How Is Newton's Law of Cooling Derived?

AI Thread Summary
Newton's Law of Cooling describes how the temperature of an object changes over time, stating that the rate of temperature change is proportional to the difference between the object's temperature and the surrounding temperature. The mathematical representation is dT/dt ∝ T - Ts, where T is the object's temperature and Ts is the surrounding temperature. The constant of proportionality, k, is introduced when the relationship is expressed as an equation: dT/dt = -k(T - Ts). Understanding "directly proportional" in this context means that as the temperature difference increases, the rate of cooling also increases. This foundational concept is essential for deriving and applying Newton's Law of Cooling effectively.
Junior Newton
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Homework Statement



Mathematically prove how Newton's Law of Cooling is derived.

Homework Equations


Newton’s Law of Cooling states that the rate of change of a temperature of an object is directly proportional to the difference between its own temperature and the temperature of its surroundings (Smith, 2010). Therefore, from this information it can be seen that the following relationship can be interpreted.

dT/dt ∝ T-Ts

Where T = temperature of the object and t = time.

The Attempt at a Solution



not sure how the k comes into it?
 
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What does the term "directly proportional" mean in mathematical terms? That should answer your question.
 
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