Question #317384

Applications of Differential Equation of Newton's law of cooling / warming

1
Expert's answer
2022-03-28T12:13:03-0400

The temperature of many objects can be modelled using a differential equation. Newton's law of cooling (or heating) states that the temperature of a body changes at a rate proportional to the difference in temperature between the body and its surroundings. It is a reasonably accurate approximation in some circumstances.


More precisely, let TT denote the temperature of an object and T0T_0 the ambient temperature. If tt denotes time, then Newton's law states that:


dTdt=k(TT0)\dfrac{dT}{dt} =-k(T-T_0)


where kk is a positive constant. Thus, if the object is much hotter than its surroundings, then TT0T-T_0 is large and positive, so dTdt\dfrac{dT} {dt} is large and negative, so the object cools quickly. If the object is only slightly hotter than its surroundings, then TT0T-T_0 is small positive, and the object cools slowly. So a cup of hot coffee will cool more quickly if you put it in the refrigerator!


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