Hawking’s Tea by Mr Ottewill
Stephen Hawking, who died earlier this year, is rightly judged to have been one of the greatest theoretical physicists of the twentieth century, if not of all time. He came up with pioneering ideas in cosmology, not least on fundamental questions such as the start of the universe, as well as making great strides in combining general relativity with quantum mechanics.
An autobiography of Hawking called ‘Stephen Hawking – A Life in Science’ by Michael White and John Gribbin, relates the following anecdote from a fellow student at his high school in St Albans:
One particular example of Stephen’s highly developed insight left a lasting impression on John McClenahan. During a sixth-form physics lesson, the teacher posed the question, “If you have a cup of tea, and you want it with milk and it’s far too hot, does it get to a drinkable temperature quicker if you put the milk in as you pour the tea, or should you allow the tea to cool down before adding the milk?” While his contemporaries were struggling with a muddle of concepts to argue the point, Stephen went straight to the heart of the matter and almost instantly announced the correct answer: “Ah! Milk in first, of course,” and then went on to give a thorough explanation of his reasoning: because a hot liquid cools more quickly than a cool one, it pays to put the milk in first, so that the liquid cools more slowly. This article considers what some of the ‘muddle of concepts’ that his contemporaries might have been looking at are, i.e. what might be supposed to be the ‘standard’ way to answer the question. In doing this we uncover a potential counter argument to Hawking’s insight.
The main physics concepts needed to tackle the problem are:
(1) The formula for the temperature of a mixture formed by combining two substances with different temperatures is:
1
1
1 1
+ 2 + 2
2 2
2
=
1
where 1 , 2 = the initial masses of the substances
1 , 2 = the specific heats of the substances* 1 , 2 = the initial temperatures of the substances
* Note: the specific heat of a substance is the amount of heat per unit mass required to raise the temperature of the substance by one degree Celsius, or in other words how difficult it is to ‘heat up’ the substance.
(2) We also need to use Newton’s law of cooling which states that:
d 𝜃 d
= −𝜆(𝜃 − 𝜃 )
where 𝜃 = the temperature of the substance
t = the time since the start of the cooling process 𝜆 = a constant of proportionality 𝜃 = the temperature of the surrounding air
It can be shown using A-level techniques that the solution to this differential equation is:
) −𝜆
𝜃 = 𝜃
+ (𝜃 0
− 𝜃
where 𝜃 0 = the temperature of the substance at the start time.
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