A-Level Physics — 8) Gravitational Fields (IP-Friendly Guide)

TL;DR
Newton's inverse-square law is the only new “tool” in this topic; every other result (\(g\), \(\varphi\), \(U_G\), escape speed, orbit radius) is crafted by algebra and energy bookkeeping that you already know from mechanics. Nail those derivations once, and the WA1-to-A-Level questions collapse into four recurring templates.

1 Newton's law of gravitation

Isaac Newton modelled gravity as a mutual, attractive, central force:

\[ F = G \frac{m_1 m_2}{r^2}. \]

• Proportional to mass product: doubling either mass doubles the force.
• Inverse-square with separation: the force drops by \(4\) x when distance doubles — the geometry of spreading field lines over a sphere.
• Universal constant \(G\): \(6.67 \times 10^{-11} \space \pu{N.m^2.kg^{-2}}\), first measured by Cavendish in 1798.

1.1 IP exam cue

List all three features (“attractive”, “inverse-square”, “proportional to masses”) for a 2-mark definition: one IP tuition classic.

2 Gravitational field strength \(g\)

Field strength is force per unit mass: \(g = F/m\). Combining with Newton's law gives

\[ g = \frac{GM}{r^2}. \tag{1} \]

Near Earth's surface, \(r\) ≈ Earth's radius, so \(g ≈ 9.81 \space \pu{m.s-2}\) and is directionally “down”.

Parent insight: “Why is \(g\) 'constant'?” — because \(r\) changes by < 0.1 % across school-lab altitudes, giving < 0.3 % drift in \(g\).

3 Gravitational potential \(\varphi\)

Definition: work done per unit mass by an external agent in bringing a small test mass from infinity to the point.

For a point mass \(M\):

\[ \varphi = -\frac{GM}{r}. \tag{2} \]

The negative sign encodes that gravity is attractive; zero potential is set at infinity.

Gradient link: \(g = -\dfrac{\mathrm{d}\varphi}{\mathrm{d}r}\) — differentiating Eq. (2) regenerates Eq. (1).

4 Gravitational potential energy \(U_G\)

For two point masses \(M\) and \(m\):

\[ U_G = -\frac{GMm}{r}. \tag{3} \]

Think of \(U_G\) as the shared energy store of the pair; separating them to infinity requires positive work equal to \(|U_G|\).

5 Escape velocity

Set “initial kinetic energy + potential energy = 0 at infinity”:

\[ \frac12 m v_e^2 - \frac{GMm}{r} = 0 \quad \Rightarrow \quad v_e = \sqrt{\frac{2GM}{r}}. \tag{4} \]

At Earth's surface \(v_e ≈ 11.2 \space \pu{km.s-1}\) — 34 x the expressway speed limit!

6 Circular orbits & centripetal acceleration

Equate gravitational force to the required centripetal force \(m v^2 / r\):

\[ \frac{GMm}{r^2} = \frac{m v^2}{r} \quad \Rightarrow \quad v = \sqrt{\frac{GM}{r}}. \tag{5} \]

Key takeaway: orbital speed halves when radius quadruples — a fast elimination step in MCQs.

7 Geostationary satellites

A geostationary satellite has

1. Orbital period = 24 h (synchronised with Earth's spin),
2. Zero inclination & eccentricity (lies above the equator),
3. Altitude ≈ 35 786 km.

Applications span weather monitoring, TV broadcast and VSAT internet.

7.1 Deriving the GEO radius

Set centripetal period \(T = 2 \pi r / v\) and combine with Eq. (5):

\[ r = \sqrt[3]{\frac{GMT^2}{4\pi^2}} ≈ 4.22 \times 10^4 \space \pu{km}. \tag{6} \]

8 Negative potential gradient shortcut

Many WA problems ask for \(g\) at a point off-axis or between bodies. Instead of re-drawing vectors, evaluate \(\varphi\) and differentiate — one line, fewer sign errors.

9 WA timing hacks (tested on IP papers)

1. Derivations first: write Eqs. (4)-(6) from memory, circle any required answer — anchors marks early.
2. Unit tagging: copy SI units alongside numbers before punching the calculator.
3. Escape-vs-orbit confusion check: escape needs \(\sqrt{2}\) factor; circular orbit does not.

10 Why parents should care

An early mastery of gravitational fields lets students pre-learn circular motion and satellite communication, compounding advantage in Term 3. We schedule a 60-min clinic right after the Topic 8 lecture — seats fill fast each semester.

11 Mini-drill (do now!)

Find the minimum launch speed for a probe released from a 400 km-high ISS-style orbit so that it never falls back to Earth.
Hint: subtract the orbital kinetic energy at 400 km from the escape kinetic energy at the same altitude.

12 Further reading

• SEAB H2 Physics 9478 syllabus (2026)
• Newton's law of universal gravitation — Wikipedia
• Save My Exams — Gravitational field of a point mass
• BYJU'S — Gravitational potential energy derivation
• NASA Space-Math — Gravity & Escape Speed
• HyperPhysics — Escape velocity
• HyperPhysics — Gravitational potential energy
• Earth Fact Sheet — NASA NSSDCA
• Basics of Space Flight — NASA Science
• CelesTrak — Geostationary orbit basics
• Natural Resources Canada — Geostationary weather satellites

Last updated 14 Jul 2025. We will refresh the guide once the 2027 draft syllabus lands.