state Newton's law of cooling.express it mathmatically.
http://www.biology.arizona.edu/biomath/tutorials/applications/cooling.html
Newton's law of cooling states that the rate of change of temperature of an object is directly proportional to the difference between its own temperature and the ambient (surrounding) temperature.
Mathematically, it can be expressed as:
dT/dt = -k (T - Ta)
Where:
- dT/dt is the rate of change of temperature with respect to time
- T is the temperature of the object
- Ta is the ambient (surrounding) temperature
- k is the cooling constant, which depends on the properties of the object and its surroundings.
To understand this equation better, let me explain how to derive it using calculus.
To start, we assume that the rate of heat loss from an object is proportional to its temperature difference with the surroundings:
dQ/dt = -k (T - Ta)
Where:
- dQ/dt is the rate of heat loss with respect to time
- k is the constant of proportionality (cooling constant)
Now, we know that the rate of heat loss (dQ/dt) is related to the rate of temperature change (dT/dt) by the specific heat capacity of the object (C) and its mass (m):
dQ/dt = -Cm dT/dt
By combining these equations, we get:
-dT/dt = (k/mC) (T - Ta)
Now, let's simplify further by replacing (k/mC) with a single constant 'k':
-dT/dt = k (T - Ta)
Finally, rearranging the equation gives us:
dT/dt = -k (T - Ta)
So, the final expression of Newton's law of cooling mathematically is:
dT/dt = -k (T - Ta)