E^2}\approx 9.8\mathrm{m}{\mathrm{s}}^{-2}$

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Why $g$ is Approximately Constant Near Earth's Surface

At a height $h$ above Earth's surface, the distance from Earth's centre is $r=R_E+h$, so:

$g_h=\frac{G{M}_E}{(R_E+h{)}^2}=\frac{G{M}_E}{R_E^2}\cdot \frac{1}{{\left(1+\frac{h}{R_E}\right)}^2}$

Using the binomial approximation for $h\ll R_E$:

$g_h\approx g_s\left(1-\frac{2h}{R_E}\right)$

The fractional change in $g$ is approximately $\frac{2h}{R_E}$.

Since Earth's radius $R_E\approx 6.4\times 10^6\mathrm{m}$, even at $h=10\mathrm{km}$:

$\frac{2h}{R_E}=\frac{2\times 10^4}{6.4\times 10^6}\approx 0.003=0.3\%$

This is negligibly small, so $g\approx 9.8\mathrm{m}{\mathrm{s}}^{-2}$ is treated as constant for everyday calculations near Earth's surface.

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Summary Table