Newton's inverse-square law is the only new “tool” in this topic; every other result (\\(g\\), \\(\varphi\\), \\(UG\\), escape speed, orbit radius) is crafted by algebra and energy bookkeeping that you already know from mechanics.
Key points
Nail those derivations once, and the \_WA1-to-A-Level questions collapse into four recurring templates.
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1 Newton's law of gravitation
2 Gravitational field strength \(g\)
Q: What does A-Level Physics: 8) Gravitational Fields Guide cover? A: From Newton's law to geostationary satellites, this post unpacks Section I Topic 8 of the 2026 H2 Physics syllabus for IP students and parents.
TL;DR Newton's inverse-square law is the only new “tool” in this topic; every other result (g, φ, \(UG\), 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.
If the question asks for force between two masses, use Newton's law of gravitation. If it asks for field strength at a point, divide force by test mass and use g=GM/r2. That distinction stops force and field answers from blurring.
Step through the rest of the mechanics/electromagnetism refresh with our free H2 Physics notes so this gravitation set feeds smoothly into circular motion, SHM, and field comparisons.
1 Newton's law of gravitation
Isaac Newton modelled gravity as a mutual, attractive, central force:
F=Gr2m1m2.
Proportional to mass product: doubling either mass doubles the force.
Inverse-square with separation: the force drops by a factor of 4 when distance doubles - the geometry of spreading field lines over a sphere.
Universal constant G:6.67×10−11N⋅m2⋅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=r2GM.(1)
Near Earth's surface, r≈ Earth's radius, so g≈9.81m⋅s−2 and is directionally “down”.
Parent insight: “Why is g 'constant'?” - because r changes by under one-tenth of a percent across school-lab altitudes, so the drift in g stays below three-tenths of a percent.
3 Gravitational potential φ
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:
φ=−rGM.(2)
The negative sign encodes that gravity is attractive; zero potential is set at infinity.
Think of UG as the shared energy store of the pair; separating them to infinity requires positive work equal to ∣UG∣.
5 Escape velocity
Set “initial kinetic energy + potential energy = 0 at infinity”:
21mve2−rGMm=0⇒ve=r2GM.(4)
At Earth's surface ve≈11.2km⋅s−1 - roughly 450 times the expressway speed limit.
6 Circular orbits & centripetal acceleration
Equate gravitational force to the required centripetal force mv2/r:
r2GMm=rmv2⇒v=rGM.(5)
Key takeaway: orbital speed halves when radius quadruples - a fast elimination step in MCQs.
7 Geostationary satellites
A geostationary satellite has
Orbital period = 24 h (synchronised with Earth's spin),
Zero inclination & eccentricity (lies above the equator),
Altitude ≈ 35 786 km.
Applications span weather monitoring, TV broadcast and VSAT internet.
7.1 Deriving the GEO radius
Set centripetal period T=v2πr and combine with Eq. (5):
r=34π2GMT2≈4.22×104km(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 φ and differentiate - one line, fewer sign errors.
9 WA timing hacks (tested on IP papers)
Derivations first: write Eqs. (4)-(6) from memory, circle any required answer - anchors marks early.
Unit tagging: copy SI units alongside numbers before punching the calculator.
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!)
A probe is already in a 400 km-high circular orbit. What minimum additional speed Δv (applied in the direction of motion) is needed for it to escape Earth? Hint: at the same radius r, orbital speed is v=rGM and escape speed is \( ve = \sqrt{\dfrac{2GM}{r}} \), so Δv=ve−v. _Answer (Earth):Δv≈3.2km⋅s−1 using r≈(6.37×106+4.00×105)m
Need structured practice on Gravitational Fields? Our H2 Physics tuition Singapore programme covers this topic with weekly problem sets and Paper 4 practical drills.
Comprehensive revision pack
9478 Section II, Topic 8 Syllabus outcomes
Candidates should be able to:
(a) recall and use Newton's law of gravitation in the form F=Gr2m1m2.
(b) derive, from Newton's law of gravitation and the definition of gravitational field strength, the field strength due to a point mass, g=Gr2M.
(c) recall and use g=Gr2M for the gravitational field strength due to a point mass to solve problems.
(d) show an understanding that near the surface of the Earth, gravitational field strength is approximately constant and equal to the acceleration of free fall.
(e) define gravitational potential at a point as the work done per unit mass by an external force in bringing a small test mass from infinity to that point.
(f) solve problems using the equation ϕ=−GrM for the gravitational potential in the field due to a point mass.
(g) show an understanding that the gravitational potential energy of a system of two point masses is UG=−GrMm.
(h) recall that gravitational field strength at a point is equal to the negative potential gradient at that point and use this to solve problems.
(i) analyse problems related to escape velocity by considering energy stores and transfers.
(j) analyse circular orbits in inverse square law fields by relating the gravitational force to the centripetal acceleration it causes.
(k) show an understanding of satellites in geostationary orbit and their applications.
Concept map (in words)
Start from the inverse-square law. Differentiate once to obtain field strength, integrate to recover potential energy. Combine with conservation of energy to handle escape and transfer problems. Link to circular motion by equating gravitational and centripetal forces. Graphs of g(r) and φ(r) reveal behaviour between masses, so practise reading them.
Key definitions & formulae
Quantity
Expression / idea
Gravitational field strength
g=r2GM (towards the mass)
Gravitational potential
φ=−rGM
Gravitational potential energy
UG=−rGMm
Escape velocity
ve=r2GM
Orbital speed (circular)
v=rGM
Orbital period
T=2πGMr3
Field from multiple masses
gtot=∑igi
Potential from multiple masses
φtot=∑iφi
Term guide
Symbol / term
Meaning
G
Universal gravitational constant
M
Mass of the primary body (e.g. planet)
m
Mass of the test object
r
Distance from the centre of mass M
g
Gravitational field strength
φ
Gravitational potential
UG
Gravitational potential energy
ve
Escape velocity at radius r
v
Orbital speed for circular motion
T
Orbital period
gtotal
Vector sum of gravitational fields from multiple masses
φtotal
Scalar sum of potentials from multiple masses
Derivations & reasoning to master
Kepler's third law: combine centripetal requirement with Newton's law to show T2∝r3.
Escape velocity: equate 21mv2 with ∣UG∣ at launch radius, emphasising independence from m.
Energy change for orbital transfers: compute ΔU and ΔEk when moving between radii (e.g., GEO to LEO).
A weather satellite moves from an orbit of radius 7.0×106m to the geostationary radius 4.2×107m. Calculate the work required per unit mass and the change in kinetic energy per unit mass.
Method: compute φ(r) at both radii to find Δφ. Use v=rGM to evaluate the kinetic energy change. Comment on the net energy input required.
Taking Earth GM≈3.99×1014m3⋅s−2,
Δφ=GM(r11−r21)≈4.75×107J⋅kg−1.
Δ(mEk)=21(r2GM−r1GM)≈−2.37×107J⋅kg−1.
So the satellite has higher (less negative) potential but lower kinetic energy in the higher orbit; net energy input per unit mass is about 2.37×107J⋅kg−1.
Worked example 2 - zero net force point
Two planets of masses 5.0×1023kg and 8.0×1023kg are separated by 3.2×108m. Find the point along the line where a small probe experiences zero net gravitational force and determine the potential there.
Strategy: set x2GM1=(d−x)2GM2 to solve for x, then sum the potentials to obtain φ.
x=1+M2/M1d=1+8/53.2×108≈1.41×108m.
So the point is ≈1.41×108m from M1 (and ≈1.79×108m from M2). The potential there is
φ=−xGM1−d−xGM2≈−5.35×105J⋅kg−1.
Practical & data tasks
Plot g vs r using spreadsheet for Earth data; compare near-surface approximation with full expression.
Use NASA orbital databases to compute periods and check T2/r3 consistency.
Model potential wells with elastic sheets or digital simulations; observe how objects move in curved spacetime analogies.
Common misconceptions & exam traps
Thinking gravitational potential is positive; remember zero at infinity, negative elsewhere.
Forgetting that potential is scalar, so contributions add algebraically even when fields oppose.
Confusing escape velocity with orbital velocity (missing 2 factor).
Neglecting the mass of the orbiting body in energy equations - mass cancels only because m appears in every term.
Quick self-check quiz
If Earth's mass doubled but radius stayed the same, how would g at the surface change? - It would double.
Where is gravitational potential zero? - At infinity (reference level by definition).
What provides the centripetal force for the Moon's orbit? - Earth's gravitational pull.
Does a satellite in higher orbit have more or less kinetic energy than one in low orbit? - Less kinetic energy but higher (less negative) total energy.
Why is gravitational potential negative? - Work must be done to separate masses to infinity (attractive interaction).
Revision workflow
Re-derive escape velocity and Kepler's law until you can do both within five minutes.
Complete two past-paper questions on satellite motion and one on zero-field points.
Sketch g(r) and φ(r) for Earth plus Moon; label key radii and explain features.
Create a mind map linking gravitational results to electrostatic analogues (Topic 14 preview).
