It depends whether you mean real or apparent contact area - because materials aren't smooth when you look at them really closely, they don't contact together nicely. In one respected book on the subject, it is described as taking one mountain range, turning it upside down and putting it onto another - only a few peaks actually touch. For further explanation, see the book "Friction and Lubrication of Solids" by Bowden & Tabor
The real area of contact was usually measured via a precision electrical resistance technique.
The area of contact between two speres with zero load is, as you say, a point. As soon as you apply load, the infinitesmal area of material deforms, and quickly yields, and the contact area increases. This load/area relationship is described by Hertzian contact theory - for a good explanation, see the book "Contact Mechanics" by Johnson.
In practical terms, this means that items like rolling element bearings with point and line contacts work under very high stresses. The maximum shear stress occurs a little way beneath the contact point, and it is here that bearing races begin to fail by fatigue - but, because this is beneath the surface, you can't see it until the crack grows up to the surface, by which time, you can hear it!!
What is the application for this rather interesting question?
Number_Cruncher
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What is the application for this rather interesting question?
I dont know, I just like to think of things like this. I was wondering how much something like a snooker ball deforms, the chance of getting a "kick" from a speck of chalk dust.
Also as you mentioned, friction at a molecular level - how smooth is smooth etc.
Do you get a tiny amount of curvature on a (say) swimming pool full of water?
If you drilled a hole through the planet and jumped down it, would you oscillate from pole to pole, gradually coming to rest at the centre?
sorry it's those dire England world cup performances , make the mind wander.
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What is the application for this rather interesting question? I dont know, I just like to think of things like this. I was wondering how much something like a snooker ball deforms, >>
Ball bearing applications?
Do you get a tiny amount of curvature on a (say) swimming pool full of water? >>
Yes,curvature of the earth in practice, the Humber Bridge main supports are about 1500 metres apart, the tops are about 12" further apart than the bases though both are verticle.
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If you drilled a hole through the planet and jumped down it, would you oscillate from pole to pole, gradually coming to rest at the centre?
Yes, if the planet were solid, allowing a stable hole to be drilled between the poles*, a test particle would indeed oscillate - in fact if there were no atmospheric drag, there would be very little damping, and the oscillation would continue for a very long time.
*I **think** that drilling the hole between the poles would avoid any complications from the Coriolis acceleration, which would otherwise **probably** make the test particle hit the sides of the hole. I need to think about this point some more, I'm not quite sure about it.
In vacuum chambers, there are some physics experiments involving predominantly gravitationally sprung torsional pendulums with ringdown times measured in months.
But, assuming that the oscillation would eventually decay, the test particle would be in free fall, or "weightless" in the centre, as the gravitational attraction from each little part of the planets material would be balanced by an equal attraction from the diamtrically opposite mass.
Number_Cruncher
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Thanks for that N C - lots to think about now. What do you for a living?
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>>What do you for a living?
Thanks for asking. I have been very lucky, and have worked in a wide variety of roles beginning with working in a number of local vehicle workshops. Then I went back to university as a mature! student, paying my way by driving HGVs. With my degree, I worked on an extremely wide variety of projects (bridges, nuclear power plant, aircraft, rail vehicles, submarines) for a company of engineering consultants. After a brief spell in a Dickensian aerospace company, I now work for a university physics department, where, alongside my project duties, I am studying for a Phd.
What do you do?
Number_Cruncher
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Um I used to draw maps, then as things became more computerised, drifted into PC support. I spend a lot of time telling surveyors they've lost all their work and wil have to do it all again.
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But, assuming that the oscillation would eventually decay, the test particle would be in free fall, or "weightless" in the centre, as the gravitational attraction from each little part of the planets material would be balanced by an equal attraction from the diamtrically opposite mass. Number_Cruncher
I suggest that the particle would not stay suspended because the gravitaional field of the earth is ever changing and is influenced by 3rd party bodies such as the moon, which is the cause of tides for instance, hence the partical would be pulled towards the sides of the shaft be cause the gravitational field would vary as the earth rotates.
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If you drilled a hole through the planet and jumped down it, would you oscillate from pole to pole, gradually coming to rest at the centre?
If the planet was The Earth the drill would be burnt to nothing long before it reached the centre.
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L\'escargot.
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If the planet was The Earth the drill would be burnt to nothing long before it reached the centre. --
Hmm - quite right - let's not bother!
However, didn't you mean to ask
If the planet were the earth...
Yes, the graviational attraction of the moon would move the particle - whether the particle hits the wall or not is determined by the size of the hole. As the particle moves away from centre, the imbalance of the earth's attraction will pull it back - at some radius, these two forces will balance. I suspect this radius is quite small because;
a) the uniform gravitational force is proportional to 1/r^2, and the particle is much closer to the earth than any other body
b) being a particle, and having no significant size, the particle does not feel the tidal component of the moon's gravitational field
If part b of my answer seems odd, then ask;
1) why there are two tides per day
2) Although the moon dominates the tides, why don't we orbit the moon? - or, equally, while the sun has a smaller effect on the tides than the moon, we orbit the sun - why isn't the sun's effect on the tides dominant?
Number_Cruncher
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Or if the earth was the size of the moon and vice versa, what would our tides be like?
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Imagine two theoretical perfect spheres. Their contact area would be a point (by definition, infinitely small).
Imagine two lightly inflated footballs being pressed against each other. Ignoring the molecular effects outlined in other answers, their contact area would be a large flat circular plane.
Somewhere between the two would come a couple of touching ball-bearings. There would be some deformation of their contact area, which would result in a circle rather than a point. A bit of maths with the elasticity of the steel would give you an idea of the size of the circle for any given force. Unfortunately, I don't have any tables available for elasticity of different types of steel.
V
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Yes, you're right Vin, that's exactly the Hertzian contact theory I mentioned.
For an idea, a typical railway vehicle has a mass of 40 tonnes - so, assuming an even distribution, 5 tonnes per wheel. The contact patch between them is about the size of a 5p coin.
Unfortunately, my information, while hopefully interesting, isn't scaleable, because Hertzian contact is helplessly non-linear.
By non-linear, I mean that at the point of initial contact, there is negligible stiffness, but as the contact area increases, so, rapidly does the stiffness. This means that you can't scale the effect - double the load doesn't double the area!
Somewhere!, I think I have some code I wrote some years ago which can estimate these contact parameters - I'll check and see if I can find it.
Incidentally Vin, you never need to worry about the elasticity of different types of steel - it all has a Young's modulus of about 210 GPa give or take a few %, and Poisson's ratio, while being an irrelevance for most structural calcs doesn't vary much between steels either.
What varies between qualities of steel is the yield stress and the % elongation before yield. Typical cheap, low grade structural steel has a yield stress of 270MPa. High quality CrMo steels such as are used in the gearboxes of the TGVs have a yield stress of about 1100 MPa, but they both have a near idential Young's modulus.
Number_Cruncher
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Does this mean that a (say) wheel on a railway carriage in motion is a perpetually changing eliptical sort of shape?
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It's circular except for the contact patch, but the contact patch shape depends upon how worn the wheel and rail are.
Number_Cruncher
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<< In practice this cannot be so, but how doyou measure the contact area between two (say) steel spheres.
You smear one of the spheres with an infinitesmally thin layer of Engineers Marking Blue and then rotate one sphere against the other, being careful to make sure that the sphere only rotates and doesn't slide. You then measure the diameter of the Blue that has transferred to the stationary sphere.
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L\'escargot.
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