Physics • Year 12 • Module 5 • Lesson 15
Gravitational Potential
Lock in the core vocabulary, the key formula $V = -GM/r$, and the properties of equipotential surfaces before tackling harder questions.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from this list: gravitational potential, gravitational potential energy, potential gradient, equipotential surface, scalar, zero at infinity, field lines, work per unit mass, centre-to-centre distance. 9 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The gravitational potential energy per unit mass at a point in a gravitational field, given by $V = -GM/r$ (units: J/kg). | |
| 1.2 | The energy stored by a specific mass $m$ in a gravitational field, given by $U = -GMm/r$ (units: J). | |
| 1.3 | The rate of change of gravitational potential with distance, $dV/dr$, which is related to field strength by $g = -dV/dr$. | |
| 1.4 | A surface on which the gravitational potential $V$ has the same value at every point; no work is done moving along it. | |
| 1.5 | A quantity that has magnitude and sign but no direction; gravitational potential is this type of quantity. | |
| 1.6 | The convention for the reference level of gravitational potential: $V = 0$ when $r \to \infty$, far from any mass. | |
| 1.7 | Lines in a gravitational field that show the direction of the gravitational force; they are always perpendicular to equipotential surfaces. | |
| 1.8 | The work done per kilogram of mass moved between two points; equal to the change in gravitational potential, $\Delta V$. | |
| 1.9 | The distance $r = R_\text{planet} + h$ measured from the centre of the central body to the object, not from the surface. |
2. True or false — with correction
Circle T or F for each statement. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 Gravitational potential $V$ is a vector quantity because it comes from the gravitational force, which is a vector. T / F
2.2 At any finite distance $r$ from a mass, the gravitational potential $V$ is always negative because gravity is attractive. T / F
2.3 Gravitational potential energy $U$ and gravitational potential $V$ are the same quantity measured in the same units. T / F
2.4 The work done moving a mass along an equipotential surface is always zero, regardless of the path taken. T / F
2.5 The gravitational field strength $g$ equals $+dV/dr$ (without a negative sign). T / F
2.6 On a $V$ vs $r$ graph, a steeper slope at a point indicates a stronger gravitational field at that point. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word is used once. 8 marks (1 per blank)
Word bank:
attractive · decreasing · energy · gradient · infinity · negative · perpendicular · scalar
Gravitational potential $V$ is a ___________ quantity: it has magnitude and sign but no direction. $V$ is always ___________ for any finite distance from a mass, because gravity is ___________ and work must be done against gravity to move a mass to infinity. By convention, $V = 0$ at ___________. The gravitational potential can be thought of as the ___________ landscape of a gravitational field — it tells us how much energy per unit mass is stored at each point. The gravitational field strength $g$ equals the negative of the potential ___________, so $g$ points in the direction of ___________ potential. Field lines are always ___________ to equipotential surfaces.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 What does the gravitational potential $V$ tell us about a point in a gravitational field, and how does it differ from gravitational potential energy $U$?
4.2 Why is the formula written as $V = -GM/r$ with a negative sign, rather than $V = +GM/r$?
4.3 What is the significance of closely spaced equipotential surfaces compared with widely spaced equipotential surfaces?
4.4 State two units in which gravitational potential can be expressed and explain why they are equivalent.
5. Build a concept map
Draw labelled arrows between the six terms below to show how they connect. Each arrow must carry a linking phrase (e.g. “is the negative gradient of”, “is perpendicular to”, “equals $U$ divided by”). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: gravitational potential $V$ · gravitational field strength $g$ · equipotential surface · field lines · work done · potential gradient.
Q1 — Term–definition match
1.1 gravitational potential • 1.2 gravitational potential energy • 1.3 potential gradient • 1.4 equipotential surface • 1.5 scalar • 1.6 zero at infinity • 1.7 field lines • 1.8 work per unit mass • 1.9 centre-to-centre distance.
Marking criteria: 1 mark per correct match.
Q2 — True / false with correction
2.1 False. Gravitational potential $V$ is a scalar quantity — it has magnitude and sign but no direction. It is gravitational field strength $\vec{g}$ that is the vector quantity derived from the gravitational force.
2.2 True. $V = -GM/r$ is always negative for finite $r > 0$ because the negative sign reflects that gravity is attractive and energy must be supplied to move the mass to infinity.
2.3 False. Gravitational potential $V$ (J/kg) is the GPE per unit mass — a property of the field at a point. Gravitational potential energy $U = mV$ (J) is the energy of a specific mass $m$ at that point. They have different units and different physical meanings.
2.4 True. Along an equipotential surface $\Delta V = 0$, so the work done per unit mass $W/m = \Delta V = 0$ regardless of the path taken, because gravitational force is conservative.
2.5 False. $g = -dV/dr$ — there is a negative sign. This ensures $g$ points toward decreasing potential (inward, toward the mass), which is the correct direction for an attractive gravitational field.
2.6 True. The slope of $V$ vs $r$ gives $dV/dr$, and $|g| = |dV/dr|$. A steeper slope means a larger potential gradient, which corresponds to a stronger gravitational field at that point.
Marking criteria: 1 mark per correct T/F; 1 mark per correct correction for each false statement.
Q3 — Cloze paragraph
In order: scalar / negative / attractive / infinity / energy / gradient / decreasing / perpendicular.
Marking criteria: 1 mark per correct blank.
Q4.1 — What $V$ tells us and how it differs from $U$
Gravitational potential $V$ at a point tells us the gravitational potential energy per unit mass that a test mass would have at that point; it is a property of the field, independent of what mass is placed there. Gravitational potential energy $U = mV$ is the energy of a specific mass $m$ placed in the field. Their units differ: $V$ in J/kg, $U$ in J.
Marking criteria: 1 mark for describing $V$ as GPE per unit mass / property of the field; 1 mark for correctly distinguishing $V$ (J/kg, field property) from $U$ (J, depends on the placed mass).
Q4.2 — Why the negative sign in $V = -GM/r$
The negative sign reflects that the gravitational force is attractive. Work must be done on a mass to move it from $r$ to infinity against no force (from $V < 0$ to $V = 0$); therefore $V$ at any finite $r$ must be below the reference level of zero, making it negative. Physically, a mass is in a “potential well” that becomes deeper as it approaches the central body.
Marking criteria: 1 mark for linking the negative sign to the attractive / binding nature of gravity; 1 mark for explaining that the reference ($V = 0$ at infinity) is above all finite values.
Q4.3 — Spacing of equipotential surfaces
Closely spaced equipotential surfaces indicate a large potential gradient over a small distance, which means a strong gravitational field in that region (since $g = -dV/dr$). Widely spaced surfaces indicate a small potential gradient and therefore a weak gravitational field.
Marking criteria: 1 mark for closely spaced = strong field (large $dV/dr$); 1 mark for widely spaced = weak field (small $dV/dr$).
Q4.4 — Two units of gravitational potential
Gravitational potential can be expressed in J/kg (joules per kilogram) or m2/s2 (metres squared per second squared). They are equivalent because $1\text{ J} = 1\text{ kg m}^2\text{/s}^2$, so dividing by kg gives $1\text{ J/kg} = 1\text{ m}^2\text{/s}^2$.
Marking criteria: 1 mark for correctly naming both units; 1 mark for a correct unit equivalence argument.
Q5 — Sample concept map
Correct maps should include arrows such as:
- potential gradient — is the rate of change of → gravitational potential $V$
- gravitational potential $V$ — negative gradient gives → gravitational field strength $g$
- equipotential surface — has constant → gravitational potential $V$
- work done — equals zero along → equipotential surface
- field lines — are perpendicular to → equipotential surface
- gravitational field strength $g$ — is tangent to → field lines
Award 1 mark per valid labelled arrow (minimum 6, maximum 6 marked). Accept equivalent correct phrasing.
Marking criteria: 1 mark per valid labelled arrow with a correct linking phrase.