(13) Gravitational Fields
Definition:
A gravitational field is a region of space in which a mass experiences a force.
Simpler version:
Any object with mass creates a gravitational field around it. Any other mass placed in that field will experience a gravitational force of attraction.
Key point: The field is always attractive — it pulls masses towards each other. There is no repulsive gravitational force.
Two Types of Gravitational Fields (Syllabus Expectation)
Cambridge expects you to distinguish between:
Type | Definition | Example |
|---|---|---|
Uniform field | Gravitational field strength gg is constant in both magnitude and direction | Near the Earth's surface (field lines are parallel and equally spaced) |
Radial (non-uniform) field | Gravitational field strength gg varies with distance from the centre; it follows an inverse-square law | Field around a point mass, a planet, or a star |
1. Uniform Gravitational Field (Near Earth's Surface)
Characteristics:
Field lines are parallel and equally spaced.
gg is constant (approximately 9.81 N kg−19.81N kg−1 or m s−2m s−2).
Force on a mass: F=mgF=mg (constant, regardless of position).
Example: A small region near the Earth's surface (e.g., a laboratory). The field lines point vertically downwards towards the Earth's centre.
2. Radial (Non-Uniform) Gravitational Field (Around a Point Mass or Planet)
Characteristics:
Field lines are radial — they point towards the centre of the mass.
The spacing between field lines increases as you move further away — indicating the field gets weaker.
gg follows an inverse-square law:
g=GMr2g=r2GM
Force on a mass: F=mg=GMmr2F=mg=r2GMm (Newton's law of gravitation).
Graphical representation: Field lines radiate inwards like the spokes of a wheel towards the centre of the planet/star.
Gravitational Field Strength (gg) — The Key Definition
This is the single most important definition in this topic. Cambridge expects you to know this word-for-word.
Gravitational field strength at a point is the force per unit mass acting on a small mass placed at that point.
Equation:
g=Fmg=mF
Units: N kg−1N kg−1 (equivalent to m s−2m s−2).
Two Expressions for gg (Memorise Both!)
Context | Equation | When to Use |
|---|---|---|
General definition | g=Fmg=mF | Definition of field strength (always true) |
Radial field (point mass/planet) | g=GMr2g=r2GM | For a planet, star, or any spherical mass — distance rr measured from centre |
Near Earth's surface | g≈9.81 N kg−1g≈9.81N kg−1 | For small heights above the surface (uniform field approximation) |
Derivation of g=GM/r2g=GM/r2 (Exam Favourite)
Step 1 – Newton's law of gravitation (force between two masses):
F=GMmr2F=r2GMm
Step 2 – Definition of gravitational field strength:
g=Fmg=mF
Step 3 – Substitute:
g=1m×GMmr2g=m1×r2GMmg=GMr2g=r2GM
Key observation: gg depends only on:
The mass MM of the object creating the field
The distance rr from its centre
It does not depend on the mass mm of the test object placed in the field.
Field Lines (Syllabus Expectation)
Cambridge expects you to be able to draw and interpret gravitational field lines.
Feature | Meaning |
|---|---|
Direction of arrow | Direction of force on a small test mass (always towards the centre of the field) |
Spacing of lines | Close lines = strong field; widely spaced = weak field |
Radial pattern | Field around a point mass (weaker further away) |
Parallel pattern | Uniform field (constant gg) |
Gravitational Field vs Electric Field (Comparison — Often Tested)
Cambridge sometimes asks you to compare gravitational and electric fields.
Feature | Gravitational Field | Electric Field |
|---|---|---|
Source | Mass | Charge |
Force | Always attractive | Attractive or repulsive |
Field strength | g=F/mg=F/m (N kg⁻¹) | E=F/QE=F/Q (N C⁻¹) |
Inverse-square law | g=GM/r2g=GM/r2 | E=kQ/r2E=kQ/r2 |
Test object | Small mass mm | Small positive charge QQ |
Can it be shielded? | ❌ No | ✅ Yes (by conductors) |
The "Small Test Mass" Keyword — Exam Trick!
Cambridge often uses the phrase "small test mass" in definitions. Why?
A small mass ensures it does not disturb the field it is measuring.
If you placed a large mass (like a second planet), it would alter the field — so the definition requires a negligible mass.
Definition for full marks:
"Gravitational field strength at a point is the force per unit mass acting on a small mass placed at that point."
Worked Example (Typical Paper 1/2)
Q: The Earth has mass 6.0×1024 kg6.0×1024kg and radius 6.4×106 m6.4×106m.
(a) Calculate gg at the Earth's surface.
(b) Calculate gg at a height of 3.6×106 m3.6×106m above the surface.
G=6.67×10−11 N m2kg−2G=6.67×10−11N m2kg−2.
Solution (a) — at surface:
r=6.4×106 mr=6.4×106mg=GMr2=6.67×10−11×6.0×1024(6.4×106)2g=r2GM=(6.4×106)26.67×10−11×6.0×1024=4.002×10144.096×1013=9.77 N kg−1=4.096×10134.002×1014=9.77N kg−1
Solution (b) — at height hh:
r=R+h=6.4×106+3.6×106=1.0×107 mr=R+h=6.4×106+3.6×106=1.0×107mg=GMr2=6.67×10−11×6.0×1024(1.0×107)2g=r2GM=(1.0×107)26.67×10−11×6.0×1024=4.002×10141.0×1014=4.00 N kg−1=1.0×10144.002×1014=4.00N kg−1
The Link Between gg and rr (Inverse-Square Relationship)
Cambridge expects you to understand the relationship between gg and distance rr:
At the surface: g=GM/R2g=GM/R2
Above the surface: g∝1/r2g∝1/r2 (decreases with distance squared)
Below the surface (inside Earth): g∝rg∝r (assuming uniform density) — but this is not in the 9702 syllabus, so don't worry about it!
Graph shape: gg against rr is a curved decreasing graph (like 1/r21/r2).
Summary Table (Gravitational Field)
Concept | Key Points |
|---|---|
Definition | Region where a mass experiences a force |
Field strength gg | Force per unit mass: g=F/mg=F/m |
Uniform field | gg constant; field lines parallel (near Earth's surface) |
Radial field | g∝1/r2g∝1/r2; field lines point towards centre (planet/star) |
Equation | g=GM/r2g=GM/r2 |
Direction | Always towards the centre of the mass creating the field |
Test mass | Must be small to avoid disturbing the field |
Units | N kg⁻¹ (or m s⁻²) |
1. Gravitational Potential Energy (EpEp or UU)
Before we define potential, let’s define the energy associated with it.
Definition (Near Earth's Surface - Uniform Field)
Gravitational potential energy is the energy stored in an object due to its position in a gravitational field.
Equation (Uniform Field):
ΔEp=mgΔhΔEp=mgΔh
Where:
ΔEpΔEp = change in gravitational potential energy (J)
mm = mass (kg)
gg = gravitational field strength (N kg⁻¹)
ΔhΔh = change in vertical height (m)
Definition (Radial Field - General Definition)
For radial fields (planets, stars), the equation mghmgh is not valid because gg changes with distance. Instead, we use the universal definition:
Ep=−GMmrEp=−rGMm
Where:
EpEp = gravitational potential energy (J)
GG = universal gravitational constant
MM = mass creating the field (e.g., Earth)
mm = mass of the object in the field
rr = distance from the centre of MM
The Crucial Point: Why is it Negative?
This is the #1 thing Cambridge tests.
Gravitational potential energy is negative because zero potential energy is defined at infinity (r=∞r=∞).
At infinity, the gravitational force is zero, so Ep=0Ep=0.
As an object moves closer to the planet (from infinity), the gravitational field does work on the object (it pulls it in).
The object loses potential energy, so EpEp becomes negative.
The more negative the value, the more tightly bound the object is to the planet.
Analogy: Think of a hole in the ground. The bottom of the hole has negative "height" relative to the ground. You have to do work to lift something out of the hole (to get it to zero).
2. Gravitational Potential (ϕϕ — Greek letter "phi")
Definition
Gravitational potential at a point is the gravitational potential energy per unit mass of a small mass placed at that point.
Equation:
ϕ=Epmϕ=mEp
Units: J kg−1J kg−1
The Two Equations for ϕϕ (Memorise Both!)
Context | Equation | When to Use |
|---|---|---|
Definition (always true) | ϕ=Ep/mϕ=Ep/m | General definition — used in questions defining the term |
Radial field (point mass/planet) | ϕ=−GMrϕ=−rGM | For a planet, star, or spherical mass — distance from centre |
Key Characteristics of Gravitational Potential
Concept | Detail |
|---|---|
Scalar | ϕϕ is a scalar quantity (no direction — just a number) |
Always negative | In a radial field (unless at infinity) |
Zero at infinity | By definition, ϕ=0ϕ=0 at r=∞r=∞ |
Increases with distance | As rr increases, ϕϕ becomes less negative (e.g., from -63 MJ/kg to -10 MJ/kg — it is "increasing") |
Exam Warning: Cambridge often asks "Does potential increase or decrease as you move away from Earth?"
Answer: It increases (goes from -63 to -10 to 0). Even though it is negative, getting closer to zero means it is increasing.
3. The Relationship Between Force and Potential (Gradient)
Cambridge expects you to know the link between gravitational field strength and gravitational potential.
Gravitational field strength is the negative gradient of the potential-distance graph.
g=−ΔϕΔrg=−ΔrΔϕ
Graphically:
On a ϕϕ vs rr graph (which curves upwards from negative towards zero), the slope (gradient) gives the value of gg.
The gradient is steepest near the planet (strong field) and flattens out at large rr (weak field).
4. Work Done and Potential Difference
This is a very common calculation in Paper 4.
Work done per unit mass moving between two points = change in potential (ΔϕΔϕ).
Δϕ=ϕf−ϕi=−GMrf−(−GMri)Δϕ=ϕf−ϕi=−rfGM−(−riGM)Δϕ=GM(1ri−1rf)Δϕ=GM(ri1−rf1)
Work done (total energy) to move mass mm:
ΔEp=mΔϕΔEp=mΔϕ
If moving away from the planet (to a larger rr):
ΔϕΔϕ is positive (less negative).
Work is done on the object (by an external force) — energy must be supplied.
If moving towards the planet (to a smaller rr):
ΔϕΔϕ is negative (more negative).
Work is done by the field — energy is released.