D.1 Gravitational fields

UNIT SYLLABUS

SL/HL 5 hours SL + 7 hours HL

Newton's law of gravitation — every mass attracts every other with a force falling off as $1/r^2$ — unifies the falling apple with the orbiting Moon. The field concept turns this action-at-a-distance into a local picture: a mass shapes the space around it with field strength $g = GM/r^2$, and other masses respond to the field where they are. Kepler's three laws then drop out as consequences. At HL the description deepens from forces to energy: gravitational potential wells, equipotential surfaces perpendicular to field lines, escape speed, and orbital mechanics precise enough to explain why atmospheric drag paradoxically speeds satellites up as they spiral down.

Guiding Questions

? How can a field be used to model an interaction between masses that never touch?
? What determines whether an object remains in orbit or escapes a gravitational field?

The radial field of a point mass: field lines point inward, equipotential surfaces (dashed) cross them at right angles, spreading further apart as the field weakens.

✓ What the IB expects you to master

State and apply Kepler's three laws of orbital motion.
Use Newton's law of gravitation $F = G\frac{m_1m_2}{r^2}$ for point (and spherically symmetric) masses.
Define field strength $g = F/m = GM/r^2$ and draw gravitational field line patterns.
HL: define gravitational potential energy of a system, $E_p = -G\frac{m_1m_2}{r}$ , assembled from infinite separation.
HL: use gravitational potential $V_g = -GM/r$ , the potential gradient $g = -\Delta V_g/\Delta r$ , and $W = m\Delta V_g$ for work done moving a mass.
HL: relate equipotential surfaces to field lines (always perpendicular; equal spacing means uniform field).
HL: derive and use escape speed $v_{esc} = \sqrt{2GM/r}$ and orbital speed $v_{orbital} = \sqrt{GM/r}$ .
HL: explain qualitatively how viscous atmospheric drag lowers an orbit while increasing orbital speed.

1 Key Formulas

Newton's law of gravitation

F = G\frac{m_{1}m_{2}}{r^{2}}

Gravitational field strength

g = \frac{F}{m} = G\frac{M}{r^{2}}

Gravitational PE (HL)

E_{p} = -G\frac{m_{1}m_{2}}{r}

Gravitational potential (HL)

V_{g} = -\frac{GM}{r}

Potential gradient (HL)

g = -\frac{\Delta V_{g}}{\Delta r}

Work done (HL)

W = m\Delta V_{g}

Escape speed (HL)

v_{\text{esc}} = \sqrt{\frac{2GM}{r}}

Orbital speed (HL)

v_{\text{orbital}} = \sqrt{\frac{GM}{r}}

2 Exam Preparation & Topic Explanations

Energy accounting in orbits

For a circular orbit, memorise the energy triple: $E_k = \frac{GMm}{2r}$ , $E_p = -\frac{GMm}{r}$ , total $E = -\frac{GMm}{2r}$ . Every orbital-transfer or escape question is bookkeeping between these.

Potential-well diagrams (V against r) answer "how much energy to escape from radius r" by reading the depth remaining to zero.

Pro Exam Strategy

Always measure $r$ from the centre of the body — surface-height traps abound.
Field strength is a vector, potential a scalar: potentials add algebraically, fields add as vectors.
Field lines never cross; equipotentials never cross; they always meet each other at 90°.
No work is done moving along an equipotential — that sentence alone often earns a mark.

3 MCQ Practice

Q1. The gravitational field strength at the surface of a planet is $g$ . At a height equal to the planet's radius above the surface, it is:

$g/2$
$g/4$
$g/8$
$g/16$

Q2. According to Kepler's third law, a planet orbiting at 4 times Earth's orbital radius has a period of:

4 years
8 years
16 years
64 years

Q3. Gravitational potential is always negative because:

Gravity always repels
The zero is defined at infinity and gravity is attractive, so work is released bringing a mass inward
It is a vector pointing towards the mass
Potential energy cannot be positive

4 Short Answer Questions

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