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towards James Clerk Maxwell 6: Newton’s universal law of gravitation and G

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Newton’s law of gravity goes like this:

{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},}

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the geometry of his time to present his time squared law, but algebraic notation, invented principally by Descartes, superseded this approach in the seventeenth century.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.

the ‘simple’ Michell/Cavendish device for measuring the mass/density of the Earth – Science!

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

Written by stewart henderson

July 14, 2019 at 4:51 pm

towards James Clerk Maxwell 4: a detour into dimensional analysis and Newton’s laws

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nutshell Newton

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

{\displaystyle {\bar {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}.}

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

Khan Academy physics

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

Written by stewart henderson

June 17, 2019 at 9:52 pm