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All revision notes Revision notes · GravityGravity revision notes
Free fall, mass and weight, the inverse-square law, the Earth, Moon and Solar System, Kepler's laws, satellites, gravitational energy, tides and black holes. Each section follows one lesson: what you need to know, key terms, the equations with units, a worked example, common mistakes and an exam tip.
1 Why Things Fall & Orbit
What you need to know
Free Fall
- With no air resistance, everything near the Earth falls with the same acceleration of free fall, g = 9.8 m/s², whatever its mass.
- Dropped from rest, its speed rises by 9.8 m/s every second: v = gt. In twice the time it falls four times as far.
- A strobe photo of a falling ball shows the gaps between the images growing by the same amount each flash: a constant acceleration.
Mass & Weight
- Mass is the amount of matter in an object, in kg; it is the same everywhere. Weight is the gravitational force on the object, in N.
- W = mg, with the gravitational field strength g = 9.8 N/kg on Earth. Twice the mass means twice the weight.
- A newton-meter measures weight. The 9.8 N/kg of the field and the 9.8 m/s² of free fall are the same number.
Orbits
- An object in orbit is in free fall: gravity pulls it towards the centre of the planet, and its sideways speed carries it round so it never reaches the ground.
- Orbital speed v = 2πr ÷ T, with r measured from the centre of the planet and T the time for one orbit (the period).
- Faster than a circular orbit, the path becomes an ellipse: the object slows as it climbs and speeds up as it falls back in. Fast enough (about 11 km/s from near the Earth) it escapes.
Capture
- To be captured into orbit round a moon or planet, a spacecraft must slow below the escape speed at its height. It fires its engine against its motion, ideally at the closest point.
Key terms
- Free fall
- motion under gravity alone, with no air resistance
- Mass
- the amount of matter in an object, in kg; the same on every world
- Weight
- the gravitational force on an object, in N
- Gravitational field strength
- the force per kilogram of mass, in N/kg; 9.8 N/kg at the Earth's surface
- Orbital period
- the time for one complete orbit, in s
Equations
| Equation | What it says, symbols and units |
|---|---|
g ≈ 9.8 m/s2 | Near the Earth's surface, an object in free fall accelerates at about 9.8 m/s² if air resistance is small.
|
W = mg | Weight is the gravitational force on a mass; on Earth g is about 9.8 N/kg.
|
v = 2πrT | Average orbital speed is the distance round the orbit divided by the time for one orbit.
|
Diagram to know
Newton's cannon: the Earth with a mountain and three paths fired sideways from it. A slow shot lands nearby, a faster one further round, and the fastest goes all the way round as a circular orbit. Gravity arrows point to the Earth's centre.
Worked example
A satellite orbits 7000 km from the centre of the Earth and goes once round every 97 minutes. Calculate its orbital speed in km/s.
- Distance once round = 2πr = 2π × 7000 = 43 982 km.
- T = 97 × 60 = 5820 s.
- v = 2πr ÷ T = 43 982 ÷ 5820 = 7.6 km/s.
Answer: 7.6 km/s
Common mistakes
- Giving weight in kilograms. Weight is a force, in newtons.
- Measuring the orbit radius from the ground instead of from the centre of the planet: add the planet's radius to the height.
- Saying there is no gravity in orbit. Gravity is what keeps the satellite going round.
Exam tip
In v = 2πr ÷ T, convert the period to seconds first. If r is in km the answer is in km/s; if r is in m it is in m/s. Say which.
AP Physics and NGSS: AP Physics 1 treats free fall with the full constant-acceleration equations and signed velocities, and explains the orbit as the gravitational force supplying the centripetal force, mv²/r.
2 Weight, Scales and Weightlessness
What you need to know
Balance or Scale?
- A balance compares masses, so it reads the same on the Moon as on the Earth. A newton-meter (spring scale) measures weight, which changes with g.
- On the Moon g is about 1.6 N/kg, so a rock keeps its mass but weighs about a sixth of its Earth weight.
The Lift
- A bathroom scale reads the push it gives you, the normal force N. In a lift, N − mg = ma, so N = m(g + a) with up positive.
- Speeding up going up (or slowing down going down) the scale reads more than your weight; speeding up going down it reads less; at a steady speed it reads your weight.
Weightless, Not Gravity-Free
- Astronauts on the Space Station are weightless, not gravity-free: at 400 km up g is still about 8.7 N/kg.
- They float because they and the station fall together, so nothing pushes on them. Their weight is still mg.
Two Kinds of Mass
- Inertial mass (how hard an object is to accelerate, from F = ma) and gravitational mass (how strongly gravity pulls it, from W = mg) are always equal. That is why everything falls with the same g.
Key terms
- Balance
- an instrument that compares masses; it reads the same anywhere
- Newton-meter
- a spring scale that measures a force, such as weight, in N
- Normal force
- the push of a surface at right angles to it; what a bathroom scale reads
- Apparent weightlessness
- the feeling of no weight when you and your surroundings fall together
Equations
| Equation | What it says, symbols and units |
|---|---|
W = mg | Weight is the gravitational force on a mass; on Earth g is about 9.8 N/kg.
|
g = Wm | Gravitational field strength is the force on each kilogram, and it equals the acceleration of free fall.
|
N = m(g + a) | The scale reading in a lift with acceleration a, taking up as positive.
Comes from F = ma; AP Physics 1 uses it directly. |
Diagram to know
A free-body diagram of a person in a lift: weight mg downwards and the normal force N upwards from the floor. Draw N longer than mg when the acceleration is upwards.
Worked example
A 70 kg student stands on a bathroom scale in a lift. Find the scale reading when the lift (a) is still, (b) speeds up upwards at 2.0 m/s², (c) slows down while going up at 2.0 m/s².
- (a) Still: N = mg = 70 × 9.8 = 686 N.
- (b) Up is positive, a = +2.0 m/s²: N = 70 × (9.8 + 2.0) = 826 N.
- (c) Slowing while going up means the acceleration is downwards, a = −2.0 m/s²: N = 70 × (9.8 − 2.0) = 546 N.
Answer: (a) 686 N (b) 826 N (c) 546 N
Common mistakes
- Saying astronauts float because there is no gravity in space.
- Using the direction of motion, not the direction of the acceleration, to decide whether the scale reads more or less.
- Saying a balance would give a different reading on the Moon. It compares masses, so it reads the same.
Exam tip
When a question gives a weight on Earth and asks for the weight on another world, find the mass first (m = W ÷ 9.8), then multiply by the new g. Mass is the one thing that does not change.
AP Physics and NGSS: AP Physics 1 uses apparent weight in lifts and on rides as a standard Newton's second law problem, and states the equivalence of inertial and gravitational mass.
3 Newton's Law of Gravitation
What you need to know
Gravity and Distance
- Every mass attracts every other mass. The pull follows an inverse-square law: twice as far from the centre gives a quarter of the pull, three times as far a ninth.
- Near the Earth, g at a distance r from the centre is 9.8 × (R ÷ r)², where R is the Earth's radius. The Moon, about 60 Earth radii away, feels 60² = 3600 times weaker pull per kilogram.
A Planet's Pull
- A planet's surface field strength grows with its mass and falls with the square of its radius. In Earth units, g = 9.8 × M ÷ R².
- On another world your mass stays the same; your weight changes with g.
Orbital Speed
- Further from the planet the pull is weaker, so an orbit there is slower and has a longer period.
- The period follows from the speed: T = 2πr ÷ v.
Signal Delay
- Radio signals and light cross space at 3.0 × 108 m/s, so the delay is t = d ÷ v. An echo, or a message and its reply, makes the trip twice.
Key terms
- Inverse-square law
- a quantity that falls as 1/r²: double the distance, a quarter of the effect
- Gravitational field
- the region round a mass where another mass feels a force
- Field strength
- the force on each kilogram at a point in the field, in N/kg
- Signal delay
- the time a radio signal takes to cross a distance, d ÷ c
Equations
| Equation | What it says, symbols and units |
|---|---|
Fg = Gm1m2r2 | Every two masses attract; double the distance and the force is a quarter. G = 6.67 × 10⁻¹¹ N m²/kg².
|
g = Wm | Gravitational field strength is the force on each kilogram, and it equals the acceleration of free fall.
|
v = 2πrT | Average orbital speed is the distance round the orbit divided by the time for one orbit.
|
g = 9.8 × MR2 | Surface field strength of a planet, with its mass and radius given in Earth units.
|
v = xt | Wave speed from how far a wave travels in a measured time, as with an echo.
|
Diagram to know
The Earth with field lines pointing radially inwards from all directions; they are closer together near the surface, where the field is stronger.
Worked example
Mars has 0.107 times the Earth's mass and 0.532 times its radius. Calculate the gravitational field strength at its surface and the weight there of a 60 kg astronaut.
- R² = 0.532² = 0.283.
- g = 9.8 × 0.107 ÷ 0.283 = 3.7 N/kg.
- W = mg = 60 × 3.7 = 222 N (her mass is still 60 kg).
Answer: 3.7 N/kg; 222 N
Common mistakes
- Halving the pull when the distance doubles. It falls to a quarter.
- Measuring distance from the surface instead of from the centre of the planet.
- Forgetting to double the distance for an echo or a there-and-back message.
Exam tip
For an inverse-square question, first write the distance as a multiple of the starting distance ("3 times as far from the centre"), then divide by that number squared. It is quicker and safer than putting values into F = Gm₁m₂/r².
AP Physics and NGSS: AP Physics 1 uses Newton's law of gravitation, F = Gm₁m₂/r² with G = 6.67 × 10−11 N m²/kg², and derives the orbital speed v = √(GM/r). NGSS HS-PS2-4 asks you to use the law to predict forces between objects.
4 Earth, Moon and the Solar System
What you need to know
Day, Night and the Sun's Path
- The Earth spins once on its axis in about 24 hours, from west to east. That gives day and night, and makes the Sun appear to rise in the east and set in the west, moving about 15° across the sky each hour.
Why We Have Seasons
- The Earth orbits the Sun in about 365 days, with its axis tilted at 23.4°. The tilt, not the distance from the Sun, causes the seasons.
- In summer your hemisphere leans towards the Sun: the Sun is higher in the sky, its light is more concentrated and the days are longer.
The Moon's Phases
- The Moon orbits the Earth in about a month. Half of it is always lit by the Sun; its phase is how much of that lit half we see from Earth.
- Phases are not caused by the Earth's shadow; that only happens in a lunar eclipse.
The Sun's Family
- The Solar System is the Sun with, in order, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune, plus dwarf planets such as Pluto, asteroids, moons and comets.
- Gravity pulled a spinning cloud of gas and dust into a disc. Small, dense rocky planets formed close to the Sun; the gas giants formed further out, where it was cold enough for ice.
Earth-Centred or Sun-Centred?
- Scientific models change with evidence: Ptolemy's Earth-centred model gave way to Copernicus's Sun-centred one. Galileo's observation that Venus shows a full set of phases fits only the Sun-centred model.
Key terms
- Axis
- the imaginary line the Earth spins about, tilted at 23.4°
- Rotation
- spinning on an axis; the Earth's takes about 24 hours
- Orbit
- the path of one body round another; the Earth's round the Sun takes about 365 days
- Phase of the Moon
- the shape of the sunlit part of the Moon seen from Earth
- Dwarf planet
- a body orbiting the Sun that is round but has not cleared its orbit, such as Pluto
Equations
| Equation | What it says, symbols and units |
|---|---|
ρ = mV | Density is the mass of each cubic metre of a material; a solid less dense than a liquid floats in it.
|
Diagram to know
The Earth at two points of its orbit on opposite sides of the Sun, with its axis tilted the same way at both. The hemisphere leaning towards the Sun has summer; mark the Sun's rays hitting it more directly.
Worked example
Jupiter has a mass of 1.90 × 1027 kg and a radius of 6.99 × 107 m. Calculate its average density and compare it with the Earth's, 5500 kg/m³.
- V = ⁴⁄₃πr³ = ⁴⁄₃ × π × (6.99 × 107)³ = 1.43 × 1024 m³.
- ρ = m ÷ V = 1.90 × 1027 ÷ 1.43 × 1024 = 1330 kg/m³ (to 3 s.f.).
- 1330 ÷ 5500 = 0.24: Jupiter is about a quarter as dense as the Earth, because it is mostly hydrogen and helium.
Answer: About 1330 kg/m³, roughly a quarter of the Earth's density
Common mistakes
- Saying summer happens because the Earth is closer to the Sun. The Earth is actually closest in January.
- Saying the Moon's phases are caused by the Earth's shadow.
- Mixing up rotation (a day) with orbit (a year).
Exam tip
For a seasons question, use the three linked points examiners look for: the axis is tilted, so in summer the Sun is higher in the sky, so the same sunlight falls on a smaller area and the days are longer.
AP Physics and NGSS: NGSS MS-ESS1-1 asks for a model of the Earth–Sun–Moon system that explains phases, eclipses and seasons. This lesson is not part of AP Physics 1.
5 Kepler's Laws
What you need to know
Elliptical Orbits
- The Sun holds 99.8% of the Solar System's mass, and its gravity keeps every planet in orbit.
- Planetary orbits are ellipses with the Sun at one focus, not at the centre (Kepler's first law). The Earth's orbit is almost a circle, which is why the distance does not cause the seasons.
Kepler's Third Law
- Kepler's third law: T² ∝ r³. With T in years and the orbit's size a in astronomical units (AU), T² = a³ for planets of the Sun.
- So a planet further out has a longer year and moves more slowly. The inner four planets are small and rocky; the outer four are giants.
Comets
- A comet has a very elongated elliptical orbit. It moves fastest when closest to the Sun and slowest when furthest away (Kepler's second law: equal areas in equal times).
- Falling inwards, gravitational potential energy turns into kinetic energy; climbing out, it turns back. The total stays the same.
Key terms
- Ellipse
- a stretched circle; an orbit's shape, with the Sun at one focus
- Astronomical unit (AU)
- the average distance from the Earth to the Sun, about 1.5 × 1011 m
- Kepler's third law
- the square of the period is proportional to the cube of the orbit's size
- Comet
- a small icy body on a highly elliptical orbit round the Sun
Equations
| Equation | What it says, symbols and units |
|---|---|
T2 ∝ r3 | For bodies orbiting the same star or planet, the further out, the longer the year.
With T in years and a in AU, for planets of the Sun: T² = a³. |
v = 2πrT | Average orbital speed is the distance round the orbit divided by the time for one orbit.
|
Diagram to know
An elongated ellipse with the Sun at one focus, near one end. Mark the comet's closest point (fastest) and furthest point (slowest), and two equal-area slices swept out in equal times.
Worked example
Mars orbits the Sun at an average distance of 1.52 AU. Use T² = a³ to find its year in Earth years and in days.
- a³ = 1.52³ = 3.51.
- T = √3.51 = 1.87 years.
- In days: 1.87398 × 365 = 684 days, close to the measured 687 days.
Answer: 1.87 Earth years, about 684 days
Common mistakes
- Putting the Sun at the centre of an elliptical orbit. It is at one focus.
- Saying a comet moves at a steady speed. It speeds up near the Sun.
- Using T² = a³ with days and kilometres. It only works in years and AU for the Sun's planets.
Exam tip
When asked why a comet's speed changes, answer in energy terms: "as it moves closer to the Sun, gravitational potential energy is transferred to kinetic energy, so it speeds up". That sentence carries the marks.
AP Physics and NGSS: AP Physics 1 links Kepler's second law to conservation of angular momentum (at the closest and furthest points, vr is the same) and derives the third law from gravity providing the centripetal force.
6 Satellites
What you need to know
Moons and Machines
- Moons are natural satellites; artificial satellites are machines placed in orbit. Both obey the same gravity.
- At one orbit radius every satellite, heavy or light, has the same speed and the same period. Low satellites slowly lose height through drag from the thin upper atmosphere.
Geostationary or Polar?
- A geostationary satellite orbits above the equator, in the direction the Earth spins, once every 24 hours, about 42 000 km from the Earth's centre (about 36 000 km up). It stays over one spot, so it is used for communications and TV.
- A low polar orbit takes about 100 minutes and passes over the poles; the Earth turns beneath it, so it can scan the whole surface for weather, mapping and monitoring.
Weigh a Planet
- Gravity provides the centripetal force on a moon or satellite, so measuring its orbit radius r and period T gives the planet's mass: M = 4π²r³ ÷ (GT²). The moon's own mass cancels out.
Key terms
- Natural satellite
- a moon orbiting a planet
- Artificial satellite
- a machine placed in orbit by people
- Geostationary orbit
- an equatorial orbit with a 24-hour period, so the satellite stays above one point
- Polar orbit
- a low orbit passing over both poles, scanning the whole Earth
Equations
| Equation | What it says, symbols and units |
|---|---|
v = 2πrT | Average orbital speed is the distance round the orbit divided by the time for one orbit.
|
v = √(GMr) | Speed of a circular orbit; it does not depend on the mass of the orbiting body.
|
M = 4π2r3GT2 | The mass of a planet from the orbit of one of its moons or satellites.
AP Physics 1. IGCSE and GCSE need only that higher orbits are slower. |
Diagram to know
The Earth with two orbits: a large circle in the equatorial plane labelled geostationary (24 h) and a small circle passing over both poles labelled low polar (about 100 min).
Worked example
A geostationary satellite orbits 42 200 km from the Earth's centre. The Earth's radius is 6371 km. Find its orbital speed and its height above the ground.
- T = 24 h = 24 × 3600 = 86 400 s.
- Once round: 2πr = 2π × 42 200 = 265 150 km.
- v = 265 150 ÷ 86 400 = 3.07 km/s.
- Height = 42 200 − 6371 = 35 829 km, about 36 000 km.
Answer: 3.07 km/s; about 36 000 km above the ground
Common mistakes
- Saying a geostationary satellite is not moving. It moves at about 3 km/s; it keeps pace with the Earth's spin.
- Saying a heavier satellite needs a different speed at the same height. Mass does not matter.
- Placing a geostationary satellite over the poles. It must be above the equator.
Exam tip
When asked to compare two orbits, give the reason as well as the result: "higher orbit, weaker gravitational field, so lower speed and longer period".
AP Physics and NGSS: AP Physics 1 sets GMm/r² equal to mv²/r to find orbital speeds and planetary masses; NGSS HS-ESS1-4 uses the same mathematics for orbiting objects in the Solar System.
7 Gravitational Energy
What you need to know
Weighing the Pull Between Two Balls
- Every pair of masses attracts: F = Gm₁m₂ ÷ r². G = 6.67 × 10−11 N m²/kg² is so small that the pull between everyday objects is tiny.
- Cavendish measured it in 1798 with lead balls and a thin wire; knowing G and g let him find the mass of the Earth.
Beyond mgh
- mgh only works close to the surface, where g is almost constant. Far from the Earth, the gravitational potential energy is U = −GMm ÷ r, with r from the Earth's centre.
- U is zero far away and negative closer in. Each kilometre further out costs less energy than the one before.
Orbit Energy and Escape
- In a circular orbit the total energy is E = −GMm ÷ (2r). A higher orbit has more (less negative) total energy but a lower speed.
- Escape speed is the launch speed that gives just enough kinetic energy to reach zero total energy: v = √(2GM ÷ r), about 11.2 km/s from the Earth's surface.
Key terms
- Gravitational constant G
- 6.67 × 10−11 N m²/kg², the constant in Newton's law of gravitation
- Gravitational potential energy
- energy due to position in a gravitational field; −GMm/r far from the surface
- Escape speed
- the minimum speed for an object to leave a planet's gravity for good, ignoring air
- Bound orbit
- an orbit with negative total energy; the object cannot escape
Equations
| Equation | What it says, symbols and units |
|---|---|
Fg = Gm1m2r2 | Every two masses attract; double the distance and the force is a quarter. G = 6.67 × 10⁻¹¹ N m²/kg².
|
U = −GMmr | Gravitational potential energy at distance r from the centre of a mass M; zero far away.
AP Physics 1 (Unit 3). |
vesc = √(2GM/r) | The speed needed at distance r to escape completely, ignoring air resistance.
|
Worked example
Calculate the escape speed from the Earth's surface. G = 6.67 × 10−11 N m²/kg², the Earth's mass is 5.97 × 1024 kg and its radius is 6.37 × 106 m (AP Physics 1).
- GM = 6.67 × 10−11 × 5.97 × 1024 = 3.98 × 1014 m³/s².
- 2GM ÷ r = 2 × 3.98 × 1014 ÷ 6.37 × 106 = 1.25 × 108 m²/s².
- v = √(1.25 × 108) = 1.12 × 104 m/s = 11.2 km/s.
Answer: 1.12 × 104 m/s (11.2 km/s)
Common mistakes
- Using r as the height above the surface in U = −GMm/r. It is the distance from the centre.
- Using mgh for a rocket going thousands of kilometres up. g gets smaller with height.
- Thinking escape speed depends on the mass of the rocket. The rocket's mass cancels.
Exam tip
Before you calculate, check which r the question gives you: "altitude" or "height" means above the surface, so add the planet's radius before using any 1/r or 1/r² equation.
AP Physics and NGSS: This lesson goes beyond IGCSE and GCSE, which use only mgh. It is AP Physics 1 (Units 2, 3 and 6) and NGSS HS-PS3-2 content.
8 Tides, Roche & Black Holes
What you need to know
Tides
- A tide is a difference in pull: the Moon pulls the near side of the Earth more strongly than the centre, and the far side less. The oceans form two bulges, so most places have two high tides a day.
- The pull itself falls as 1/r², but the tidal stretch falls as 1/r³: halve the distance and the tides are 8 times stronger.
The Roche Limit
- Close enough to a planet, the tidal stretch is stronger than the gravity holding a moon together, and the moon breaks up into a ring. That distance is the Roche limit.
- For a loose rubble moon with the same density as the planet it is about 2.44 planet radii from the planet's centre; a solid moon, held by its own strength, can come closer (about 1.26 radii).
Black Holes
- If a mass is crushed until the escape speed from its surface reaches the speed of light, it becomes a black hole. Its event horizon has radius r = 2GM ÷ c², about 3 km for each Sun's worth of mass.
- Far away, a black hole pulls like any other mass: the same mass at the same distance gives the same orbit.
Key terms
- Tidal force
- the difference in gravitational pull across an object; it stretches the object
- Roche limit
- the closest a moon can orbit before tides tear it apart
- Black hole
- a mass so compact that nothing, not even light, can escape from inside its event horizon
- Event horizon
- the boundary of a black hole, at the Schwarzschild radius 2GM/c²
Equations
| Equation | What it says, symbols and units |
|---|---|
rs = 2GMc2 | The radius of a black hole's event horizon: where the escape speed equals the speed of light.
|
tidal stretch ∝ Mr3 | The tidal effect of a body of mass M at distance r falls with the cube of the distance. |
Diagram to know
The Earth with the Moon to one side. Draw the ocean as an oval stretched along the Earth–Moon line, with one bulge facing the Moon and one on the far side, and three arrows of decreasing length showing the Moon's pull on the near side, the centre and the far side.
Worked example
Find the radius of the event horizon of a black hole with the Sun's mass, 1.99 × 1030 kg. G = 6.67 × 10−11 N m²/kg² and c = 3.00 × 108 m/s.
- 2GM = 2 × 6.67 × 10−11 × 1.99 × 1030 = 2.65 × 1020.
- c² = (3.00 × 108)² = 9.00 × 1016.
- r = 2.65 × 1020 ÷ 9.00 × 1016 = 2950 m, about 3 km (to 3 s.f.).
Answer: About 2950 m, roughly 3 km
Common mistakes
- Saying there is only one tidal bulge, facing the Moon. There is a second on the far side.
- Saying a black hole sucks in everything around it. Far away it pulls like any other mass.
- Using 1/r² for tides. The tidal stretch falls as 1/r³.
Exam tip
To explain the far-side bulge, compare three pulls: near side, centre and far side. The Earth's centre is pulled towards the Moon more than the far-side water, so that water is left behind as a bulge.
AP Physics and NGSS: Tides, the Roche limit and black holes are beyond IGCSE, GCSE and AP Physics 1; NGSS HS-PS2-4 uses them as applications of Newton's law of gravitation, and AP Physics 1 meets escape speed in Unit 6.
Written in Physiworld's own words. The course tags come from the same mapping as the syllabus maps; equations are on the IGCSE, GCSE, AP Physics 1 and NGSS formula sheets. Values use g = 9.8 N/kg.
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