Answer to Question #317384 in Differential Equations for Almas

Question

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

Answer 1

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 $T$ denote the temperature of an object and $T_0$ the ambient temperature. If $t$ denotes time, then Newton's law states that:

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

where $k$ is a positive constant. Thus, if the object is much hotter than its surroundings, then $T-T_0$ is large and positive, so $\dfrac{dT} {dt}$ is large and negative, so the object cools quickly. If the object is only slightly hotter than its surroundings, then $T-T_0$ 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!