Change and calculus
1701
Newton’s law of cooling
Isaac Newton (published anonymously)
Why does tea cool quickly at first, then so slowly?
Newton heated a block of iron red-hot and timed how it cooled. He found it lost heat at a rate proportional to how much hotter it was than the air around it. It was one of the first laws written as a rate of change: a differential equation.
First, make a guess
A hot cup of tea in a cool room cools fastest…
- Latest cup after 10 minutes
- 58.4 °C
- Halfway to room temperature after
- 11.6 min
Every dash points the way temperature heads at that moment: fast when far from room temperature, slowly when close. Any cup just follows the dashes. Euler’s method follows them in straight steps, which is how computers solve equations they cannot solve exactly; big steps drift off the true curve.
How they figured it out
In 1701 Newton published, anonymously, a temperature scale running from freezing water to red-hot iron. Ordinary thermometers could not measure that high, so he heated a block of iron and timed it cooling in a steady breeze.
He assumed the iron lost heat at a rate proportional to how much hotter it was than the air. Then the time taken told him the temperature. His law of cooling began as a clever measuring trick.
The equation
The hotter something is compared with its surroundings, the faster it cools.
See it in real life
Milk now or milk later?
You pour a coffee but won’t drink it for ten minutes. Add the milk now, or just before drinking? Cooling is fastest when the gap to room temperature is biggest, so adding milk first lowers that gap and the coffee loses heat more slowly: milk now keeps it warmer.
Where engineers use it
Thermal design
Engineers size heat sinks, insulation and battery cooling with this equation.
Forensics
Body temperature cooling estimates time of death.
Worked example: When is the tea drinkable?
- Tea at 90 °C, room at 20 °C, per minute.
- Target 60 °C:
- minutes.
Where it came from
- The invention of calculus 1665–1684
Led to
Newton wrote cooling as a rate of change: one of the first differential equations used on real measurements.
What it led to
- The age of the Earth 1956
Same technique
Both follow exponential decay: the rate of change is proportional to how much is left.
Where the comparison stops: Cooling is smooth because trillions of molecules average out. Decay is random atom by atom, and only exponential on average.