understand why g is approximately constant for small changes in height near the Earth’s surface

13 Gravitational Field of a Point Mass – Near‑Surface Approximation

1. Learning Objectives (AO1–AO3)

• Define the gravitational field g as the force per unit test mass and describe its radial‑inward direction.
• State Newton’s law of universal gravitation and derive the field strength g = GM/r².
• Introduce gravitational potential ϕ = –GM/r and relate it to gravitational potential energy.
• Use a first‑order Taylor (linear) expansion to justify treating g as constant for small height changes near the Earth’s surface.
• Apply the constant‑g approximation in practical problems (projectile motion, engineering) and recognise its limits.
• Design and evaluate a simple experiment for measuring g (AO3 – experimental skills).

2. Gravitational Field (Syllabus outcome 13.1)

The gravitational field at a point in space is the force experienced by a test particle of mass m divided by that mass:

\[

\mathbf{g} = \frac{\mathbf{F}}{m}\qquad\bigl[\mathbf{g}\bigr]=\mathrm{N\,kg^{-1}}=\mathrm{m\,s^{-2}} .

\]

• It is a vector directed radially inwards towards the source mass.
• For a point mass (or the Earth treated as a point) the magnitude depends only on the distance r from the centre.

Figure 1 shows the field‑line pattern for a point mass.

3. Newton’s Law of Universal Gravitation (13.2)

For two point masses m₁ and m₂ separated by a distance r:

\[

F = G\frac{m{1}m{2}}{r^{2}},\qquad G = 6.674\times10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}} .

\]

Dividing the force by the test mass m₂ gives the gravitational field produced by m₁:

4. Gravitational Field of a Point Mass (13.3)

Let M be the mass of the Earth and m a small test mass. Substituting into Newton’s law and dividing by m yields

\[

\boxed{g(r)=\frac{GM}{r^{2}}}

\]

where r is the distance from the Earth’s centre.

The product GM is the standard gravitational parameter; many exam questions give this value directly (GM = 3.986×10¹⁴ m³ s⁻²).

5. Gravitational Potential (13.4)

• Potential (ϕ) is the work done per unit mass in bringing a test mass from infinity to a point at distance r:

\[

\boxed{\;\phi(r) = -\frac{GM}{r}\;}

\]

• The gravitational potential energy of a mass m at that point is

\[

U = m\phi = -\frac{GMm}{r}.

\]

Example (h = 10 km): with r = R_{\oplus}+10 000 m and GM = 3.986×10¹⁴ m³ s⁻²,

\[

\phi = -\frac{3.986\times10^{14}}{6.371\times10^{6}+10^{4}}

\approx -6.26\times10^{7}\ \mathrm{J\,kg^{-1}} .

\]

6. Approximation Near the Earth’s Surface

For everyday problems the height h above the surface satisfies h \ll R{\oplus} (R{\oplus}=6.371×10⁶ m). Write

\[

r = R_{\oplus}+h,\qquad

g(h)=\frac{GM}{(R_{\oplus}+h)^{2}} .

\]

6.1 First‑Order (Linear) Taylor Expansion

Expanding about h = 0 and retaining only the linear term:

\[

\begin{aligned}

g(h) &= \frac{GM}{R{\oplus}^{2}}\left(1+\frac{h}{R{\oplus}}\right)^{-2} \\

&\approx g{0}\Bigl[1-2\frac{h}{R{\oplus}}\Bigr],

\end{aligned}

\]

where g₀ = GM/R_{\oplus}^{2} \approx 9.80665 m s⁻².

Assumption: \(|h| \ll R{\oplus}\) so that terms of order \((h/R{\oplus})^{2}\) and higher are negligible.

6.2 Validity Range

• Relative error < 0.1 % for h ≤ 3 km.
• Relative error ≈ 1 % for h ≈ 30 km.
• For h ≥ 100 km the linear approximation is no longer reliable; the full expression must be used.

6.3 Numerical Illustration

Values are rounded to five significant figures, matching the convention used in Cambridge examinations.

7. Practical Applications (AO2)

1. Projectile motion – In most AS/A‑Level problems the vertical acceleration can be taken as a constant g = 9.81 m s⁻² because the height travelled is far smaller than R_{\oplus}.
2. Engineering calculations – Design of buildings, bridges, short‑range rockets, and sports equipment assumes a constant g for the same reason.
3. Satellite orbits (link to later A‑Level topic “Motion in a circle”) – Equating centripetal force to gravitational force gives the orbital speed for a circular orbit:

\[

\frac{mv^{2}}{r}= \frac{GMm}{r^{2}}\;\Longrightarrow\;

\boxed{v = \sqrt{\frac{GM}{r}} } .

\]

This result follows directly from the same inverse‑square law used in Sections 3–4.

8. Experimental Determination of g (AO3)

A simple classroom method uses a vertical electro‑optical (photogate) timer:

1. Measure the fall time t for a known distance s (e.g., 0.5 m).
2. Assuming constant acceleration, use \(s = \tfrac{1}{2}gt^{2}\) to calculate

\[

g = \frac{2s}{t^{2}} .

\]

3. Repeat for several heights, plot s versus t², and obtain g from the slope (linear fitting, error analysis).

Typical result: g = 9.78 ± 0.05 m s⁻², demonstrating both the near‑surface constancy of g and experimental uncertainties.

9. Key Constants (exam data sheet)

10. Summary

The gravitational field of a point mass follows the inverse‑square law, \(g = GM/r^{2}\). Near the Earth’s surface the change in distance from the centre is tiny when the height h satisfies h \ll R_{\oplus}\). A first‑order Taylor expansion gives

\[

g(h) \approx g{0}\Bigl(1-2\frac{h}{R{\oplus}}\Bigr).

\]

The correction term is typically less than 0.01 % for heights up to a few kilometres, justifying the use of a constant \(g \approx 9.81\ \mathrm{m\,s^{-2}}\) in most AS/A‑Level calculations. For larger altitudes, satellite work, or high‑precision engineering, the full expression must be retained. The same concepts underpin gravitational potential, orbital motion, and can be investigated experimentally through simple free‑fall measurements.