The shape of the lesson
By the end of the lesson, learners can
- Climbing out of the well: Gravitational potential is the work done per kilogram to bring a mass from infinity to a point.
- Potential, and potential energy: For a point mass the potential is ϕ = −GM / r, and the potential energy of a mass m placed there is Eₙ = mϕ = −GMm / r.
- Newton's law: F = Gm₁m₂ / r², with uniform spheres treated as point masses at their centres.
- Field strength and potential: g = GM / r² and ϕ = −GM / r, so g is the gradient of ϕ.
- Orbits: gravity is the centripetal force, GMm / r² = mv² / r, giving v = √(GM / r) and T = 2π√(r³ / GM).
One picture every idea returns to
Gravitational potential is the work done per unit mass to bring a test mass from infinity, ϕ = −GM / r, always negative. The potential energy of a mass m is Eₙ = mϕ = −GMm / r.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Always negative: potential.. Gravitational potential measures the work per kilogram to escape to infinity. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 17 min | Teach and check: Climbing out of the well | Gravitational potential is the work done per kilogram to bring a mass from infinity to a point. Project tab 3 of the lab (Potential well and escape (3D)) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Potential, and potential energy | For a point mass the potential is ϕ = −GM / r, and the potential energy of a mass m placed there is Eₙ = mϕ = −GMm / r. | Corners |
| 30 to 43 min | Check and plenary | The quick-check questions from the student lesson, answered on mini whiteboards one at a time; re-teach the one most get wrong, then a random call on the last. | Individual, then whole class |
| 43 to 45 min | Exit and homework | Set the worksheet below (4 questions, 48 marks) from the topic practice set. Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Potential, and potential energy, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Corners
Put one answer or viewpoint in each corner of the room. Students move to the corner they agree with, justify the choice to a partner there, then hear one voice from each corner before the reveal.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
No examiner trap is recorded for this lesson. Use the quiz key below to spot the distractor most of the class picks.
Gravitational potential: topic worksheet
4 questions, 48 marks. Written in the style of the papers; no past paper question is reproduced. These questions cover the whole topic, so sibling lessons share them.
A space probe of mass m = 1200 kg moves in the gravitational field of a planet of mass M = 3.0 × 10²⁵ kg. Fig. 3.1 shows how the gravitational potential φ varies with distance r from the centre of the planet.
(a) Define gravitational potential at a point. [2]
(b) Calculate the gravitational potential at a point r₁ = 8.0 × 10⁶ m from the centre of the planet. [2]
(c) Calculate the gravitational potential energy of the probe at this point. [2]
(d) The probe moves outward from r₁ = 8.0 × 10⁶ m to r₂ = 2.0 × 10⁷ m. Calculate the change in its potential energy, and state whether work is done on or by the probe. [3]
(e) Explain why values of gravitational potential are always negative. [2]
(f) State how the potential φ changes as r increases without limit. [2]
This question is about gravitational potential and potential energy near the Earth (M = 6.0 × 10²⁴ kg).
(a) Define gravitational potential at a point. [2]
(b) Explain why gravitational potential is always negative. [2]
(c) Write the expression for the gravitational potential energy of a mass m at distance r from a mass M. [1]
(d) Calculate the gravitational potential energy of a 500 kg satellite at r = 7.0 × 10⁶ m. [2]
(e) Calculate the work done to raise the satellite to r = 1.4 × 10⁷ m. [3]
(f) State how the gravitational potential varies with distance r. [1]
A satellite of mass 500 kg orbits the Earth (M = 6.0 × 10²⁴ kg) in a circular orbit of radius r = 7.0 × 10⁶ m. This question connects the orbital motion of Q6 with the energy ideas of Q8.
(a) Write expressions, in terms of G, M, m and r, for the kinetic energy and the gravitational potential energy of the satellite. [2]
(b) Show that the total energy of the satellite is E = −GMm / 2r. [2]
(c) Calculate the kinetic energy, the potential energy and the total energy of this satellite. [3]
(d) The satellite is moved to a higher orbit. State and explain what happens to its kinetic energy and to its total energy. [3]
(e) Explain how a satellite in a higher orbit can move more slowly yet have a greater total energy. [2]
Fig. 10.1 shows how the gravitational potential φ near a planet varies with distance r from its centre. This question connects potential (Q8) with field strength (Q4).
(a) State the relationship between gravitational field strength and gravitational potential. [2]
(b) Explain how the field strength at a point can be obtained from the φ-r graph. [1]
(c) Compare how field strength and potential each vary with distance from a point mass, referring to their different powers of r. [3]
(d) State the relationship between gravitational field lines and equipotential surfaces. [2]
(e) The potential at the planet’s surface is −6.3 × 10⁷ J kg⁻¹. Calculate the energy needed to remove a 2.0 kg mass completely from the surface to infinity. [3]
(f) State the value of the gravitational potential at infinity. [1]
Answer key and marking notes
- (a)The gravitational potential at a point is the work done per unit mass ✓ in bringing a small test mass from infinity to that point ✓
- (b)φ = −GM / r = −(6.67 × 10⁻¹¹ × 3.0 × 10²⁵) / 8.0 × 10⁶ ✓ = −2.5 × 10⁸ J kg⁻¹ ✓
- (c)Eₕ = mφ = 1200 × (−2.5 × 10⁸) ✓ = −3.0 × 10¹¹ J ✓
- (d)φ₂ = −GM / r₂ = −1.0 × 10⁸ J kg⁻¹, so Eₕ₂ = 1200 × (−1.0 × 10⁸) = −1.2 × 10¹¹ J ✓. ΔEₕ = −1.2 × 10¹¹ − (−3.0 × 10¹¹) = +1.8 × 10¹¹ J ✓. The potential energy increases, so work is done on the probe (against the gravitational field) ✓
- (e)The zero of potential is defined at infinity ✓; gravity is attractive, so the field does positive work on a mass brought in from infinity, leaving the potential below zero everywhere, hence negative ✓
- (f)As r increases, φ = −GM / r increases (becomes less negative) ✓ and tends toward zero as r tends to infinity ✓
- (a)The work done per unit mass in bringing a small test mass from infinity to the point ✓✓
- (b)The zero of potential is taken at infinity, and gravity is attractive, so the field does positive work bringing a mass inward, leaving the potential below zero everywhere ✓✓
- (c)Eₙ = −GMm/r ✓
- (d)Eₙ = −(6.67 × 10⁻¹¹ × 6.0 × 10²⁴ × 500)/(7.0 × 10⁶) ✓ = −2.9 × 10¹⁰ J ✓
- (e)At r = 1.4 × 10⁷ m (double), Eₙ = −1.43 × 10¹⁰ J ✓; work done = ΔEₙ = (−1.43 × 10¹⁰) − (−2.86 × 10¹⁰) ✓ = +1.4 × 10¹⁰ J ✓
- (f)φ = −GM/r, so its magnitude varies as 1/r, rising toward zero as r increases ✓
- (a)KE = ½mv² = GMm/2r (using v² = GM/r) ✓; Eₙ = −GMm/r ✓
- (b)E = KE + Eₙ = GMm/2r − GMm/r = −GMm/2r ✓✓
- (c)KE = +1.4 × 10¹⁰ J ✓; Eₙ = −2.9 × 10¹⁰ J ✓; total E = −1.4 × 10¹⁰ J ✓
- (d)At larger r the speed is smaller, so the kinetic energy decreases ✓; the total energy E = −GMm/2r increases (becomes less negative) ✓✓
- (e)The gain in potential energy is larger than the loss in kinetic energy, so the total energy rises toward zero even though the satellite is slower ✓✓
- (a)The field strength is the negative gradient of the potential: g = −dφ/dr ✓✓
- (b)It is the magnitude of the gradient (steepness) of the φ-r graph at that point ✓
- (c)Field strength varies as 1/r² (an inverse-square fall-off, steeper) ✓; potential varies as 1/r (a gentler fall-off) ✓; so the potential reaches further before becoming negligible ✓
- (d)Field lines are perpendicular to equipotential surfaces ✓ and point from high to low potential (toward the planet) ✓
- (e)Energy = m(φ∞ − φₙ) = 2.0 × (0 − (−6.3 × 10⁷)) ✓✓ = 1.3 × 10⁸ J ✓
- (f)Zero ✓
The lesson's own checks as a quiz
The quick-check and exam-style questions from the student lesson, printable with the key.
The gravitational potential at a point is the work done per unit mass in bringing a small test mass:
- Afrom infinity to that point
- Bfrom the surface to that point
- Cfrom that point to the centre
- Dthrough a distance of one metre
Gravitational potential ϕ = −GM / r is negative because:
- Athe zero is taken at infinity and gravity is attractive
- Bmass can be negative
- Cthe distance r is negative
- Dthe constant G is negative
The gravitational potential energy of a mass m at a point where the potential is ϕ is:
- AEₙ = mϕ = −GMm / r
- BEₙ = ϕ / m
- CEₙ = +GMm / r
- DEₙ = mgh in all cases
As a probe moves to a larger distance r from a planet, its gravitational potential energy:
- Aincreases, becoming less negative
- Bdecreases, becoming more negative
- Cstays constant
- Dimmediately becomes zero
Quiz key
Potential is defined relative to infinity, where it is zero: the work per unit mass to bring a test mass from infinity to the point.
With ϕ = 0 at infinity and an attractive force, the field does work on a mass brought inward, leaving the potential below zero everywhere.
Potential energy is mass times potential: Eₙ = mϕ = −GMm / r, negative and rising toward zero as r grows.
Eₙ = −GMm / r rises toward zero as r increases, so moving outward increases the potential energy; work is done against gravity.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.