Does a ring contract faster than a disc?

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SUMMARY

The discussion centers on the contraction behavior of a ring compared to a disc made of the same material, particularly in the context of a lock mechanism. It is established that both the ring and disc will contract by the same percentage based on their coefficient of thermal expansion, assuming they are made from identical materials. The geometry of the ring does not inherently allow it to contract faster than the disc; rather, any perceived difference in behavior may arise from mechanical interactions or corrosion within the lock. The mathematical relationships governing the volume and circumference changes of both shapes are also explored.

PREREQUISITES
  • Understanding of thermal expansion and the coefficient of expansion
  • Basic knowledge of geometry related to volume and circumference calculations
  • Familiarity with Poisson's Ratio and its implications in material science
  • Concepts of mechanical interactions in lock mechanisms
NEXT STEPS
  • Research the coefficient of thermal expansion for various materials
  • Learn about the mathematical principles governing volume and circumference changes in geometric shapes
  • Explore the implications of Poisson's Ratio in material deformation
  • Investigate common mechanical issues in lock mechanisms, including corrosion effects
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Engineers, physicists, locksmiths, and anyone interested in the principles of material contraction and mechanical interactions in locking mechanisms.

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Household physics question: Before I left town for 3 weeks the lock on my apartment door was loose in its encasement. I had to hold it in place while turning the key or the inner disc would rotate uselessly inside the outer ring:

http://scott-shepherd.com/share/forums/lock.jpg

When I came back after Thanksgiving the lock was fixed. So, assuming no opportunistic handyman came by, I think it must have tightened as a result of contraction in the cold weather. Does this make sense? Is it a demonstrable principle that a ring contracts faster than the circumference of a disc of the same material?
 
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Well, I am not 100% sure that a disk has more "freedom" to contract than a disk since it is hollow and no matter in the inside to oppose it (even though this matter is also contracting)...but another explanation to your problem is that the two pieces could very well be made of different materials even though they appear very similar.
 
No. If the ring and the disk are made of the same material they will contract by the same percentage which is equal to their coefficient of expansion. The space between them will also shrink. It's possible that something in the lock is now rubbing against the ring and that's why it doesn't spin freely any more.

http://en.wikipedia.org/wiki/Coefficient_of_expansion
 
Yes the volume changes by the same coefficient (if the materials really are the same in both). I was wondering if there was something about the geometry of the ring that might make its inner circumference contract differently than the outer circumference of the disc.

With the disc it seems safe to assume that if its volume shrinks by a proportion P, its radius shrinks by cuberoot(P) and so its circumference by 3√P2. (This presumes that the height and radius contract in equal proportion.)

volume(disc) = hπr2
P * volume = (cuberoot(P)h) * π * (cuberoot(P)r)2

With the ring I'm not so sure. Probably the outer and inner radii contract equally, at the same proportion as the radius of the disc, but one could also effectively decrease the volume by increasing the inner radius and decreasing the outer. Not that this would answer my question, since it would make the lock looser, but it shows that there's more than one way to imagine the shrinkage of the ring...

volume(ring) = hπ(r22-r12)

A related question, probably not applicable to the lock: do solids expand liquidly? Like if a cylinder of an expanding solid material x was surrounded by an adamant material y that was not expanding, would the cylinder "squeeze out", expanding through its height to achieve the volume increase while its radius remained constant?
 
There is a "bit" of "liquidly" as you call it.

Depending on the strength ratio of the inner cylinder respect to the outer cylinder, the inner cylinder will make the outer one expand a bit, too...so, that the inner does not have to lengthen too much...

...but, yes, if a cylinder is being squeezed in the radial direction, it will grow a bit length-wise (in the axial direction)...look up Poisson's Ratio
 
Couldn't you put the whole thing down to a bit of corrosion inside the lock?
 

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