Archive for the ‘Newton’ Category
towards James Clerk Maxwell 6: Newton’s universal law of gravitation and G
Newton’s law of gravity goes like this:
where F is the force of gravitational attraction, G is the constant of proportionality or gravitational constant, m is an entity, particle or object with a particular mass, and r is the distance between the centres of mass of the two entities, particles or objects.
What’s the relation between all this and Maxwell’s electromagnetic work? Good question – to me, it’s about putting physics on a mathematical footing. Newton set us on this path more than anyone. The task I’ve set myself is to understand all this from the beginning, with little or no mathematical expertise.
The law of gravity, in its un-mathematical form, says that every object of mass attracts every other massive object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
It seems often to be put about that Newton was revolutionary because he was the first to wonder why objects fell to the ground. This is unlikely, and Newton wasn’t the first to infer an inverse square law in relation to such falling. Two Italian experimenters, Francesco Grimaldi and Giovanni Riccioli, investigated the free fall (where no force acts besides gravity) of objects between 1640 and 1650, and noted that the distance of the fall was proportional to the time taken. Galileo had previously conducted free fall experiments and found that objects fell with uniform acceleration – an acceleration that is proportional to the square of the elapsed time. Nor was he the first to find a time-squared relationship. The point of all this is that science doesn’t proceed via revolutions proceeding from one brilliant person (which shouldn’t diminish Galileo or Newton’s genius). The more you find out about it, the more incremental and fascinatingly collaborative and confirmative over time it is.
Galileo used the geometry of his time to present his time squared law, but algebraic notation, invented principally by Descartes, superseded this approach in the seventeenth century.
What about the gravitational constant? This appears to be a long and complicated mathematical story. I think it tries to answer the question – why do objects fall to Earth at such and such a rate of acceleration? But I’m not sure. The rate of acceleration would have been easy enough to measure – it’s approximately 9.8 m/sec2. This rate would appear to be caused by the mass of the Earth. The Moon has a fraction of Earth’s mass, and I believe the gravitational force it exerts is approximately one sixth that of Earth. It has been measured as 1.62 m/sec² (for Mars it’s 3.71).
It’s frustratingly difficult to get an explanation online of what the gravitational constant (G) is or really means – without very quickly getting into complex (for me) mathematics. Tantalisingly, Wikipedia tells us that the aforementioned Grimaldi and Riccioli ‘made [an attempted] calculation of the gravitational constant by recording the oscillations of a pendulum’, which means nothing to me. Clearly though, there must be some relationship between G and the mass of the Earth, though how this can be ascertained via pendulums is beyond me. Anyway, on with the struggle.
We do have a number for G, or ‘Big G’, as it’s called (explanation to come), and it’s a very very small number, indicating that, considering that the multiplied masses divided by the square of the distance between them then get multiplied by G, gravitation is mostly a very small force, and only comes into play when we’re talking about Big Stuff, like stars and planets, and presumably whole galaxies. Anyway here’s the actual number:
G = 0.0000000000667408, or 6.67408 × 10-11
I got the number from this useful video, though of course it’s easily available on the net. Now, my guess is that this ‘Big G’ is specific to the mass of the Earth, whereas small g is variable depending on which mass you’re referring to. In other words, G is one of the set of numbers in g. We’ll see if that’s true.
Now, looking again at the original equation, F stands for force, measured in newtons, m for mass, measured in kilograms, and and r for distance in metres (these are the SI units for mass and distance). The above-mentioned video ‘explains’ that the newtons on one side of the equation are not equivalent to the metres and kilograms squared on the other side, and G is introduced to somehow get newtons onto both sides of the equation. This has thrown me into confusion again. The video goes on to explain how G was used by Einstein in relativity and by Max Planck to calculate the Planck length (the smallest possible measure of length). Eek, I’m hoping I’m just experiencing the storm before the calm of comprehension.
So, to persist. This G value above isn’t, and apparently cannot be, precise. That number is ‘the average of the upper and lower limit’, so it has an uncertainty of plus or minus 0.00031 x 10-11, which is apparently a seriously high level of uncertainty for physicists. The reason for this uncertainty, apparently, is that gravitational attraction is everywhere, existing between every particle of mass, so there’s a signal/noise problem in trying to isolate any two particles from all the others. It also can’t be calculated precisely through indirect relation to the other forces (electromagnetism, the strong nuclear force and the weak nuclear force), because no relationship, or compatibility, has been found between gravity and those other three forces.
The video ends frustratingly, but providing me with a touch of enlightenment. G is described as a ‘fundamental value’, which means we don’t know why it has the value it does. It is just a value ‘found experimentally’. This at least tells me it has nothing to do with the mass of the Earth, and I was quite wrong about Big G and small g – it’s the other way round, which makes sense, Big G being the universal gravitational constant, small g pertaining to the Earth’s gravitational force-field.
Newton himself didn’t try to measure G, but this quote from Wikipedia is sort of informative:
In the Principia, Newton considered the possibility of measuring gravity’s strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he estimated the order of magnitude of the constant when he surmised that “the mean density of the earth might be five or six times as great as the density of water”
Pendulums again. I don’t quite get it, but the reference to the density of the Earth, which of course relates to its mass, means that the mass of the Earth comes back into question when considering this constant. The struggle continues.
I’ll finish by considering a famous experiment conducted in 1798 by arguably the most eccentric scientist in history, the brilliant Henry Cavendish (hugely admired, by the way, by Maxwell). I’m hoping it will further enlighten me. For Cavendish’s eccentricities, go to any online biography, but I’ll just focus here on the experiment. First, here’s a simplification of Newton’s law: F = GMm/R2, in which M is the larger mass (e.g. the Earth), and m the smaller mass, e.g a person. What Cavendish was trying to ascertain was nothing less than the mass and density of the Earth. In doing so, he came very close – within 1% – of the value for G. Essentially, all that has followed are minor adjustments to that value.
The essential item in Cavendish’s experiment was a torsion balance, a wooden bar suspended horizontally at its centre by a wire or length of fibre. The experimental design was that of a colleague, John Michell, who died before carrying out the experiment. Two small lead balls were suspended, one from each end of the bar. Two larger lead balls were suspended separately at a specific distance – about 23cms – from the smaller balls. The idea was to measure the faint gravitational attraction between the smaller balls and the larger ones.
Wikipedia does a far better job than I could in explaining the process:
The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the Earth to be calculated, using Newton’s law of gravitation.
To fully understand this, I’d have to understand more about torque, and how it’s measured. Clearly this weak interaction is too small to be measured directly – the key is in the torque. Unfortunately I’m still a way from fully comprehending this experiment, and so much else, but I will persist.
References
https://en.wikipedia.org/wiki/Newton’s_law_of_universal_gravitation
https://en.wikipedia.org/wiki/Gravitational_constant
https://energyeducation.ca/encyclopedia/Gravitational_constant
https://en.wikipedia.org/wiki/Cavendish_experiment
https://www.school-for-champions.com/science/gravitation_cavendish_experiment.htm#.XSrCrS3L1QI
Go to youtube for a number of useful videos on the gravitational constant
towards James Clerk Maxwell 4: a detour into dimensional analysis and Newton’s laws
Canto: Getting back to J C Maxwell, I’m trying to learn some basic physics, which may or may not be relevant to electromagnetism, but which may help me to get in the zone, so to speak.
Jacinta: Yes, we’re both trying to brush up on physics terms and calculations. For example, acceleration is change in velocity over time, which is hard to put in notation form in a blog post, but I can steal it from elsewhere
in which the triangle represents ‘change in’. Now velocity is a vector quantity, therefore so is acceleration – it’s a particular magnitude in a particular direction. So imagine a car that goes from stationary to, say 50 kms/hour in 5 seconds, what’s the acceleration? According to the formula, it’s 50 – 0 kph/5 seconds, or 10kph/sec, which we can write out as a change of velocity of ten kilometres per hour per second.
Canto: So every second, the velocity of the car is increasing by 10 kilometres per hour. I’m trying to picture that. It’s quite hard.
Jacinta: Okay while you’re doing that, let’s introduce dimensional analysis, so that we reduce everything to the same dimension, sort of. I mean, we have hours and seconds here, so let’s take it all to seconds. I won’t be able to do this properly without an equation-writing plug-in, which I can’t work out how to get without paying. Anyhow..
10 kms/hour.second.1/3600 hour/second. Cancelling out the hours, you get 10 kms/3600 seconds squared, or 1/360 km/s2
Canto: I wonder if there’s a way of hand-writing equations in the blog, that’d be more fun and easy. So can you briefly explain dimensional analysis?
Jacinta: Well physical quantities are often measured in different units – for example, quantities of time – time is called the base quantity – are measured in seconds, hours, days etc. So, it’s just a matter of getting such measurements to be commensurate, so that an equation can be simplified – all in seconds, or all in metres, when they can be. Though actually it’s more complicated than that, and I’ve probably got it wrong.
Canto: So talking of brushing up on stuff, or actually knowing about stuff for the first time, I thought it might be good to go back to Newton, his three laws of motion, in written and mathematical form.
Jacinta: Go ahead.
Canto: Well, the first law, which really comes from Galileo, is often called the law of inertia. Newton formulated it this way, in the Principia (translated from Latin):
Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.
And as Sal Khan and others point out, Newton is talking about an unbalanced force, one that isn’t matched by an equal and opposite force (which would be a balanced force – see Newton’s third law). This law doesn’t come with a mathematical formula.
The second law, which I filched from The Physics Classroom, can be stated thus:
The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
It’s famous formula is this:
Fnet = m • a
It can be written different ways, for example simply F =m.a, or with the vector sign (an arrow) above force (F) and acceleration (a), showing the same direction, but it’s certainly important to explain net force here. It’s essentially the sum of all the forces acting on the mass, in vector or directional terms. It’s this net force that produces the acceleration.
So to the third law, and this is how Newton presented it, again translated from Latin:
To every action there is always an equal and opposite reaction: or the forces of two bodies are always equal and are directed in opposite directions.
It’s often stated in this ‘wise proverb’ sort of way: ‘for every action there’s an equal and opposite reaction’.
Jacinta: What goes around comes around.
Canto: That’s more of a wise-guy thing. Anyway, the best formula for the third law is:
FA = −FB
where force A is the action and force B the reaction. This law is sort of counter-intuitive and also sort of obvious at the same time! I think it’s the most brilliant law. Sal Khan gives a nice extra-terrestrial example of how you might utilise it. Imagine you’re in outer space and you’ve been cut off from your spaceship and are accelerating away from it. To save yourself, take something massive, if you can, something on your suit or a tool you’re carrying, and push it hard away from you in the opposite direction to the ship, and this should send you accelerating back to the ship. But make sure your aim is true!
Jacinta: Okay, so this seems to have taken us absolutely no closer to Maxwell’s equations.
Canto: Well, yes and no. It makes us think of forces and energy, albeit of a different kind, and it makes us think in a logical, semi-mathematical way. but we’ve certainly got a long way to go…
References
https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion
https://www.physicsclassroom.com/class/newtlaws/Lesson-1/Newton-s-First-Law
https://www.livescience.com/46558-laws-of-motion.html
https://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
why are our days getting longer?
I’ve just finished reading a book by the Welsh biologist and science communicator Steve Jones entitled Coral; a pessimist in paradise, which covers a helluva lot of ground and makes me feel inadequate as most science writers do, but one of the many things he has taught me about – something I didn’t know that I didn’t know – is that the days are getting longer, in an inexorable process of rotational slowing. This fact, and the reasons behind it, were further confirmed for me today in an episode of an elegant little podcast out of the University of Houston, called The engines of our ingenuity. I just happened to be browsing through the science and scepticism podcasts on my TV, and I sampled a few curiously titled ones…
Let me backtrack a bit. I’m very very poor (from an affluent western perspective of course) but I received a HD TV from my neighbour recently as part of a complicated deal, and now I can watch free-to-air channels I didn’t have access to before, and what’s more I’ve managed to buy a device which I’m sure many people out there know all about, called an Apple TV, which is so cheap that even I can afford it without too much suffering (what’s a few days without food? it’ll probably extend my lifespan). So now I can explore an almost endless variety of podcasts, vodcasts and classic film noir movies on youtube. That reminds me, one of the podcasts I’ve listened to, the Brain Science Podcast, was all about brain fitness – at least the episode I tuned into was – and inter alia the interviewee informed us that just about the worst thing for the brain was sitting around all day watching TV – Apple or no Apple, presumably…
Anyway I listened to this informative and also charmingly poetic three-minute episode of The engines of our ingenuity, entitled ‘How far the moon?’, narrated and presumably written by Dr John Lienhard. So I’ll share the info, if not the poetry, here.
Our earth spins at a pretty well constant rate because of the forces that set it in motion in the first place and because of Newton’s first law of motion which, put simply, states that an object will stay in the same state (resting or in motion) unless an external force acts on it. A ball spinning in the air will slow down because of air friction, but the earth is spinning in a vacuum, essentially – there’s nothing to slow it down.
Well, not quite. The earth is slowing down, and all in accordance with Newtonian physics. And it’s all due to the moon. Each day is about a twelfth of a second longer than it was when the Egyptians built the pyramids. Doesn’t sound that much, but 4000 years is a mere blip in geological and cosmological time. The moon drags at the earth gravitationally, creating high tides and low tides at a regular rate, and slowing our rate of rotation. But our earth has a much greater influence on the moon than vice versa, the moon having only an eightieth of earth’s mass. This gravitational effect slowed down the moon’s spin until it was in synch with the earth, and locked into the earth’s movement like a dancer being swung around by its partner. And so the moon faces us always. The slowing down of the earth due to the moon’s influence had the effect of loosening the embrace – the moon is slowly moving away from us. Just as a spinning dancer or skater extends her arms out to slow down or pulls her limbs in to speed up. The moon moves away from us so that our combined rotational inertia remains constant. The distance between earth and moon, and the speed at which the moon moves away from us, is being measured thanks to an instrument, placed on the moon by Apollo astronauts, which reflects laser beams from earth. Through measuring the time taken for the beam to return, we know that the moon is moving away from us at a little under 4 cms a year. Back in the dim distant past, days lasted only 12 hours, and the moon was half of today’s distance from us. This has affected the shape of the earth, which is gradually becoming more spherical. The earth’s diameter is at its greatest at the equator and at its smallest at the poles, because of centrifugal forces operating against the force of gravity…
Okay, let me get clearer on this, with the help of this source, among others. Isaac Newton accepted the mathematics and the accuracy of Kepler’s laws of planetary motion, but the great unanswered question was why planets – and moons – traced out these orbits. Newton’s own first law stated that an object will continue in its trajectory (that is, in a straight line) or in its resting state, unless some external force acted upon it to speed it up or slow it down. This state is called a state of inertia. Clearly planets and moons were being acted upon by some force, which could only be exerted by the object being orbited. This force might be called a centripetal force, though that doesn’t explain it in this case. If you swing a stone around on the end of a string, you apply a force to the stone to keep it going, but the string, and your hand holding the string, exerts a force on the string to keep it ‘in orbit’. Its motion will be circular, providing you keep your hand still, because the length of the string is constant. But there’s nothing obvious attaching the moon to the earth. Newton pondered this for some time, until one day the apple dropped.
I’m thinking that, if the moon is moving away from us, its orbit can’t be entirely circular, it must be spiralling outwards, ever so slightly. In any case, the moon pulls the earth out of shape, and that is due to a centrifugal force that balances the centripetal force exerted by the earth on the moon. The moon is moving away due to a reduction in both these forces, and a slowing of the earth’s rotation, and hence of the moon’s orbit.
But sadly, it gets more complicated than that! This is the Newtonian explanation of how these forces operate, but it doesn’t really answer the why question. I’m not going to go deeply into that here – as if I could – but I’ll end with a quote from an astronomer’s explanation, not so much about the earth’s slowing, but about the moon’s behaviour, in terms of Newtonian and then Einsteinian physics:
First case: – Why does the Moon orbit the Earth? It just does. And you can understand how it does by analyzing the forces on the Moon caused by its orbit and finding the forces pushing in and out are equal.
Second case: – Why does the Moon orbit the Earth? Because the Earth distorts spacetime in the vicinity of the Moon, and causes it to orbit the Earth the way it does and the balance of forces to come out the way it does.
So why do massive objects distort space-time? Apparently they just do?