Computing Work with Integral Calculus!

[@binbots]

Love this channel. Wish I had your expertise so I could graph this equation x/ln(x)-e^(1/ln(x) compared to the actual number of primes.

[@MathVisualProofs]

You can get there to check it out. Try Desmos?

[@binbots]

@ I use Desmos. I checked this equation all the way up to 10^308. It works amazing. But I’m an amateur. So I don’t know what to do with it next. Just want to get it out there.

[@MrGustavier]

This is awesome ! I had only seen mathematical stuff on your channel so far, this is the first time I see physical stuff. Did I simply miss it (do you have other physical stuff on your channel) or is it indeed your first ?

[@MathVisualProofs]

Pretty much the first real physics-y video. I do have one from a long time ago where I use moments of mass to prove the mean inequalities. I also have a short, which comes from a longer video, about using centers of mass to prove the sum of the first n positive integers formula.

[@MrGustavier]

@@MathVisualProofs That's great ! Do you plan on doing thermodynamical stuff ? Electromagnetic stuff ? General relativity stuff ? Quantum mechanics stuff ?

[@JasonOvalles]

Since the cable is uniform along its length, you can massively simplify this problem so it doesn't require calculus (the total mass of the cable is 5*50=250 kg, the force acts on its center of mass which is at (0+50)/2=25 meters, so the total work done is 250*25*9.8=61,250). I think the power of calculus becomes a lot clearer when thinking of something that isn't entirely uniform throughout (maybe an elevator where 50 kgs of construction material is unloaded on each floor? Or a tank of water that is leaking at a certain rate while being raised?). You could argue that you can still calculate the center of mass, but, well, that's still a calculus problem now! (I know this is for your Calculus course so it makes sense to start with a simple problem, even if it doesn't necessarily require calculus, as a foundation).

[@MathVisualProofs]

For sure. This is an intro as you noted so people can use similar techniques for more complicated situations :) but always good when you know you got the right answer because other techniques match!

[@in-depthpen-dent]

Could you do a video on the visual proof for standard deviation?thank you

[@incendiohawk1725]

I love these sort of problems with physical applications of integration. It is always so satisfying when you realise that you are just adding up lots of tiny pieces and it turns into an integral. The only thing I am missing is why we can’t just treat the rope as 1 object and calculate the work done for that object. Is it assumed that the sections of rope that reach the top just disappear so that a force is no longer applied to them; or do we just ignore the extra work done to lift those sections further, as would happen if lifting the cable from a point higher than the top of the building, as they aren’t part of the quantity asked for in the question; or is it something else that I haven’t thought of?

[@MathVisualProofs]

Your first thought is right. Not all parts of rope move same distance. Once one pice gets to the top we stop moving it any distance so no work is done on that bit anymore.

[@RabanRaphael]

Like the way you explained can you help me out to explain the line integral coz it gives me headaches

[@jovidmunosibshoev2502]

Please, more vedios great channel

[@MathVisualProofs]

Thanks!

[@Newworld-r7j]

In the cable problem should the density me kg/m^3 and i think you assumed the cable only in lenth so the density given in the question is actually mass per unit length right?

[@MathVisualProofs]

Yes. If you wanted to do it more realistically you want mass per volume. Then you need to compute the volume of the slice. In this problem we just assume the cable mass depends only on length to help (you can assume the cable is thin enough that maybe the extra scale factor is negligible).

[@freitchetsleimwor2406]

Compute work? I sure hope it does!!!

[@AbhishekKumarSingh688]

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