P4 Gravity and orbits
use Newton's gravity to explain orbits, weightlessness and tides.
Before you begin: P3 · Energy and momentum.
Videos coming soon
Three videos (part 1: gravity and weighing the Earth; part 2: orbits and Kepler's laws; part 3: weightlessness, tides and where Newton breaks) and a worked-example video (the space station's orbit) are made, and will appear here once they're uploaded to YouTube.
Watch first
Notes available, video soon.
Notes
In this chapter
- Use Newton's law of universal gravitation, \(F = Gm_1m_2/r^2\), and the inverse-square law, and say why \(g = GM/R^2\) is the same for every falling object.
- Explain how Cavendish's torsion balance weighed the Earth, and use \(M = gR^2/G\).
- Treat an orbit as a fall that keeps missing; find circular orbital speed \(v = \sqrt{GM/r}\) and the period, and the escape speed \(\sqrt{2GM/R}\).
- State Kepler's three laws and see where each comes from; use \(T^2 \propto a^3\) to compare orbits and to weigh the thing in the middle.
- Explain weightlessness as free fall, and tides as differences in the Moon's (and Sun's) pull.
- Say why Mercury's orbit showed that Newton's gravity is not the last word.
The apple and the Moon
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Draw a sideways velocity and an inward gravitational acceleration at one point of an orbit. Consider a short forward step followed by a small inward deflection.
MAKE THE CONNECTION
Key idea
The force that pulls an apple down is the one that holds the Moon in orbit.
The apple and the Moon
Worked number: the Moon test
If gravity gets weaker with the square of the distance, the pull at the Moon, 60 Earth radii away, should be \(9.81/60^2 = 0.0027\) m/s².
The Moon's own motion (\(a = 4\pi^2 r/T^2\), with \(T = 27.3\) days and \(r = 3.84\times10^8\) m) gives \(0.00272\) m/s². They agree to about 1%.
The apple and the Moon
An inverse-square law is what you get when something spreads over a sphere: at twice the distance, the area is four times as big and each patch gets a quarter.
The apple and the Moon
The apple and the Moon
Myth
"There is no gravity in space." At the height of the space station (about 408 km), gravity is still about 88% of its value on the ground. Gravity gets weaker with distance, but it never stops.
Checkpoint: The apple and the Moon (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: The apple and the Moon (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Universal gravitation
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Set the orbit model's radius multiplier to 2 and then 4. Read the gravitational-acceleration ratio, holding the central mass fixed, and relate it to force on a fixed orbiting mass.
Orbital radius: 1 ×
- Gravitational acceleration
- 1 ×
- Orbital speed
- 1 ×
- Orbital period
- 1 ×
Model assumptions
Fixed central mass. All values are ratios to a reference circular orbit; radius is measured from the centre. Diagram is schematic.
MAKE THE CONNECTION
Key idea
Every mass pulls on every other mass, with a force that falls with the square of the distance between their centres.
Universal gravitation
- \(r\) is the distance between the centres of the two things.
- The force on each is the same size (Newton's third law).
- Double the distance and the force falls to a quarter; double one mass and it doubles.
Universal gravitation
Worked number: two people
Two 70 kg people 1 m apart pull on each other with only \(3.3\times10^{-7}\) N, because \(G\) is tiny: gravity only becomes obvious when one mass is huge.
Universal gravitation
At the surface of a planet of mass \(M\) and radius \(R\), the pull on a mass \(m\) is \(GMm/R^2\), and \(F = ma\) gives \(a = g = GM/R^2\). The \(m\) cancels, so everything falls at the same rate (apart from air). The Moon's \(g\) is \(1.62\) m/s², Mars's \(3.73\) m/s².
Checkpoint: Universal gravitation (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Universal gravitation (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Weighing the Earth
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Rearrange g = GM/R² for M. Keep the radius measured from Earth's centre, attach units to G and verify that the result has units of mass.
MAKE THE CONNECTION
Key idea
Once you know \(G\), the pull of gravity at the surface tells you the mass of the Earth.
Henry Cavendish used a torsion balance (a rod with lead balls hung on a wire) and measured the tiny pull of two big lead balls on two small ones; the twist of the wire gave the force.
- He reported the Earth's density, about \(5.48\) times that of water (the modern value is \(5.51\)), within about 1%.
- With the density and the radius, \(M = \rho\cdot\tfrac43\pi R^3 = 5.97\times10^{24}\) kg; once \(G\) is known, \(M = gR^2/G\) gives the same.
- Since the average is 5.5 and surface rock only 2.7, the inside must be much denser than rock: a metal core.
Checkpoint: Weighing the Earth (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Weighing the Earth (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Orbits
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
In the model, distinguish the tangential velocity arrow from the direction towards the central mass. Draw the inward change between two nearby tangent vectors.
Orbital radius: 1 ×
- Gravitational acceleration
- 1 ×
- Orbital speed
- 1 ×
- Orbital period
- 1 ×
Model assumptions
Fixed central mass. All values are ratios to a reference circular orbit; radius is measured from the centre. Diagram is schematic.
MAKE THE CONNECTION
Key idea
An orbit is a fall that keeps missing.
The Earth's surface curves away about 5 m for every 8 km, and an object falls \(\tfrac12 g t^2 = 4.9\) m in 1 s; so a ball fired sideways at about 8 km/s falls as fast as the ground curves away.
Orbits
Checkpoint: Orbits (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Orbits (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Orbital speed and period
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Increase the circular orbit's radius while monitoring both speed and period. Explain why a lower speed around a longer path means a much longer period.
Orbital radius: 1 ×
- Gravitational acceleration
- 1 ×
- Orbital speed
- 1 ×
- Orbital period
- 1 ×
Model assumptions
Fixed central mass. All values are ratios to a reference circular orbit; radius is measured from the centre. Diagram is schematic.
MAKE THE CONNECTION
Key idea
In a circular orbit gravity is the centripetal force, which fixes the speed: higher orbits are slower.
For a circular orbit of radius \(r\) (from the Earth's centre), gravity supplies the centripetal force:
Orbital speed and period
The orbiting mass cancels, and higher orbits are slower.
Orbital speed and period
Orbital speed and period
Worked numbers: the space station and geostationary orbit
At the space station (\(r = 6779\) km), \(v = 7.67\) km/s and \(T = 92.6\) min. At \(r = 4R\) the speed is half the ground-skimming 7.9 km/s.
A geostationary orbit has \(T\) = one sidereal day (86 164 s): \(r = 42\,164\) km from the Earth's centre, 35 786 km above the equator, so a satellite stays over one spot.
Orbital speed and period
Myth
"An orbiting object has escaped gravity." It hasn't: it is falling all the time. The higher the orbit, the slower it goes.
Checkpoint: Orbital speed and period (1 of 3)
Question 1 of 3 Commit to an answer first, then look.
Checkpoint: Orbital speed and period (2 of 3)
Question 2 of 3 Commit to an answer first, then look.
Checkpoint: Orbital speed and period (3 of 3)
Question 3 of 3 Commit to an answer first, then look.
Escape speed
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Compare ½mv² with GMm/r. Solve for the minimum escape speed, cancel the test mass and divide by the circular-speed expression.
MAKE THE CONNECTION
Key idea
Escape is the speed that leaves nothing left over at infinity: \(\sqrt2\) times the circular orbital speed.
The gravitational potential energy is \(-GMm/r\) (negative, zero at infinity). To reach infinity with nothing left over, \(\tfrac12 mv^2 - GMm/R = 0\):
Escape speed
- From the Earth, 11.2 km/s; the Moon, 2.4; Mars, 5.0; Jupiter's cloud tops, about 60.
- A rocket can keep burning, so it need not reach this speed at launch.
Checkpoint: Escape speed (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Escape speed (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Kepler's three laws
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Sketch a visibly elliptical orbit with the Sun at one focus. Draw equal-area narrow sectors near and far from the Sun and compare their arc lengths.
MAKE THE CONNECTION
Key idea
Planets move on ellipses, sweep equal areas in equal times, and have periods that grow as \(a^{3/2}\).
- Ellipses. Planets move on ellipses with the Sun at one focus (not the middle). An ellipse has semi-major axis \(a\) and eccentricity \(e\) (0 for a circle). The Earth's \(e\) is only 0.017.
- Equal areas. A line from the Sun to a planet sweeps equal areas in equal times, so a planet moves faster near the Sun. At the closest and farthest points, \(r_1v_1 = r_2v_2\) (this is conservation of angular momentum: see P5).
Kepler's three laws
Kepler's three laws
- Periods. \(T^2 \propto a^3\). With \(T\) in years and \(a\) in AU, \(T^2 = a^3\): Mars at 1.524 AU has \(T = \sqrt{1.524^3} = 1.88\) years (687 days).
Kepler's three laws
Kepler's three laws
Myth
"Summer is warm because the Earth is closer to the Sun." In the northern hemisphere the Earth is closest to the Sun in early January. The seasons come from the tilt of the Earth's axis.
Checkpoint: Kepler's three laws (1 of 3)
Question 1 of 3 Commit to an answer first, then look.
Checkpoint: Kepler's three laws (2 of 3)
Question 2 of 3 Commit to an answer first, then look.
Checkpoint: Kepler's three laws (3 of 3)
Question 3 of 3 Commit to an answer first, then look.
Weighing the Sun
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Rearrange T² = 4π²a³/(GM) for M. Identify which measurements an astronomer needs and check how using days instead of seconds would affect a calculation with SI G.
MAKE THE CONNECTION
Key idea
Newton's version of the third law, \(T^2 = 4\pi^2a^3/(GM)\), weighs whatever sits in the middle of an orbit.
Newton's circular-orbit argument gives \(4\pi^2a^2/T^2 = GM/a\), i.e. \(T^2 = \dfrac{4\pi^2}{GM}a^3\).
Weighing the Sun
Worked number: weighing the Sun
Rearranged, it weighs the thing in the middle: \(M = 4\pi^2a^3/(GT^2)\), which gives the Sun as \(1.99\times10^{30}\) kg (333 000 Earths).
Checkpoint: Weighing the Sun (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Weighing the Sun (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Weightlessness
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Draw force diagrams for an astronaut standing on Earth and freely orbiting. Remove the support force in the second diagram while retaining gravity.
MAKE THE CONNECTION
Key idea
Weightlessness is free fall, not an absence of gravity.
What a bathroom scale reads is the normal force, not the pull of gravity.
Weightlessness
Worked numbers: the lift
In a lift accelerating upwards at \(a\), \(N = m(g+a)\): for a 70 kg person at 2 m/s², 827 N instead of 687 N.
If the lift falls freely, \(N = m(g-g) = 0\).
Weightlessness
The space station is falling around the Earth, and so is everything inside it, so nothing pushes on anything: free fall, not "no gravity". (Astronomers say microgravity, since the weightlessness is not quite perfect.) The same cancellation, the equality of the \(m\) in \(F = ma\) and the \(m\) in the law of gravity, was the seed of Einstein's general relativity.
Checkpoint: Weightlessness (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Weightlessness (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Tides
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
Draw the Moon's pull at Earth's near side, centre and far side. Subtract the centre's acceleration from each side and inspect the relative directions.
MAKE THE CONNECTION
Key idea
Tides come from the difference in the Moon's pull across the Earth, not from the pull itself.
The Moon's pull is not the same all over the Earth: stronger on the near side, weaker on the far side. Measured against the pull on the Earth's centre, the difference is a stretching, about \(2GMR/d^3\), which is tiny (about \(10^{-7}\) of \(g\)), but the sideways part pushes water along the surface towards the Earth-Moon line, making two bulges: a high tide on both the near and the far side.
- A high tide comes every 12 h 25 min, because the Moon moves on while the Earth turns.
- Tides go as \(M/d^3\): the Sun's tide is only 0.46 of the Moon's, even though its pull is far larger, because it is so far away.
- At new and full Moon the two add (spring tides); at the quarters they partly cancel (neap tides), a ratio of \(1.46/0.54 \approx 2.7\).
Checkpoint: Tides (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Tides (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Where Newton breaks
FIRST, MAKE A PREDICTION
TRY IT / FOLLOW THE REASONING
List three possible sources of an orbital mismatch and an observation that would test each. Then compare the roles of Neptune's discovery and Mercury's remaining precession.
MAKE THE CONNECTION
Key idea
A tiny error in a prediction can point to a new planet, or a new theory.
Newton's law predicted Uranus wrongly, and Le Verrier calculated where an unseen planet should be (Neptune, found in 1846). Mercury's orbit also precesses, by an extra 43 arcseconds per century that Newton's law (with the other planets' pulls) cannot explain; a proposed planet "Vulcan" was never confirmed. In 1915, Einstein's general relativity gave exactly 43.
Checkpoint: Where Newton breaks (1 of 2)
Question 1 of 2 Commit to an answer first, then look.
Checkpoint: Where Newton breaks (2 of 2)
Question 2 of 2 Commit to an answer first, then look.
Key terms
Key terms
- Inverse-square law
- A force or intensity that falls to a quarter when the distance doubles, \(\propto 1/r^2\).
- Universal gravitation
- Every pair of masses attracts with \(F = Gm_1m_2/r^2\).
- Gravitational constant (\(G\))
- \(6.674\times10^{-11}\) N m²/kg², the strength of gravity.
- Torsion balance
- A rod hung on a thin wire; its twist measures a tiny force.
Key terms
- Orbital speed
- \(v = \sqrt{GM/r}\) for a circular orbit of radius \(r\).
- Escape speed
- \(\sqrt{2GM/R}\): the speed from which an object just reaches infinity.
- Ellipse, focus, eccentricity
- The shape of a bound orbit; the Sun sits at a focus; \(e\) measures how stretched it is.
- Perihelion, aphelion
- The closest and farthest points of an orbit around the Sun.
Key terms
- Geostationary orbit
- An orbit of period one day above the equator, so a satellite stays over one place.
- Normal force
- The push of a surface on an object; it is what a scale reads.
- Free fall
- Motion with only gravity acting; it feels weightless.
- Tidal force
- The difference in the pull of a body across an object, which stretches it.
Key terms
- Spring and neap tides
- The largest tides (Sun and Moon in line) and the smallest (at right angles).
History and sources
History
Where a primary text wasn't read, the sources are encyclopedic or institutional pages, and the wording below is cautious.
Newton, the apple and the Moon (1726, about 1666)
Newton, the apple and the Moon (1726, about 1666). William Stukeley recorded in 1752 a conversation with Newton on 15 April 1726 in which Newton said the idea of gravitation was prompted by a falling apple; no source says it hit him. Newton later said that around 1666 he compared the Moon's fall with falling bodies on Earth; the first try came out a little off, and he returned to it in 1684. Sources: Stukeley's Memoirs (Newton Project); Whiteside, Newton's early thoughts on planetary motion (1964).
Hooke, Halley and the Principia (1679–1687)
Hooke, Halley and the Principia (1679–1687). Hooke and Newton exchanged letters in 1679–80 about falling bodies and the force law; Hooke later claimed a share of the inverse-square idea and Newton disputed it. In 1684 Halley travelled to Cambridge to ask what path an inverse-square force would produce; Newton said an ellipse, and went on to write the Principia (1687), which Halley saw through the press and paid to print. Sources: Philosophiae Naturalis Principia Mathematica; MacTutor: Halley.
Cavendish and the Earth (1797–98)
Cavendish and the Earth (1797–98). Henry Cavendish used a torsion balance designed by John Michell, who died before he could use it. "Experiments to determine the Density of the Earth", Philosophical Transactions 88 (1798). The symbol \(G\) was popularised by C. V. Boys in 1894. In 1774 Maskelyne measured how much Schiehallion tugged a plumb line. Sources: Royal Society: Making Science; Cavendish experiment; Schiehallion experiment.
Newton's cannon (1680s, published 1728)
Newton's cannon (1680s, published 1728). In a text drafted in the 1680s and published after his death, Newton imagined a projectile fired sideways from a high mountain: at the right speed it never lands. Source: Newton's cannonball.
Tycho and Kepler (1576–1619)
Tycho and Kepler (1576–1619). Tycho Brahe's naked-eye instruments were good to about one arcminute. Kepler took over his observations of Mars in 1601; his best circular model missed by 8 arcminutes, and he trusted Tycho. Astronomia nova (1609) gave the ellipse and equal areas; the third law, found in 1618, is in Harmonices mundi (1619). Sources: MacTutor: Brahe; MacTutor: Kepler; Astronomia nova.
Halley's comet (1705)
Halley's comet (1705). Using Newton's laws, Halley wrote that the comets of 1531, 1607 and 1682 were one comet and "dare venture to foretell" its return in 1758. He died in 1742; Johann Palitzsch spotted it on Christmas Day 1758. Sources: Halley, A Synopsis of the Astronomy of Comets (Smithsonian Libraries); Halley's Comet.
Sputnik, Gagarin, Voyager (1957–2012)
Sputnik, Gagarin, Voyager (1957–2012). Sputnik 1 was launched on 4 October 1957; Yuri Gagarin made one orbit in about 108 minutes on 12 April 1961; Voyager 1 crossed the heliopause on 25 August 2012. Sources: NASA: Dawn of the space age; NASA: Yuri Gagarin, first human in space; NASA: Voyager 1.
Tides (1609–1687)
Tides (1609–1687). Kepler suggested in 1609 that the Moon raises the tides; Galileo rejected the idea in his 1632 Dialogue, arguing for the Earth's motion; Newton showed in 1687 that the Moon and Sun pull more on the near side than the far side. Sources: Galileo Project: Galileo's theory of the tides; Theory of tides.
Apollo 15 and Einstein (1907, 1971)
Apollo 15 and Einstein (1907, 1971). On 2 August 1971, David Scott dropped a hammer and a feather together on the Moon, and they landed together. In 1907, Einstein realised that a person in free fall would not feel their own weight, "the happiest thought of my life". Sources: NASA: the Apollo 15 hammer-feather drop; Equivalence principle.
Neptune and Vulcan (1846–1915)
Neptune and Vulcan (1846–1915). Le Verrier predicted where an unseen planet should be; Galle and d'Arrest found Neptune within about a degree on 23 September 1846 (John Couch Adams had worked on the same problem). In 1859 Le Verrier found Mercury's orbit turns about 38 arcseconds a century more than Newton allows (later refined to 43); Vulcan, reported by the amateur Lescarbault, was never confirmed. On 18 November 1915 Einstein showed general relativity gives 43. Sources: Discovery of Neptune; MacTutor: Le Verrier; Tests of general relativity.
Worked example: the space station's orbit
The problem
Problem. The International Space Station flies in a nearly circular orbit about 408 km above the ground. How long does one orbit take? Use only \(g = 9.81\) m/s² and the Earth's radius, \(R = 6371\) km (no \(G\), no mass of the Earth).
Solution
- The picture. What counts is the distance from the Earth's centre: \(r = R + h = 6371 + 408 = 6779\) km.
- The trick. For a circular orbit, \(v^2 = GM/r\). At the surface, \(g = GM/R^2\), so \(GM = gR^2\). Then \(v^2 = gR^2/r\).
- Speed. \(v^2 = 9.81 \times (6.371\times10^6)^2 / 6.779\times10^6 = 5.874\times10^7\), so \(v = 7664\) m/s \(= 7.66\) km/s (about 27 600 km/h).
Solution
- Period. \(T = 2\pi r/v = 2\pi \times 6.779\times10^6/7664 = 5558\) s \(= 92.6\) min.
- Check. The real period is about 93 minutes. Orbits per day: \(1440/92.6 = 15.5\), so astronauts see about 16 sunrises a day.
- Stretch. For a period of one sidereal day (86 164 s), \(r^3 = gR^2T^2/4\pi^2\) gives \(r = 42\,150\) km, a height of 35 780 km: the geostationary orbit (35 786 km with the equatorial radius, 6378 km).
Practice
Practice
Use \(g = 9.81\) m/s², \(G = 6.674\times10^{-11}\) N m²/kg², the Earth's mass \(5.972\times10^{24}\) kg and radius 6371 km unless told otherwise.
★ Warm-ups
1.1 What is the strength of gravity 2000 km above the Earth's surface, in m/s²? Then give it as a fraction of the surface value.
Problem 1.2 (★ Warm-ups)
1.2 At what height above the surface is gravity half its value on the ground? Give the height in kilometres.
Problem 1.3 (★ Warm-ups)
1.3 By what factor does the force between two bodies change if the distance between their centres is doubled and one mass is tripled?
Problem 1.4 (★ Warm-ups)
1.4 The Moon takes 27.3 days to circle the Earth at a distance of \(3.84\times10^8\) m. Find its centripetal acceleration \(4\pi^2r/T^2\), in m/s², to repeat the Moon test.
★★ Standard
2.1 Find the escape speed from Mars (\(M = 6.417\times10^{23}\) kg, \(R = 3389.5\) km), in km/s. Then find the Moon's (\(M = 7.342\times10^{22}\) kg, \(R = 1737.4\) km).
Problem 2.2 (★★ Standard)
2.2 A satellite orbits at the height of the space station (\(r = 6779\) km). Find its orbital speed, in km/s. Then find the escape speed from there, and show that their ratio is \(\sqrt2\).
Problem 2.3 (★★ Standard)
2.3 How much kinetic energy does a 1000 kg probe need at the ground to escape the Earth in one push (ignoring air)? Give it in gigajoules (GJ).
Problem 2.4 (★★ Standard)
2.4 A satellite orbits 600 km above the ground. Find its period, in minutes. Then find its speed.
★★★ Stretch
3.1 Jupiter's moon Io orbits at a mean distance of \(4.217\times10^8\) m with a period of 1.769 days. Use Kepler's third law in Newton's form to find Jupiter's mass, in units of \(10^{24}\) kg. Then compare it with the Earth's.
Hint
\(M = 4\pi^2a^3/(GT^2)\). Convert the period to seconds first.
Problem 3.2 (★★★ Stretch)
3.2 Using Jupiter's mass \(1.898\times10^{27}\) kg and cloud-top radius of \(6.991\times10^7\) m, find the strength of gravity there, in m/s². Then find the escape speed.
Problem 3.3 (★★★ Stretch)
3.3 A 60 kg person stands on a scale in a lift. What does the scale read, in newtons, when the lift accelerates upwards at 1.5 m/s²? Then find the reading when it accelerates downwards at 1.5 m/s², and when the cable breaks.
Problem 3.4 (★★★ Stretch)
3.4 The comet Halley has a semi-major axis of 17.8 AU. Use \(T^2 = a^3\) (years and AU) to find its period, in years.
Problem 3.5 (★★★ Stretch)
3.5 Tides go as \(M/d^3\). The Sun has \(2.709\times10^7\) times the Moon's mass and is 389.2 times as far away. How strong is the Sun's tide compared with the Moon's, as a ratio?
Check yourself
Answer these out loud, from memory. Then try again in a few days.
Check yourself
- What is the law of universal gravitation, and why is it called an inverse-square law?
Check yourself
- How did the Moon test show that the apple and the Moon obey the same law?
Check yourself
- Why do all objects fall at the same rate on a given planet?
Check yourself
- How did Cavendish's balance lead to the mass of the Earth?
Check yourself
- Why is an orbit a fall that keeps missing, and what sets the speed of a circular orbit?
Check yourself
- Why does a higher circular orbit go more slowly?
Check yourself
- What is the escape speed, and why is it \(\sqrt2\) times the orbital speed?
Check yourself
- What are Kepler's three laws, and what physics lies behind each?
Check yourself
- How can the period and size of an orbit give the mass of the central body?
Check yourself
- Why are astronauts weightless on the space station, if gravity is still 88% as strong?
Check yourself
- What does a bathroom scale really measure?
Check yourself
- Why does the Moon cause two high tides a day, and why is the Sun's tide smaller than the Moon's?
Check yourself
- What is the difference between spring and neap tides?
Check yourself
- What did Mercury's orbit show about Newton's gravity?
Summary
Summary
KEEP THE THREAD GOING
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Further learning
Free textbook: OpenStax, University Physics Volume 1, Chapter 13: Gravitation (found by search 2026-10-16; section pages for Newton's law, orbits, Kepler's laws and tides).
Free course: MIT OpenCourseWare, 8.01SC Classical Mechanics (found by search 2026-10-16; includes a lesson on the universal law of gravitation and orbital circular motion).
Simulations: PhET: Gravity and Orbits (move the Sun, Earth, Moon and space station and watch the orbits; found by search 2026-10-16) · PhET: Kepler's Laws (found by search 2026-10-16).
Next: P5 · Rotation and spin · Progress