Dunkin' Donuts
 

A torus of iron, with a diameter of 50cm and a hole diameter of 5cm, is placed in a furnace. As it heats up does the hole get bigger, smaller, or stay the same size?

I dunno. How exactly does iron expand due to heat? If I have a 1 inch by 1 inch by 10 inch bar of iron that expands due to heat, does the bar retain a 1x1x10 ratio?

If we're just scaling the torus, the hole will get bigger

Thermal expansion for Iron is positive (11.8 µm·m[sup]−1[/sup]·K[sup]−1[/sup]). Assuming the heating is uniform then all dimensions will expand by an equal ratio. That is, until the thing melts at 1811 K, at which point all bets are off.

[quote=Orgasmic Troll;195531]...If I have a 1 inch by 1 inch by 10 inch bar of iron that expands due to heat, does the bar retain a 1x1x10 ratio?[/quote]

I would think it would.

As for the iron torus with the hole, the hole size would decrease, but not proportionally to the expansion of the outer diameter of the torus.

I have an idea. I have a very large iron nut which I found many years ago. Measuring it with a precision caliper, the outer flats read 3.569" side to side. The inner thread diameter is 2.082". On a postal scale, it weights just over four pounds. I could heat this to 500°F in my oven, saturating for a few hours, and see how it changes, if it changes...

[QUOTE=storm5510;195783]I have an idea. I have a very large iron nut which I found many years ago. Measuring it with a precision caliper, the outer flats read 3.569" side to side. The inner thread diameter is 2.082". On a postal scale, it weights just over four pounds. I could heat this to 500°F in my oven, saturating for a few hours, and see how it changes, if it changes...[/QUOTE]

Assuming I understand retina's post (only by dimensional analysis!), the expected change is about 0.010" (or 0.011" if you measured it right after taking out of the freezer), so it should be doable.

[QUOTE=storm5510;195783]As for the iron torus with the hole, the hole size would decrease, but not proportionally to the expansion of the outer diameter of the torus.[/QUOTE]

This is a common mistake. People think piecemeal about the metal near the inner edge expanding, and think it must expand into the hole, making the hole smaller. Blacksmiths know better - a really tight fit is sometimes made by putting the "plug" on dry ice to make it smaller and heating the hole (OK, so the metal around the hole for you pedants) to make it larger. Tongs a sledge hammer finish the task.

Think of it like this - assume the torus is created from a disk by removing the center. Take two disks. On one of them cut the hole and then heat the disk. On the other heat the disk and then cut the hole. The final results will be the same size.

[quote=wblipp;195798]This is a common mistake. People think piecemeal about the metal near the inner edge expanding, and think it must expand into the hole, making the hole smaller. Blacksmiths know better - a really tight fit is sometimes made by putting the "plug" on dry ice to make it smaller and heating the hole (OK, so the metal around the hole for you pedants) to make it larger. Tongs a sledge hammer finish the task.

Think of it like this - assume the torus is created from a disk by removing the center. Take two disks. On one of them cut the hole and then heat the disk. On the other heat the disk and then cut the hole. The final results will be the same size.[/quote]
THX William. I was going to say something similar: heating an iron ring
then let it contract round a wheel-sledgehammer-barrel.
Better be quicker of the mark --- Paul beat me to Olber's paradox
re the Speed of Light thread.

I am trying to summon up the enthusiasm to put Mart_r out
of his misery in "Mysterious Connection" (puzzles), but so far I
find myself unable to maintain a sufficiently straight face:smile:

David

[QUOTE=wblipp;195798]Think of it like this - assume the torus is created from a disk by removing the center. Take two disks. On one of them cut the hole and then heat the disk. On the other heat the disk and then cut the hole. The final results will be the same size.[/QUOTE]

I don't follow this part of your explanation. How do you "cut the hole"?
(What determines the diameter of the hole that is formed?)

If you cut a hole of exactly 1cm diameter in cold metal, it will be slightly larger when the metal is heated.

However, if you cut the hole to the same dimension in the hot metal, you will remove less material. The hole will be 1 cm. That is not "the same".

- - -

Now, let's extend the question:

If, rather than looking at the "steady state" condition of uniform temperature throughout the metal, we start with cold metal and place if in a hot atmosphere. As the metal heats, what happens to the diameter of the hole over time?

[quote=CRGreathouse;195796]Assuming I understand retina's post (only by dimensional analysis!), the expected change is about 0.010" (or 0.011" if you measured it right after taking out of the freezer), so it should be doable.[/quote]

Um... It was an oven. I'll get it started. Once it reaches 500°F, I'll saturate it for three hours, minimum.

[QUOTE=wblipp;195798]Blacksmiths know better - a really tight fit is sometimes made by putting the "plug" on dry ice to make it smaller and heating the hole (OK, so the metal around the hole for you pedants) to make it larger. Tongs a sledge hammer finish the task.[/QUOTE]That is also how the tyres where changed on steam locomotives. They had special rings that had flames that they used to heat them up.

[QUOTE=wblipp;195798]Blacksmiths know better - a really tight fit is sometimes made by putting the "plug" on dry ice to make it smaller and heating the hole (OK, so the metal around the hole for you pedants) to make it larger. Tongs a sledge hammer finish the task.[/QUOTE]Back when I was a grad student and had access to a machine shop, I repaired the university astronomical society's telescope mount. A ring bearing needed replacing, so I machined a piece of tubing to be a few thou (a hundred microns or thereabouts for the metric minded) too large in external diameter for the hole in which it was to fit. The bearing was dropped into a bucket of liquid nitrogen until it stopped fizzing then pulled out (with tongs!) and dropped into the hole. As soon as it warmed up the fit was so tight that the bearing was completely immovable. The component into which it fitted was aluminium and so malleable enough to take the pressure without damage.


Paul

When I first read this puzzle (years ago) the answer mentioned a cooper heating a metal ring to make it easier to get it on a barrel. That's why I chose such an unusual sized torus; as a red herring. (And in the hope that some might think the tiny hole would be 'overwhelmed' by the the torus in some way.)

Just like that!

[quote=wblipp;195798]...Blacksmiths know better - a really tight fit is sometimes made by putting the "plug" on dry ice to make it smaller and heating the hole (OK, so the metal around the hole for you [B]pedants[/B]) to make it larger. Tongs a sledge hammer finish the task.[/quote]

[I]"A [B]pedant[/B] is a person who is overly concerned with formalism and precision, or who makes a show of his learning."

[/I]Hmm, Oh well.

I'll have to repeat the experiment, if I can repair my calipers. I dropped them and the needle came off the dial. They're quite old.

[QUOTE=storm5510;195816]Um... It was an oven. I'll get it started. Once it reaches 500°F, I'll saturate it for three hours, minimum.[/QUOTE]

I meant what I wrote. If you take it from 30 F to 500 F, you'd expect about 0.011; from 70 F to 500 F, more like 0.010.

[quote=CRGreathouse;195873]I meant what I wrote. If you take it from 30 F to 500 F, you'd expect about 0.011; from 70 F to 500 F, more like 0.010.[/quote]

We posted at the same time. How about starting around 0°F. My freezer usually runs in that range. Measure at zero and then again at 500?

[QUOTE=storm5510;195874]We posted at the same time. How about starting around 0°F. My freezer usually runs in that range. Measure at zero and then again at 500?[/QUOTE]

It would be about the same -- 0.011, maybe 0.012. I think 0.010 is the best you could reasonably hope to measure with the difficulties of taking it out of the {freezer|oven} and having it come toward room temperature.

I managed to repair my calipers. I slightly heated the needle hub to make it hold onto the dial post. As I said, it's an old mechanical type.

I don't know how accurately I can measure this object. It is not machine-grade by any means. There is not a common dimension between any two sides. I suspect it came from a piece of farm machinery.

I will pick a pair of sides and mark them and see what I can come up with.

[QUOTE=storm5510;195912]I managed to repair my calipers. I slightly heated the needle hub to make it hold onto the dial post. As I said, it's an old mechanical type.

I don't know how accurately I can measure this object. It is not machine-grade by any means. There is not a common dimension between any two sides. I suspect it came from a piece of farm machinery.

I will pick a pair of sides and mark them and see what I can come up with.[/QUOTE]If your object is not very regularly shaped, you must make sure you take your second measure in exactly the same place as the first one. You will also have another problem measuring the object with your calliper : it is meant to be used at a "normal" temperature, once the object is heated it will transfer heat to the calliper and the measure will not be accurate.

The question is why measure ? Instead of thinking of what happens to a disk, think of a ring of thin wire. It will lengthen when heated and the hole will grow.

Jacob

[QUOTE=S485122;195922]The question is why measure ?[/QUOTE]

Why not? I expect the hole will grow, but I'd like to see how closely the expansion matches the prediction. Also, measuring the outside as well as the inside gives a check on the measurement.

[quote=S485122;195922]...You will also have another problem measuring the object with your calliper : it is meant to be used at a "normal" temperature, once the object is heated it will transfer heat to the calliper and the measure will not be accurate...[/quote]

I considered this, and the reason I will do the measurements as rapidly as I can to avoid as much temperature transfer as possible.

I marked a spot on the bottom side to measure at because of the dimensional irregularities.

I have the cold measurements complete. The freezer cycles from 0°F up to 9°F. I have the nut sitting out and it is almost snow white from frozen condensation. I will do the hot portion later today or this evening.