Gravitational fields and potential energy: WACE Year 12 Physics Unit 3

What this dot point is asking

WACE wants you to describe gravity as a field, calculate field strength, and analyse the energy changes when a mass moves within it. The field model lets you predict the force on any mass placed at a point without re-deriving it each time.

Field strength

The gravitational field strength at a point is the force per unit mass placed there:

Numerically this equals the free-fall acceleration in $\text{m s}^{-2}$, which is why $g$ near Earth's surface is $9.8$ in both units. Field strength is a vector pointing toward the source mass.

Representing the field

Field lines show the direction a small test mass would be pushed and, by their spacing, the strength. Around a single planet the lines are radial and inward, spreading out with distance so the field weakens as $1/r^2$. Close to a small patch of surface the lines are nearly parallel and evenly spaced, which is why we treat the field as uniform for everyday problems.

Potential energy near a surface

When the field is treated as uniform, lifting a mass $m$ through a height $\Delta h$ changes its gravitational potential energy by

Raising a mass increases $E_p$; letting it fall converts that energy to kinetic energy. This near-surface model is what WACE expects for projectile and ramp problems, where $g$ barely changes over the heights involved.

Work done and energy conservation

The work done by you against gravity to raise a mass equals the increase in potential energy, and the work done by gravity on a falling mass equals the loss in potential energy. Combined with the kinetic energy $E_k=\tfrac{1}{2}mv^2$, energy conservation gives launch and landing speeds directly. For a mass falling from rest through height $h$, $mgh=\tfrac{1}{2}mv^2$, so $v=\sqrt{2gh}$, independent of mass.

Why the near-surface model works

The formula $\Delta E_p=mg\Delta h$ is only an approximation of the true field, which weakens as $1/r^2$ with distance from the planet's centre. It is accurate when the height change is tiny compared with the planet's radius, so $g$ barely changes. For Earth, climbing $1\ \text{km}$ changes $g$ by only about $0.03\%$, so the uniform-field model is excellent for projectiles, ramps and buildings. It breaks down for satellites and deep-space probes, where the full field treatment and the inverse-square variation of $g$ must be used instead. Knowing exactly when the simple model is valid is itself an exam-worthy point.

Field strength versus potential energy

Keep the two ideas distinct: field strength is force per unit mass (a property of the field at a point), while potential energy depends on the mass placed there and its position. A region can have a strong field yet a chosen mass have little potential energy if it has barely moved.

Reading energy graphs

WACE often presents the energy story as graphs against position. For a mass dropped from rest, a graph of gravitational potential energy against height falls linearly (near a surface) while kinetic energy rises by the same amount, so the total mechanical energy stays a flat horizontal line, the signature of energy conservation with no losses. If the graph of total energy slopes downward, energy is being lost, usually to friction or air resistance as heat. Being able to read which curve is kinetic, which is potential and which is the conserved total, and to interpret a sloping total as a loss mechanism, is a recurring skill in extended-response questions.