Goldilocks and the thermal characteristics of porridge

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Goldilocks and the thermal characteristics of porridge

Post by Erik_Kowal » Tue Oct 28, 2008 8:43 am

In the nursery tale of Goldilocks and the three bears, one scene puzzles me.

When Goldilocks tastes the bears' porridge, Daddy Bear's is too hot, Mother Bear's is too cold, and Baby Bear's is 'just right'.

Common sense suggests that it was all cooked in the same saucepan, so can anyone account for the divergent temperatures of the porridge in the bowls?
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Re: Goldilocks and the thermal characteristics of porridge

Post by Tony Farg » Tue Oct 28, 2008 9:44 am

I suspect it has to do with the amount of cream or milk stirred-in by the owners before going to do whatever bears do in the woods.
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Re: Goldilocks and the thermal characteristics of porridge

Post by russcable » Tue Oct 28, 2008 2:25 pm

The bowls are different sizes (and perhaps made differently), and the porridge was served at different times. Momma Bear served herself first in the pretty bone china (thin) bowl and just as she started to eat, Baby Bear became hungry so while she was attending to Baby Bear, getting the baby's bib on, etc her porridge became stone cold before anyone else's was even served. Poppa came in after the baby had stopped fussing (so he wouldn't have to help ;-) ) Poppa Bear's bowl is big and thick and the large serving retains the heat for a long time.
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Re: Goldilocks and the thermal characteristics of porridge

Post by Erik_Kowal » Tue Oct 28, 2008 8:09 pm

Thanks to you both for those possible explanations.

But I do see a snag with the explanation that Momma Bear's porridge grew cold because she was too busy making sure Baby Bear ate his porridge to feed herself: surely Baby Bear's plate would be empty, because he would pretty much have finished eating?

And if Poppa Bear only served himself after Baby Bear was done, that too implies that there was little or no porridge left in Baby Bear's bowl.

On balance, Tony's explanation seems more plausible. Or is there something I've missed?
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Re: Goldilocks and the thermal characteristics of porridge

Post by russcable » Tue Oct 28, 2008 9:38 pm

I didn't say Momma Bear fed Baby Bear but that she was getting him ready to eat. Nor did I say that Poppa Bear waited until Baby Bear was done eating but that he was waiting for him to be settled down.

Even with serving BB and PB at the same time, it's only necessary to come up with a reason why Momma's is cold.
1) MB serves herself (perhaps thinking that she will have time to eat before BB wakes up).
2) MB is interrupted by BB (perhaps she goes to get him out of bed, change him, etc.)
3) MB serves BB and PB at the same time.
4) All three leave.
5) GL comes in to find MB's is cold because of extended time, BB's is warm because though it was just served the small serving cools more quickly than PB's large serving which is still hot.
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Re: Goldilocks and the thermal characteristics of porridge

Post by trolley » Tue Oct 28, 2008 9:58 pm

But, wait. Did they not decide to go for a walk after the porridge was dished up for the specific purpose of allowing it time to cool. This implies that all three bowls were "too hot" at that point. Either there was a cold draft from an open window that, given their relative positions on the table, caused them to cool at different rates or there was a difference in the composition of the bowls.
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Re: Goldilocks and the thermal characteristics of porridge

Post by Ken Greenwald » Tue Oct 28, 2008 10:21 pm

But wait a second. The baby, the mom, and the pop being of increasing size, would probably have been served increasing sized portions, but presumably in the same size bowl. If it is assumed that the exposed surface area of each serving of porridge is not that different (the more vertical the sides of the bowl the truer this will be), but even if it isn’t, the most significant factor for a difference in cooling effect between the bowls would probably be the effect of the mass of the bowl, which would naturally lead to the observed gradation in porridge temperatures. Yes! (<:)
____________________

Ken – October 28, 2008
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Re: Goldilocks and the thermal characteristics of porridge

Post by Erik_Kowal » Tue Oct 28, 2008 11:24 pm

Not so fast, Ken! :-)

If the facts you state are correct, then we should expect Baby Bear's portion to be the coldest, Momma Bear's to be middling warm ('just right'?), and Poppa Bear's portion to be the hottest.

I'm now beginning to warm to Russ's theory, although the explanations put forward by Trolley and Tony also work for me.

How can we decide definitively which one is right?
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Re: Goldilocks and the thermal characteristics of porridge

Post by Ken Greenwald » Wed Oct 29, 2008 5:17 am

Erik, I said would probably serve increasing size portions. But that shows what I know. The typical caring mama bear, who is in charge of the cooking (in the typical sexist bear household), would of course arrange things in her self-sacrificing way so that her baby bear’s porridge temperature was just right and so that the little bruin would grow big and strong. She did this by giving herself the smallest portion and daddy the largest. [[How’s that for wiggling out of a bad situation?]] (<;)
____________________

Ken – October 28. 2008
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Re: Goldilocks and the thermal characteristics of porridge

Post by trolley » Wed Oct 29, 2008 7:30 am

Just bearable.
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Re: Goldilocks and the thermal characteristics of porridge

Post by Erik_Kowal » Wed Oct 29, 2008 7:38 am

Slightly better than ursinine.
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Re: Goldilocks and the thermal characteristics of porridge

Post by Tony Farg » Wed Oct 29, 2008 10:37 am

Anyway, it's "porage" for the true believer.
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Re: Goldilocks and the thermal characteristics of porridge

Post by Erik_Kowal » Wed Oct 29, 2008 10:55 am

True believer in what, Tony?

On reflecting on the previous contributions I am now less convinced by arguments based on differences between the bowls. For instance, the thin bone china bowl that Russ hypotheses for Momma Bear will absorb less heat from its contents as it tries to reach a temperature equilibrium with those contents; on the other hand, the bowl's thinness means that its outer surface will be hotter, so the temperature gradient between itself and its surroundings will be greater, and its rate of heat loss will be more substantial than for a bowl with a greater thermal capacity.

In other words, there are two mechanisms at work in Momma Bear's bowl which will tend to work in opposite directions with respect to the temperature of the porridge it contains.

Similarly, the greater thermal capacity of Poppa Bear's bowl will lead to an initially rapid drop in the temperature of the porridge as it starts to give up its heat to the bowl, but the bowl's surface temperature will be lower and heat will be lost more slowly to the surroundings, thanks to the flatter temperature gradient in relation to those surroundings compared with Momma Bear's bowl. However, if Poppa Bear's bowl was significantly hotter than the others to begin with -- for instance because it, alone of all the bowls, had just come out of the dishwasher -- this would greatly decrease the rate of its absorption of the heat from the porridge; its porridge would therefore stay hotter for longer.

There must be a mathematical function which describes the thermal characteristics and behaviour for any given bowl/porridge complex. Presumably, in order to insert the necessary variables we will need to know -- among others -- exactly what the size, shape and composition of the bowls were, as well as the volume of the porridge, the initial temperatures of the bowls, the porridge and the bowls' surroundings, plus any complicating factors such as the presence of draughts. Only then can we attempt to calculate the relative rates of cooling of Momma Bear's, Baby Bear's and Poppa Bear's porridge.

But I had better stop here, before I start to lose interest in this fascinating conundrum.
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Re: Goldilocks and the thermal characteristics of porridge

Post by Bobinwales » Wed Oct 29, 2008 12:36 pm

And why were Mummy and Daddy Bear sleeping in separate beds? Was chauvanistic Daddy not happy with his porridge? Or was Mummy fed up with his constant moaning about it?
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Re: Goldilocks and the thermal characteristics of porridge

Post by Erik_Kowal » Wed Oct 29, 2008 5:05 pm

Another excellent question.

Oh dear -- where will all this end? Goldilocks' bowl of porridge is turning out to be quite the Pandora's box!
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