Newton's Law of Cooling Problem

[code.master]

I did a problem where I was asked to model a simple cooling graph. No big deal. I got my model and found my constant values by using initial values.

Now comes the sticky part. I am asked to create a model similar to the previous simple cooling problem for a more complex system. Keep in mind this is a first course in DE and its only the second week.

The problem is now that the ambient temperature is fluctuating. I have established a sinusoidal function to model the ambient temperature and have begun setting up my model. I am given no initial conditions of the object, I am told that no matter what the initial values are, the function becomes closer and closer to a specific harmonic function with the same period as the ambient temperatures harmonic function.

I am asked explicitly to give a model for the temp of the object with respect to time. Now, pardon me for not using LaTeX, I am pretty new here... but this is my model as of now.

dT/dt = k*(T - 80 + 30*cos(2*pi*t/24))

Ill be honest, I suck at algebra. But am I supposed to carry this further and develop anything else? I am given no k value, no initial temps, no max/min temps, no intersection points (dT/dt = 0) and basically nothing else. I am assuming my book is just wanting me to set up the basics of the DE and the 'modeling' part was in getting the [80 + 30*cos(2*pi*t/24)] part. What do you think?

 

[balakrishnan_v]

Just take the Fourier transform and invert it.You will get
$$T=80+\frac{720k}{\sqrt{576k^2+4\pi^2}}Sin\left(\frac{2\pi t}{24}-tan^{-1}\left(\frac{12k}{\pi}\right)\right)$$

 

[tacman]

It seems like you have a good grasp on the basics of Newton's Law of Cooling and how to model a simple cooling graph. It's great that you were able to find your constant values and set up your model.

Now, for the more complex system, it's important to keep in mind that the ambient temperature is fluctuating. This means that the constant values you found for the simple cooling graph may not be accurate for this new system.

In order to account for the changing ambient temperature, you have correctly introduced a sinusoidal function to model it. Your model for the temperature of the object with respect to time looks good, but as you mentioned, you are missing some important information such as the k value and initial conditions.

Without these values, it may be difficult to fully solve the problem and obtain a specific model for the temperature of the object. However, you can still continue with what you have and see if you can make any further progress. It's possible that your book just wants you to set up the basics of the DE and the modeling part was in finding the sinusoidal function for the ambient temperature.

In any case, it's important to communicate your concerns with your instructor and see if they can provide any additional guidance or information to help you solve the problem. Don't worry if you're not great at algebra, with practice and guidance, you'll improve. Just keep at it and don't be afraid to ask for help when needed. Good luck with your problem!