IBDP Physics HL Cheat sheet - D.1 Gravitational fields | IBDP Physics Cheatsheets

Core ideas and laws

Gravitational fields describe how a mass experiences an attractive force due to another mass.
Newton’s universal law of gravitation for point masses: $F = G\dfrac{m_1 m_2}{r^2}$ .
Use this law when bodies can be treated as point masses, or when dealing with spherically symmetric bodies from outside the body.
A spherical body of uniform density acts as if all its mass were concentrated at its centre.
Near a planet’s surface, the field can be approximated as a uniform field with almost constant $g$ .
Far from a mass, the field is a radial field and falls with inverse square dependence: $g \propto \dfrac{1}{r^2}$ .

This diagram combines all three of Kepler’s laws in one figure. It shows elliptical orbits, equal areas swept in equal times, and the period–distance relationship. Use it to connect the geometry of orbits to the motion of planets and satellites. Source

Kepler’s laws of orbital motion

First law: planets move in elliptical orbits with the Sun at one focus.
Second law: a line joining the planet to the Sun sweeps out equal areas in equal times.
Third law: for bodies orbiting the same central mass, $T^2 \propto r^3$ for circular orbits or more generally with semi-major axis.
The second law means an orbiting body moves faster when closer to the central mass and slower when farther away.
In IB calculations for this topic, orbital motion is usually limited to circular orbits.

Gravitational field strength

Gravitational field strength is the force per unit mass on a small test mass: $g = \dfrac{F}{m}$ .
Around a spherical mass $M$ : $g = G\dfrac{M}{r^2}$ .
Unit of $g$ : N kg $^{-1}$ , which is equivalent to m s $^{-2}$ .
The direction of $g$ is always towards the centre of the mass producing the field.
Resultant field strength may be found by vector addition; in this syllabus, this is restricted to points along the line joining two masses.

This image shows gravitational field lines and associated equipotential contours for the Earth–Moon system. The denser blue field lines indicate stronger field, while the red contours represent equal potential. It is useful for visualizing how field direction and potential are related. Source

Field lines and field patterns

Gravitational field lines show the direction a small test mass would move.
Field lines always point towards mass, because gravity is always attractive.
Closer spacing of field lines means a stronger field.
Around an isolated spherical mass, field lines are radial and directed inwards.
In a uniform field near a planet’s surface, field lines are parallel and equally spaced.
Field lines never cross.

HL only: gravitational potential and potential energy

Gravitational potential energy is the work done to assemble masses from infinite separation.
For two masses: $E_p = -G\dfrac{m_1 m_2}{r}$ .
The negative sign means the system is bound; energy must be supplied to separate the masses to infinity.
Gravitational potential is the work done per unit mass to bring a small mass from infinity to a point: $V_g = -G\dfrac{M}{r}$ .
Unit of gravitational potential: J kg $^{-1}$ .
More negative $V_g$ means the mass is more strongly bound.
Relationship between field and potential: $g = -\dfrac{\Delta V_g}{\Delta r}$ .
Work done moving a mass in a field: $W = m\Delta V_g$ .
No work is done when moving along an equipotential surface.

This graph shows the gravitational potential due to Earth, the Moon, and their combined system. It helps explain why potential is negative and how the total potential varies between two masses. Use it to connect $V_g$ , $g$ , and the idea of a bound system. Source

HL only: equipotential surfaces

Equipotential surfaces join points of the same gravitational potential.
They are always perpendicular to gravitational field lines.
Where equipotential surfaces are closer together, the potential gradient is steeper and the field is stronger.
For a spherical mass, equipotential surfaces are concentric spheres.
Mapping a field using potential is an important practical and exam skill.

Circular orbits

In a circular orbit, gravity provides the centripetal force.
Set $G\dfrac{Mm}{r^2} = \dfrac{mv^2}{r}$ to derive orbit equations.
Orbital speed: $v_{\text{orbital}} = \sqrt{\dfrac{GM}{r}}$ .
A satellite in a larger orbit moves at a lower speed.
For a given orbit radius, the orbital speed is independent of the satellite’s mass.
If radius increases, period increases and speed decreases.

Figure 13.12 shows a satellite in circular orbit with gravity directed toward the centre and velocity tangent to the orbit. Figure 13.13 shows how increasing launch speed changes trajectories from falling back to Earth to achieving orbit. These are excellent for understanding why gravity supplies the centripetal force in orbit. Source

HL only: escape speed and orbital energy ideas

Escape speed is the minimum speed needed to reach infinite separation with zero kinetic energy remaining.
Escape speed: $v_{\text{esc}} = \sqrt{\dfrac{2GM}{r}}$ .
Therefore, at the same radius: $v_{\text{esc}} = \sqrt{2},v_{\text{orbital}}$ .
To move a satellite to a higher orbit, energy must be supplied because the total energy becomes less negative.
Gravitational potential energy becomes less negative as $r$ increases.
Questions may involve changes in energy when satellites change orbit or escape from a planet.

Atmospheric drag on satellites

A small viscous drag force due to the atmosphere causes a satellite’s orbit to decay.
The satellite gradually loses mechanical energy.
Its orbital height decreases over time.
As the orbit shrinks, the satellite’s orbital speed increases because $v_{\text{orbital}} = \sqrt{GM/r}$ .
This can seem counterintuitive: losing energy can still mean moving faster in a lower orbit.

This scale diagram compares low Earth orbit, medium Earth orbit, GEO, and the ISS altitude. It helps students relate orbit radius to orbital speed, period, and the importance of atmospheric drag in lower orbits. Lower orbits are closer to the atmosphere and are therefore more affected by drag. Source

Exam traps and high-yield reminders

Use distance from the centre of the planet, not altitude above the surface, in $F = G\dfrac{m_1m_2}{r^2}$ , $g = G\dfrac{M}{r^2}$ , $v_{\text{orbital}}$ , and $v_{\text{esc}}$ .
Do not confuse gravitational field strength $g$ with the constant $g \approx 9.81,\text{m s}^{-2}$ near Earth’s surface only.
Potential is a scalar; field strength is a vector.
Potential energy depends on both masses; potential depends on the source mass only.
Remember that gravitational potential is zero at infinity.
A more negative total energy means the satellite is more strongly bound.

Checklist: can you do this?

State and apply Newton’s law of gravitation and Kepler’s laws.
Calculate $g$ , $V_g$ , $E_p$ , $v_{\text{orbital}}$ , and $v_{\text{esc}}$ .
Sketch and interpret gravitational field lines and equipotential surfaces.
Explain why gravity provides the centripetal force for circular orbits.
Interpret what happens to speed, energy, and height when a satellite changes orbit or experiences drag.

Written by:

Dr Shubhi Khandelwal

Qualified Dentist and Expert Science Educator

Shubhi is a seasoned educational specialist with a sharp focus on IB, A-level, GCSE, AP, and MCAT sciences. With 6+ years of expertise, she excels in advanced curriculum guidance and creating precise educational resources, ensuring expert instruction and deep student comprehension of complex science concepts.