Ch 2 · 13 Gravitational Fields · A Level Physics 9702
Topic 2 of 2 in Circular Motion and Gravitation — A Level Physics 9702 — 59 lessons.
Introduction
The same force that makes an apple fall keeps the Moon in orbit and holds galaxies together. Newton's law of gravitation, and the ideas of field strength and potential, let us calculate orbits, satellite speeds and the energy needed to escape a planet. (Syllabus topic 13: sections 13.1–13.4.)
Key ideas — 13.1 Gravitational fields
A gravitational field is a field of force: a region in which a mass experiences a force.
🧑🏫 Teacher's explanation
Imagine you are walking through a crowd. If the crowd is very dense, you feel a strong 'push' from all sides. If the crowd is sparse, the push is weak. A gravitational field is like that 'push'—it is a region of space where any mass placed there will feel a pull towards the center of a larger mass.
📖 Definitions
⚠️ Common errors
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🎯 Exam question
(a) Show that the gravitational field strength g at distance r is given by g = GM/r^2. [2] (b) A second satellite is moved from distance r to distance 2r. Calculate the change in gravitational potential Δφ per unit mass. [2] [4 marks]
Key ideas — 13.2 (a) Newton's law of gravitation
For points outside a uniform sphere, the sphere's mass can be treated as a point mass at its centre.
🧑🏫 Teacher's explanation
Imagine you are holding two magnets. The closer you bring them, the harder they pull on each other, and if you double the distance, that pull doesn't just halve—it drops to a quarter of what it was. Gravity works in a very similar way. It is an 'inverse square law', meaning the strength of the pull depends heavily on how far apart the objects are.
📖 Definitions
🔍 Worked example — Easy
Calculate the gravitational force between two spheres, each of mass 5.0 × 10^24 kg, separated by a distance of 7.0 × 10^6 m. Use G = 6.67 × 10^-11 N m^2 kg^-2.
🔍 Worked example — Standard
A satellite of mass 1200 kg orbits a planet of mass 6.0 × 10^24 kg. Calculate the force of gravity acting on the satellite when it is 4.0 × 10^7 m from the center of the planet.
🔍 Worked example — Exam-style
The Earth has a mass of 5.98 × 10^24 kg and a radius of 6.37 × 10^6 m. Calculate the gravitational force on a 75.0 kg astronaut standing on the surface.
⚠️ Common errors
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🎯 Exam question
Key ideas — 13.2 (b) Circular orbits
For a satellite of mass m in a circular orbit of radius r around a planet of mass M, the gravitational force provides the centripetal force:
🧑🏫 Teacher's explanation
Imagine you are swinging a ball on a string around your head. The string provides the tension that keeps the ball moving in a circle; without it, the ball would fly off in a straight line. In space, for a satellite orbiting a planet, there is no string. Instead, the 'tension' is provided by the gravitational pull of the planet. This pull acts as the centripetal force, constantly pulling the satellite toward the center of the planet, which changes its direction but not its speed, keeping it in a circular path.
📖 Definitions
🔍 Worked example — Easy
A satellite of mass m orbits a planet of mass M at a distance r. Show that the orbital speed v is independent of m.
⚠️ Common errors
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🎯 Exam question
Key ideas — 13.2 (c) Geostationary orbits
A geostationary satellite stays above the same point on the Earth's surface. It must:
🧑🏫 Teacher's explanation
Imagine you are standing on a moving carousel. If you want to hover in the air and always stay directly above the same seat on the carousel, you have to fly in a circle around the center of the carousel at exactly the same speed that the carousel is spinning. If you fly too fast, you'll pull ahead; too slow, and you'll fall behind.
📖 Definitions
🔍 Worked example — Easy
State the three specific conditions required for a satellite to be in a geostationary orbit.
🔍 Worked example — Standard
Calculate the orbital radius r of a geostationary satellite. Use M = 5.98 × 10^24 kg, G = 6.67 × 10^-11 N m^2 kg^-2, and T = 24 × 3600 s.
🔍 Worked example — Exam-style
A satellite is in a geostationary orbit. Calculate its altitude above the Earth's surface. Assume Earth's radius R = 6.37 × 10^6 m and r = 4.22 × 10^7 m.
⚠️ Common errors
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🎯 Exam question
(a) State the orbital period of the satellite. [1] (b) Explain why the satellite must be positioned directly above the Equator. [2] (c) Calculate the orbital radius of the satellite. Assume the mass of the Earth is 6.0 × 10^24 kg, G = 6.7 × 10^-11 N m^2 kg^-2, and the period is 24 hours. [3] [6 marks]
Key ideas — 13.3 Field strength of a point mass
Derivation. A small mass m at distance r from a point mass M feels a force F = GMm/r^2. By definition g = F/m, so
🧑🏫 Teacher's explanation
Imagine you are standing on a giant, invisible trampoline that is stretched tight. If you place a heavy bowling ball in the middle, the fabric curves. The 'steepness' of that curve at any point tells you how hard a marble would be pulled toward the center. In physics, this 'steepness' is the gravitational field strength, g. The closer you get to the bowling ball, the steeper the curve becomes, meaning the force increases.
📖 Definitions
🔍 Worked example — Standard
A satellite of mass m = 500 kg orbits a planet at a distance r = 1.0 imes 10^7 m from the planet's center. Calculate the gravitational potential φ at the satellite's position. (Assume M = 5.0 imes 10^24 kg and G = 6.67 × 10^-11 N m^2 kg^-2)
⚠️ Common errors
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🎯 Exam question
Key ideas — 13.4 Gravitational potential
The gravitational potential φ at a point is the work done per unit mass in bringing a small test mass from infinity to that point.
🧑🏫 Teacher's explanation
Imagine you are trying to pull a heavy ball away from a massive magnet. As you move the ball further away, you are fighting the pull of the magnet. If you were to move the ball all the way to 'infinity' (a place where the magnet's pull is zero), you would have to put in a huge amount of energy.
📖 Definitions
🔍 Worked example — Easy
Calculate the gravitational potential at a point 2.0 × 10^6 m from the center of a planet of mass 6.0 × 10^24 kg. Use G = 6.67 × 10^-11 N m^2 kg^-2.
🔍 Worked example — Exam-style
A mass m is moved from a distance r_1 = 4.0 × 10^6 m to r_2 = 5.0 × 10^6 m from a planet of mass M = 6.0 × 10^24 kg. Calculate the change in gravitational potential energy Δ E_P. (Assume m = 10 kg, G = 6.67 × 10^-11 N m^2 kg^-2)
⚠️ Common errors
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🎯 Exam question
(a) Define the gravitational potential at a point at distance r from the center of the planet. [2] (b) Show that the change in gravitational potential energy when the satellite moves from radius r to radius 2r is given by Δ E_P = GMm/2r. [3] [5 marks]
Worked example
Example 1: field strength. Find g at the surface of Mars (M = 6.4 × 10^23 kg, r = 3.4 × 10^6 m).
🔍 Worked example — Easy
Calculate the gravitational field strength g at the surface of Mars. Given: Mass of Mars M = 6.4 × 10^23 kg, Radius r = 3.4 × 10^6 m, and G = 6.67 × 10^-11 N m^2 kg^-2.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) Field strength: force per unit mass on a small test mass; potential: work done per unit mass in bringing a small test mass from infinity to the point. (2) F = GMm/r^2 and g = F/m, so g = GM/r^2. (3) GMm/r^2 = mv^2/r; m cancels, leaving v = √(GM/r). (4) Period 24 hours; west to east; above the Equator. (5) Twice the distance from the centre → 9.81/4 = 2.45 N kg⁻¹. (6) Potential is zero at infinity and the force is attractive, so work is done by the field bringing a mass in; potential decreases (becomes negative) as you approach the mass.