Newton's Law of Cooling // Separable ODE Example

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How can we model the change in temperature of an object, like a cup of tea? In this video we explore Newton's Law of Cooling, which is a separable differential equation. We are going to do a bit of experimental data collection (measuring the temps of my coffee mug) along the way to see how good the model is. To solve this ODE we use the method of separation of variables. However, because there are multiple constants involved, we need to record temperatures at at least two different times to be able to solve for them.
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[@question_chaser]

Sir, you made more sense in 10 minutes than my 6 week of my D.E. lectures. Thanks for your commitment. Best,

[@NuclearPhysix]

I left university some decades ago. I do not recall anyone giving a practical use case for differential equations as you have just done Dr Trefor. You are amazing

[@DrTrefor]

This is too often the case!

[@cardinalityofaset4992]

Unfortunately, this is a sad reality; however, you can't really blame professors of mathematics, since they did not study physics directly, in the vast majority of cases. The fact that you solve a differential equation doesn't imply that it matches the reality - you can get mathematically correct answer, but physically, the result could be way off the experimental value or could be a complete nonsense at the first place. This is something mathematicians can not really justify unless they studied theoretical physics as applied maths. Furthemore, if you knew about the existence of Maxwell's equations, Schrödinger's equation or Hilbert-Einstein equations of general relativity, which are all famous diff. equations, you could have figured out in your head that most things in our universe are governed by differential equations. And so, it is quite simple to construct your own differential equation that would describe something physical (whenever you try to describe some model like perfect oscilator,pendulum,some kind of litter, usually only highschool mathematics and physics is required) Differential equation actually arrise just when you start studying physics in high school (particulary in kinematics) though they are incredibely easy to solve, since most of them are just solving simple integrals (or taking time derivatives). Nevertheless, you can complicate things if you add friction or air resistance. In that case, you need to use methods like the separation of variables, variation of constants, particular solution etc to solve them (dipending on the type of flow - sometimes you can only approximate the value if it is too complicated).

[@jlee3361]

How did you know I was taking Diff eq this sem. Impeccable timing. Bravo, sir, bravo.

[@DrTrefor]

Did it just for you lol

[@jlee3361]

@@DrTrefor dawwwww profesorrrr you shouldn't have

[@pinkpopcorn805]

This video is sooooo good, I saw ODE for the first time today, couldn't understand why we were using it or how useful could it be, but my math teacher was like "It would be good for you all to go home and do some research, examples and see how important this topic is for an engineer", in my country education is very hard to get and even the good ones (I am having now lol) isn't even close to extraordinary, it should be more like you did in the video, but we as students are not prepared enough (we don't have good exposure) and even having one or two good teachers isn't enough to have an amazing learning, a lot have to be done by ourselves and this isn't something cultural in my country (people do cheat a lot or a not real interested in studying at all), learning English makes possible to me to get this good stuff for free, I feel so lucky to cross your channel because I have internet and a clear background in English, so, thank you so much!!!

[@forresthu6204]

I really like the way you are performing physical experiment while teaching on how to solve Newton Cooling formuar. It really make the math relevent to our day to day life, and it is fun to compare experiment result to model prediction.

[@pradeepattri8375]

You're my favourite person to learn and most of UNDERSTAND differential equations.

[@raymondlancaster3355]

This is really good instruction. This is the first video I have seen by this gentleman but my guess is he does quality work consistently.

[@DrTrefor]

Thank you!

[@prabeshgautam4112]

Man, that was terrific teaching. I have known this for years and have watched tons of videos but this is the clearest anyone ever explained it.

[@items365]

That was great sir a cup of coffee made my sepereable differential solution

[@prydt]

I just HAVE to comment and say that these videos and your channel in general is amazing!!! Keep up the awesome work!!!!

[@DrTrefor]

Thanks so much!

[@georgesadler7830]

Professor Bazett, thank you for a solid analysis of the classical Newton's Law of Cooling in Differential Equations. This is a great experimental/analytical law by one of the founders of Calculus.

[@pennyyang9213]

So much better than Prof I have in my uni today about ODE almost die for catching the blackboard and the small volume

[@akif7699]

Using real life scenarios is much more understandable for us, thanks Dr. !

[@j.o.5957]

Me like. So that's why we use e^ so much in ODE's. That's also where the extra constants comes from, e^-k, by using logarithms. I also like how you model the situation. Question to self: what is a situation that I could model this way? I got a coup of instant noodles, it takes 5 minutes to cook. I could probably estimate some function of how soft it becomes, using time t, temperature and a way to measure softness (like how much it'll stretch).

[@simran19]

I was looking for an ODE and PDE course and I am glad you have started ODE. If you don't mind could you please upload videos faster 😅? I love your teaching skills.

[@DrTrefor]

haha, going to have a better schedule pretty soon lol, been a slow start:D

[@jonoliahjohn6800]

This video was so fun to watch. I've been trying to figure out how to do this ODE for more than a few hours now 😅. This explanation just made it plain as day for me

[@axbowf4924]

Thank u. Recently found ur channel through the logic class. I am glad that I found your channel. Ur videos will be helping me a lot.❤

[@DrTrefor]

Glad to hear that!

[@3laaHisham]

I really can’t imagine myself born 1 year earlier and having to study D.E. without Dr. Trevor… Allah bless you

[@olivioarmandocordeirojunio8280]

Thanks a lot Prof. Trefor. Your explanations are awesome.

[@synapticmemoryseepage4447]

Great example of a DE in action! I love it.

[@Alannnn14]

Your videos are pure gold. Thank you

[@DrTrefor]

Thank you!

[@ahmdkalef2252]

a great demo for DE example.

[@mohfa1806]

thank you ...keed doing what you are doing...educating humanity...thx again

[@pooriamoghadam1492]

that was great i have been feeling the ODE with this experiment

[@ChefFarisMom3]

Thank you sir.. I am sitting in my class and listening to you 😃😃😃

[@footbol956]

Thank you Dr. You helped me finish my assignment ODE assignment.

[@DrTrefor]

Glad I could help!

[@gilernt]

great video! just a small correction - i think at the end when you used the model to predict the temperature of the tea after a certain time, you originally said the reading was taken at 5 minutes, but used 7 in the equation. T(5) = 143.8 here, which is only about %0.77 off the measured value of 142.7; maybe you just used 7 or maybe it's a total coincidence that after 7 minutes the measured temperature almost perfectly matches the 5-minute theoretical, but either way, i learned a lot from this video, so thank you!

[@liveandlearn515]

Learned a lot from the playlist! Thank you! You might need 2 more tea bags for the tea , for it to be strong enough :) :) :) :) 00:13

[@MisterBinx]

Took heat transfer a couple years ago and I wanted the temperature profile as a function of x. I have to find my old text. Heat transfer was a cool class.

[@aggualaqisaaqs3272]

Even though the model wasn’t perfectly accurate, the fact that we were able to predict to within ~5% just by taking a measurement and moving some symbols around on a piece of paper is remarkable. Never gets old. I don’t think many people today appreciate just how incredible it is that we humans somehow figured out how to gain access to nature’s secrets just by using our brains.

[@JimKnoxAimbie]

You're a legend! Excellent work.

[@chg691]

Thank you sir.. For this amazing way of presentation

[@stephaniesmith3428]

Just started the Spring quarter at UCSD and was beginning to feel overwhelmed in DE. These helped so much and I'm so happy you started this playlist.

[@DrTrefor]

Glad it helped!

[@Escarii66]

Thanks for this video, I really understand maths better when I see it applied! If you don't mind me asking, why don't we need the Constant on the left-hand side integration? Can we ignore it because constants on both sides are effectively a single (different) constant on one side?

[@robconrad5956]

It's because a number A can be written as a number B either by +/- a number C or by multiply A by a number D. For example, if A = 100, B = 25, C = 75, and D = .25. Then, (A-C)=(B)=(A*D). Hope that is helpful.

[@BitwiseMobile]

I hated differential equations until I took EE classes and saw them at work. We were taught Ohm's and Faraday's laws (along with Kirchoff's) prior to seeing the ODE at work, so when you derive Ohm's and Faraday's laws from that a huge light bulb goes off.

[@DrTrefor]

I agree, we often just do differential euations with no applications and it feels like WHY

[@Nabuwetunabuwetuissisikuyai]

Sorry but at around 3:02 how is heat change an exponential equation, i thought its more like addition like the rise un temperature is an add up or sum of the rate of change an the initial temperature and the gradient remains constant well like maybe approximately constant if we are talking about the same medium but if the rate of change of the different temperatures is different in both then you have a parametric equation, because i don't understand how the rate is reliant on the intial difference in temperature

[@andrewbouc6015]

Thank you for your content!

[@colorx6030]

Ah, so that's how constants from all those Physics equations are derived. That's neat. Also, I still can't understand why the left hand side on when we're doing a separation of variables does not have a +C while the right hand side does.

[@kevinboykin8943]

It does. Both Integrals produce an unknown constant or some number. To simplify the overall equation, we can move and combine the left hand side C with the right hand side C. So now we just have the + C on the right hand side only.

[@lauramolloy4626]

great video, shocking cup of tea

[@ntvonline9480]

When I took Duffy-Q (decades ago), my professor introduced the Law of Cooling differently. Here’s the problem: Bonzo and Ronald Reagan were in a coffee shop. When their coffee arrived, Reagan added cream to his drink. Ten minutes later, Bonzo added cream to his coffee. Who’s coffee was hotter at ten minutes? I recall the question, but not the solution.

[@DrTrefor]

oh I like that way of presenting it, actually I think I might that an activity for my students!

[@arj123sub]

Great video, Dr Bazett !!

[@antoniojimenez7087]

You are great!!! Please keep on this!!!

[@hannananan9427]

This was too cool, thank you for posting this!!

[@phyarth8082]

Coffee mug time depended coefficient is also dependable from temperature, (natural convection phenomena) to make experiment k constant from temperature need air fan that blows air with constant flow rate.

[@skipintro9988]

Take my love, professor!

[@continnum_radhe-radhe]

Thanks a lot sir 🔥🔥🔥

[@muhammadnoumankhan330]

i calculated in parallel to you, i got T = 142.336 F (Calculated) A = 68F C or D = 93, K = 0.044 T(5) = 142.7 (Experimental) T(5) = 142.336 (Calculated)

[@omribgg]

Great video thank you for that

[@ac-jk9mz]

This video is fun to watch thank you

[@alabikayode9193]

How long will it take a 0.75dm3 of water at 25°C to ice up if suspended in an environment with a temperature of 49°C ?

[@stewy0013]

“…just because the thermometer you’re going to see in a moment [small chuckle to himself] is going to be in Fahrenheit as well.” SAVAGE!😂

[@harsh5128]

Awesome sir

[@yasamindev]

I was so excited for the moment of truth

[@LuskyMJ]

4:29 I don't get why the right side gets a constant but the left side doesn't.

[@ailphaune]

You can add the constant on the left side too but you would get: ln(T-A) + C1 = -kt + C2 and by switching side you would get: ln(T-A) = -kt + C2 - C1 but C2 - C1 is a constant too so you could perhaps call it C which lands you back to what is shown in the video: ln(T-A) = -kt + C

[@TalesFragoso]

Thank you!

[@MathAdam]

This is why true mathematicians never drink their tea hot. Every mother on the block when Dr. Trefor was 6 years old: "Kids! Come inside! Dinner's getting cold." Dr. Trefor's mum: "Trefor! dT/dt = -k(T-A)"

[@DrTrefor]

Lol

[@paulg444]

please provide a link to the fundamental physics of the dT/dt prop DeltaT

[@taotaotan5671]

This is awesome! This video really covered the ODE in a very interesting and tangible way! A side question: is it possible to add an experimental error term into ODE? For example, in linear regression, we have a gaussian error of N(0, sigma).

[@FreeMarketSwine]

Given multiple data points, solving for k would be as simple as taking an average of the values of k determined from each measurement?

[@norencezyron.bedasua]

I love this video!

[@cesarmoreno987y]

very helpful video

[@yaturengeyeemile2001]

urumwarimu w,umuhanga ndagushimiye ubutaha uzakore no kuri application of Electrical Circuits nkoreshesheje ururimi kavukire kuko nkunda igihugu cyajye.

[@wc3815]

The equation my school gave me is T - A = De^kt not -kt (but in the question, it is said that k is a negative constant) I dont get why it is negative here but not negative in my school..

[@pipertripp]

So now you've got to redo the experiment with a better ambient temp. I might have to find a decent thermometer around here and try it out for myself. Do you have a recommendation for some sort of text on the subject of building mathematical models? I would love to work through loads of examples, both analytical and numerical. Seems like with model building lots of practice would be a very good thing (so basically like everything else worth doing).

[@DrTrefor]

I really love this open source modelling textbook: github.com/bigfatbernie/IBLmodellingDEs

[@pipertripp]

@@DrTrefor noice! Thanks much. Downloading it now! I'm plowing through a massive numerical methods for engineers book ATM, but this one will be fun. I need to do a good review of ODEs and Calculus first. Then I need to really learn some basic stats, and I'm rubbish at linear algebra (that will change, but I'm struggling to get motivated). I really need to quit my job, lol. I should have been an aristocrat with loads of free time and all of this would be so much easier to manage.

[@DurlabhBiswas-x2s]

love this

[@wakeawake7167]

What is D in Newton's law of cooling?

[@Alex-nm8xn]

A question on this was in my course's book! Brilliant video. Do you ever plan to create a series on Partial Differential Equations?

[@DrTrefor]

Thank you! Yes this differential equations course is intended to have the following sections: 1st order ODEs, higher order ODEs, Laplace Transform, Fourier Series, Series Solutions, and PDEs. So probably won't be coming out until april/may if I'm being honest but will come.

[@Alex-nm8xn]

@@DrTrefor I'm more than willing to wait, looking very forwards to it all :))

[@taka4059]

How do we know that the constant of proportionality is actually constant?

[@DrTrefor]

We don’t! This is just a model. It’s validity is only proven by comparison to the real world

[@noahbarrow7979]

Hey Dr. Trefor, I re-watched this video again this morning and had a thought. What would have happened had we not multiplied our k by the difference between the temp of our cup and the ambient temp (T-A). For instance, what if we had gone ahead and selected a difference that wouldn't have been as steep as the selected one was. Indded we would have still gotten a solution and would have still been able to solve for k, right? How would we have known though that our solution and our k didn't match physical reality? I guess in the real world, I'd have then just iterated and eventually might have selected T-A, but in a math class would I be marked wrong if I had selected something other than what you selected? Or am i missing the point entirely haha? I guess it's about seeing what our percent error of a given model is vs. direct measurements we collect-though I also suppose not every problem we model will also have a thermometer. Also, I'm actually taking a thermo class right now (fascinating, incredible stuff) but was your reasoning about the larger difference between T and A based upon thermo principles or the fact that a steeper slope would yield juicier, more differentiable results? I'm asking because I really want to tune up my reasoning skills for problems like this. Thanks again!

[@DrTrefor]

To me a math model isn't "right" or "wrong", it's more or less useful. For instance not only could we have chosen different differences, we didn't have to make it linear. Maybe it should be (T-A)^2 or square root of T-A or any other number of things. Basically all we are doing is trying to think of something that seems reasonable, and the ultimate proof is not mathematical, its how well it models reality. So in this case it makes sense that the rate of change of temperature should have something to do with the difference between where we are right now and what the ambient temperature of the room is, this is a reasonable intuition. But I haven't proven that this is the best or only model of this scenario! I actually teach a math modelling course as well, and in it I typically don't mark students right or wrong too much based on their models per se, but based on their analysis and critiques of those models.

[@noahbarrow7979]

@@DrTrefor as always, you've given me a lucid, imaginative response. Thank you for being so responsive as well-you're setting new land-speed records! This is fascinating stuff. I'm really looking forward to the working and thinking through the entire playlist.

[@watermelon8057]

Why do we not care about the absolute value around the T - A when taking the derivative? I do not understand, is T - A always positive?

[@royaltelevision9]

Awesome

[@gonzaloob]

still dont understand why you put -k instead of just k tho. Like what does the - refer to??

[@whenyoudiporeosinmilktheyb8329]

You can also leave the - away but then k would be equal to -0,0409 instead of 0,0409. The function is inversely proportional

[@physicsandsciencelectures2787]

Tea is pathetic ... Hahahaha.. Thanks for the beautiful lecture.....

[@ΚωνσταντίνοςΛαζαρίδης-ξ9ι]

thanks!

[@gitgosc7075]

great!

[@sirwongalot7918]

Im doing an IA on how the surface area of a liquid affects the rate of change of temperature of it. Do you know if I could use this equation to derive an equation between the variables? I wanted to swap out Ambient temperature for the final temperature of the liquid after a set period of time as that is the final temperature. Would that be ok?

[@trexbattle]

Bruh I have an exam in this class on the 11th and u posted a cuppa vids

[@DrTrefor]

haha good luck! I'll get some more out by then but only a couple:D

[@trexbattle]

@@DrTrefor thx doc

[@AkashKumar-ff6mx]

Thanks sir

[@michaelrooze278]

What if you only have the initial temperature? Can you still solve it?

[@DrTrefor]

If you only have one data point htat isn't enough to solve both unknown constants.

[@heera_ai]

Are good modelers bad experimentalists?

[@kaifanzheng2686]

nice!!!!!

[@DrTrefor]

Thank you!!

[@shinkwrloggie7579]

I think the link to the textbook is not working :( By the way the content of your videos is great! I am preparing my 1st year Analysis exams and your channel is being very helpful :) Thank you so much!

[@DrTrefor]

Weird! The link is working for me, maybe the server glitched for a moment?

[@amanpandey1316]

sir, what if the value of k is not zero at t=0? I Think if we have different value of k the Ans may be differ.

[@kevinboykin8943]

K isn't zero when t=0, BUT, k times zero is zero.

[@KevHnh]

Great video as always. Expect nothing less from a man who was friends with xxxtentacion

[@DrTrefor]

haha i forgot about that!

[@abdullahe3674]

bless You Prof:)

[@JGambrosia]

Haha i love this so much.

[@lighteightshadow]

Nice video Dr. Bazett, I was looking for a differential equations video case on mixing of brine solution in 2 tanks and I think it would be great if you create one. Thanks

[@thistora929]

Dangh, didnt knoww ln can eliminate e. Math just got buffed

[@saadayad202]

❤

[@bhathiyaranasinghe6637]

[@gobyg-major2057]

Shouldn’t it be t of 5 instead of 7? If it was 7 minutes then the math should work but it was 5 minutes.....

[@stephaniesmith3428]

I think Dr. Bazett means 5 minutes after he took the temperature at 2 minutes. A total of 7 minutes after the initial temperature.

[@GatoArthur]

U JUST SAVED MY LIFE Thank you soooooooo much for your efforts!🥲