1. The problem statement, all variables and given/known data Please see attachment. 2. Relevant equations 3. The attempt at a solution Not really a homework problem. This is 1 of my exam question which I believe I had the right answer but my professor insists that he has the right one. I started by equating work done, W = integral F ds, with limit of integration from 0 to Lo. My professor's integral has limits from Lo to 2Lo. If I am correct, please tell me how I can explain it to my professor. If not, explain to me why. Thank you very much.
It seems clear that s goes from 0 to l_{o}. But are you sure your prof didn't simply substitute ❲s+l_{o}❳with another variable, say, x, where x goes from l_{o} to 2.l_{o}?
No substitution were made. He did it everything the same way except for the limits of integration. How can I explain to him?
The problem text says that s is the length the band is stretched by beyond the unstretched length Lo. ehild
If you get a rubber band, does it already sit at full length or is it laid in some weird oblong shape? In my mind, you would be doing some small amount of work to have the rubber-band laid out in a way to measure its full, unstretched, initial length. From this perspective, you then begin to stretch the rubber-band twice its length, so technically the force stretching the rubber band starts at the initial length and then commences at twice its length. This would be the distanced used to integrate the work done by the force on the rubber band.
I am sorry, but in this case, are you saying that my prof's limit of integration is correct? I understand that you need reference length for the force to be acting, but putting the initial limit of integration as something other than 0 would mean that you have stretched it beyond its natural length.(since 's' in the 'ds' is the amount of stretch)
Not sure what crashdirty86 is saying, but everyone else so far, me included, agrees with you: s goes from 0 to L_{0}.
Putting the shoe on the other foot? If trying to convince a disbelieving student, I might try this approach: Suppose you nominate some realistic data values (for F_{o}, l_{o}, and s) and plot the graph F vs. s. (Surely your prof would agree that the work done is represented by the area under the graph.) Then all that remains is to have his show whether his integral evaluates to the same answer as that graphical area, or yours. But, most likely he is just having a senior moment. After a restful night's sleep he'll probably smack his forehead and wonder what on earth he was dreaming about to say it so wrong.
F equals zero when s equals zero so don't both sets of limits give the same answer? If F is finite when s is zero then the professors answer would be correct. However, F is not finite for zero s so for this particular problem it doesn't matter which of the two sets of limits is used. I think I may be missing something here.
Is your professor someone for whom English is not his native language? " ... s is the distance the rubber band is stretched beyond its natural length." could be confusing, though comparison with any ordinary spring should clear it up in his mind.