## How long does it take to heat a spherical cow?

March 31, 2009 by Ben Lillie

ZapperZ points to this outreach question from the APS:

How long would you have to yell to heat a cup of coffee?

It’s a neat question. The idea is that sound transfers energy. That energy will hit the coffee and dissipate as heat, so by yelling at a cup of coffee you could heat it up. Except of course that you can’t. The coffee will be losing heat as you yell at it, and will cool from hot to room-temperature in a couple hours. And yet, the APS site gives this answer:

In other words to heat up a quarter liter of coffee 50 C it would take: 1 year, 7 months, 26 days, 20 hours, 26 minutes and 40 seconds

How did they get that? The answer, of course, is that they used a spherical cow.

We’re assuming things like a perfect transfer of energy from your yell to the coffee, a perfectly insulated cup that will never let any heat escape, and an unending even stream of energy.

“Perfect insulation” is the key there. There’s no coffee cup in existence that comes even remotely close.

The annoying part is that this is an interesting question, and gets across a couple good ideas, but it finishes with a result that is clearly, and intuitively, wrong. The calculation itself is a great example of something physicists do all the time, make an estimate of some interesting quantity. It’s incredibly useful; make some simplifications, and see what you get. The problem is that at the end of the calculation you have to think about the answer to see if it makes sense. If not, one of the assumptions is probably wrong. By asking “How long would it take the Earth to cool?”, Lord Kelvin deduced that the earth could be no more than 400 million years old. Of course, it’s older than that — by about 4 billion years. It turns out he ignored (because he didn’t know about it) the internal heating from radioactive decay.

In this case, a simple “Of course, in real life coffee cools in a few hours, so this wouldn’t actually work” at the end would have summed it all up nicely. It’s not a big deal, but when you’re trying to teach some physics it doesn’t help to present an answer that makes no sense.

(Tom Swanson points out that they got the number too small by a factor of 10. That also serves to point out the absurdity of presenting the answer to minutes and seconds.)

### Like this:

Like Loading...

*Related*

## Leave a Reply