I Getting the sign right with the work-energy theorem

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The discussion revolves around the application of the work-energy theorem in a specific physics problem, highlighting confusion over the sign convention used in the equation W = Tinitial - Tfinal. One participant notes that using W = Tfinal - Tinitial leads to nonsensical results, indicating a sign error in the textbook. Additionally, the power equation P = \mathcal{E}^2/R is discussed, revealing another sign error that offsets the first. The blurred print in the source material adds to the confusion, particularly regarding the definition of work done on the rod. Overall, the conversation emphasizes the common issue of sign errors in physics problems.
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getting the sign right with the work-energy theorem
This is problem 3 in section 2.3.4 from Conquering the Physics GRE by Kahn and Anderson:

problem.JPG

And here is the solution from the book:
solution.JPG

The point of confusion for me is that they use the work-energy theorem in the form W = Tinitial - Tfinal, instead of the other way around. If I were to do this problem, I would write W = Tfinal - Tinitial, but then I end up with a negative time, which does not make sense. What am I missing?
 
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You are right that the work performed on the rod should be ##W = \dfrac 1 2 m \dot x^2 - \dfrac 1 2 m \dot x_0^2##. So, the text has a sign error here.

But ##P = \mathcal{E}^2/R## gives the rate at which KE of the rod is transformed into Joule heat in ##R##. So, ##P## equals the rate of loss of KE of the rod. Thus, ##P = -\dfrac {d}{dt} (KE)_{\rm rod}= - \dfrac {dW} {dt}##. So, the text has an additional sign error here that compensates for the first error.

The print in the margin of the image is blurred, so I'm assuming ##W## is the work done on the rod.
 
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Thanks a lot! I constantly run into problems with minus signs. I think they make up 98% of all the mistakes that I make. It's very frustrating. And yes, the text says "the work performed on the rod". Sorry about the bad quality of the scan. I scanned the page several times and this is the best that I could do.
 
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