Physics • Year 12 • Module 5 • Lesson 14
Gravitational Potential Energy
Lock in the key formula U = −GMm/r, understand the physical meaning of the negative sign, and practise the vocabulary before tackling harder questions.
1. Term–definition match
The definitions below are shuffled. In the right-hand column write the matching term from the list: gravitational potential energy, zero-at-infinity convention, binding energy, centre-to-centre distance, near-surface approximation, work done by gravity, gravitational field, escape velocity, bound system, negative GPE. 10 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The energy stored in a gravitational field; defined as U = −GMm/r for a mass m at distance r from mass M. | |
| 1.2 | The convention that sets GPE equal to zero when two masses are infinitely far apart. | |
| 1.3 | The minimum energy required to completely free an object from a gravitational field; equal to |U| at that location. | |
| 1.4 | The distance measured from the centre of one body to the centre of another; equals Rplanet + h for orbiting objects. | |
| 1.5 | The formula ΔU ≈ mgh, which is only valid when the height h is much less than Earth’s radius. | |
| 1.6 | The energy transferred by the gravitational force when a mass moves between two points; W = GMm(1/r1 − 1/r2). | |
| 1.7 | A region of space in which a mass experiences a gravitational force due to the presence of another mass. | |
| 1.8 | The minimum launch speed needed for an object to escape a planet’s gravity entirely, without further propulsion. | |
| 1.9 | A system in which an object does not have enough energy to escape to infinity; characterised by negative total energy. | |
| 1.10 | The value of GPE for any finite separation of two masses; it indicates the object has less energy than at infinite separation. |
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 energy is always positive for any two masses separated by a finite distance. T / F
2.2 The zero reference point for gravitational potential energy is defined at Earth’s surface. T / F
2.3 As a satellite moves to a higher orbit, its gravitational potential energy increases (becomes less negative). T / F
2.4 When using U = −GMm/r, the variable r represents the height above Earth’s surface. T / F
2.5 The formula ΔU = mgh is a valid approximation only when the height h is much less than Earth’s radius. T / F
2.6 The negative sign in U = −GMm/r is simply a mathematical convention and has no physical meaning. 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 · binding · bound · convention · infinity · negative · positive · zero
Gravitational potential energy is always ___________ for any finite separation because gravity is an ___________ force. The reference point of GPE is set to ___________ at ___________, which is a ___________ rather than a natural physical zero. Any object at a finite distance therefore has a ___________ total energy — it is said to be ___________ to the central mass. The magnitude |U| represents the ___________ energy: the minimum energy an external agent must supply to free the object.
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 negative sign in U = −GMm/r tell us physically about the gravitational interaction?
4.2 Why must the variable r in U = −GMm/r be the centre-to-centre distance rather than the altitude h above the surface?
4.3 State the condition under which ΔU ≈ mgh is a valid approximation and explain why this condition is necessary.
4.4 What does it mean physically for an object’s total mechanical energy to be negative in a gravitational field?
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 defined at”, “indicates”, “is an approximation for”). Aim for at least 6 labelled arrows. 6 marks (1 per valid labelled arrow)
Supplied terms: gravitational potential energy · negative sign · zero at infinity · binding energy · mgh · bound system.
Q1 — Term–definition match
1.1 gravitational potential energy • 1.2 zero-at-infinity convention • 1.3 binding energy • 1.4 centre-to-centre distance • 1.5 near-surface approximation • 1.6 work done by gravity • 1.7 gravitational field • 1.8 escape velocity • 1.9 bound system • 1.10 negative GPE.
Q2 — True / false with correction
2.1 False. GPE is always negative for any finite separation because gravity is attractive and U = −GMm/r; it only reaches zero at infinite separation.
2.2 False. The zero reference for GPE is defined at infinity (r → ∞), not at Earth’s surface. This is the zero-at-infinity convention.
2.3 True.
2.4 False. r is the centre-to-centre distance from Earth’s centre to the object, equal to REarth + h; using h alone would give an incorrect (and much smaller) value.
2.5 True.
2.6 False. The negative sign has real physical meaning: it encodes that gravity is attractive, the system is bound, and energy must be added to separate the masses. It is essential for determining whether an object is bound or unbound.
Q3 — Cloze paragraph
In order: negative / attractive / zero / infinity / convention / negative / bound / binding.
Q4.1 — Physical meaning of the negative sign
The negative sign indicates that gravity is an attractive force and the system is bound: work must be done against the field to separate the masses. An object at any finite r has less energy than at infinite separation, so energy must be added to free it.
Q4.2 — Why r is centre-to-centre distance
The formula U = −GMm/r treats both bodies as point masses and r measures the distance between their centres of mass. Using altitude h ignores Earth’s radius (6.37 × 106 m), which would greatly underestimate the true separation and give an incorrect (far too negative) value of U.
Q4.3 — Condition for the mgh approximation
The approximation is valid when h ≪ R (height much less than Earth’s radius). This is necessary because the derivation uses the approximation R + h ≈ R; at large h the approximation fails because g decreases significantly with altitude, yet mgh treats g as constant.
Q4.4 — Meaning of negative total energy
A negative total mechanical energy (KE + GPE < 0) means the object is bound: it does not have enough kinetic energy to escape to infinity. It will remain gravitationally captured by the central mass. Only if total energy ≥ 0 can the object escape.
Q5 — Sample concept map
Award 1 mark per valid labelled arrow, minimum 6. Correct arrows include:
- gravitational potential energy — is defined as zero at → zero at infinity
- negative sign — indicates → bound system
- binding energy — equals the magnitude of → gravitational potential energy
- mgh — is a near-surface approximation for → gravitational potential energy
- bound system — has → negative sign (negative total energy)
- zero at infinity — makes → gravitational potential energy (negative at all finite r)