Hobbits, Wizards and a nice cup of tea

Classical Mechanics Level 4

Bilbo and Gandalf are having tea together. Their mugs are similar, with Gandalf's 2.5 times longer in each dimension. After Bilbo has poured the hot water (up to the brim in both the mugs) and the mugs have warmed up to be in thermal equilibrium with the tea inside them, they are each at a temperature of $90^ \circ C$ . But, Bilbo and Gandalf prefer to drink their tea at $80^ \circ C$ .

After 2 minutes, Bilbo's tea has cooled enough for him to start drinking it. How much longer (in minutes) Gandalf has to wait to drink his own tea after Bilbo has started drinking his tea?

The answer is 3.

1 solution

Andrew Normand
Feb 7, 2017

The rate of heat loss from the tea is proportional to surface area. Since the objects are similar in shape, surface area is proportional to $L^2$ where L is the length or radius of the mug.

The heat loss required to change the temperature of the mug is proportional to volume - and so proportional to $L^3$ .

Cooling rate is therefore proportional to $\frac{L^2}{L^3}$ or just $\frac{1}{L}$

Since Gandalf's mug is 2.5 times taller and wider, it will take 2.5 times longer to cool. Gandalf will have to wait another 3 minutes after Bilbo to drink his tea.

Instead of "Their mugs are identical, save that ...", I think a much clearer phrasing is "their mugs are similar, with Gandalf's mug 2.5 times as large".

Calvin Lin Staff - 4 years, 4 months ago

Perhaps, yes. I was trying to avoid any ambiguity about whether I was referring to volume or length. I've edited, going for a compromise.

Andrew Normand - 4 years, 4 months ago

Hobbits, Wizards and a nice cup of tea

The answer is 3.

1 solution

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