key term - Newton’s law of cooling

Definition

Newton’s law of cooling states that the rate at which an object changes temperature is proportional to the difference between its own temperature and the ambient temperature. It is commonly expressed using a first-order separable differential equation.

5 Must Know Facts For Your Next Test

1. The differential equation for Newton's law of cooling is $\frac{dT}{dt} = -k(T - T_{env})$, where $T$ is the object's temperature, $T_{env}$ is the ambient temperature, and $k$ is a positive constant.
2. The solution to the differential equation involves exponential decay: $T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$, where $T_0$ is the initial temperature.
3. Newton's law of cooling can be used to model various real-world processes including cooling of hot liquids, warming of cold objects, and even some biological processes.
4. It assumes that the ambient temperature remains constant during the time period considered.
5. The law provides a good approximation for small differences in temperature but may not be accurate for large differences or when other heat transfer mechanisms are significant.

Review Questions

Related terms

Exponential Decay:

A process where quantities decrease at a rate proportional to their current value, often modeled with functions like $y(t) = y_0 e^{-kt}$.

Separable Equations:

A type of differential equation that can be separated into two integrals: one involving only dependent variables and one involving only independent variables.

Ambient Temperature: The surrounding temperature or environment in which an object exists.