Rigorous Explanation of dW in Problem 121

In summary, the problem states that for a curve, dW = (kgcosΦ + mgsinΦ) ds, where ds is the differential element along the curve. They have found a way to solve for the work done against friction by breaking the curve into N segments and summing up the work done along the segments.
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EddiePhys
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In the solution of this problem(121), dW = (kmgcosΦ + mgsinΦ) ds, where ds is the differential element along the curve. Now they have done kmg dscosΦ + mg ds(sinΦ) = kmgdx +mgdy. Makes sense intuitively, but I want to know how this is rigorous. What I'm thinking is, the curve is broken into N elements of the same length along the curve over which a Riemann sum gives a line integral. But here cosΦ is varying, so wouldn't all the Δx's and Δy's not be of the same length? And if your explanation is that as Δx tends to zero the two become the same, then why can't we simply treat all differential elements as the same since they all tend to zero after all?

Sorry for the rambling at the end. But can someone just show me why such an operation is rigorous?
 
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The lengths of your segments don't have to be the same for you to sum them up. It's easier if you take equal sized horizontal steps, but let the vertical step vary.
 
  • #3
If they don't actually tell you the shape of the curve then it must be irrelevant (?). The answer must be independent of shape.
Without being too rigorous, I would suggest the work done would have to be mgh + kmgl . That would be PE gained + resultant work against friction, equivalent to dragging it along the ground first or last. Your infinitesimal approach is 'better behaved' and I think that, if you examine what cos and sin theta represent, in terms of δx and δy, I think that things should fall out when you work out the work along the hypotenuse of an infinitesimal sloping section. You will need to get your hands dirty with some trig and basic friction. You have to believe it will all fall out nicely and it will. :nb)
 
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1. What is the purpose of "Rigorous Explanation of dW" in Problem 121?

The purpose of providing a rigorous explanation of dW in Problem 121 is to ensure that the solution to the problem is logically sound and based on solid mathematical principles. It helps to eliminate any potential errors or flaws in the reasoning process and provides a clear understanding of the concept being discussed.

2. Why is it important to include a rigorous explanation of dW in scientific problems?

Including a rigorous explanation of dW in scientific problems is important because it allows for a more thorough understanding of the problem and its solution. It also helps to establish the validity and reliability of the results obtained, making them more credible and useful for future research.

3. What is the definition of dW in the context of Problem 121?

In Problem 121, dW refers to the infinitesimal amount of work done on a system, which is calculated by multiplying the force applied by the displacement of the system in the direction of the force. It is an important concept in physics and thermodynamics, and understanding its rigorous explanation is crucial for solving the problem correctly.

4. How can one ensure that their explanation of dW in Problem 121 is rigorous?

To ensure a rigorous explanation of dW in Problem 121, one should break down the problem into smaller steps and clearly define each variable and its relationship to the others. They should also use precise language and mathematical notation to support their reasoning and provide logical justifications for their calculations.

5. Are there any common mistakes or misconceptions when explaining dW in Problem 121?

One common mistake when explaining dW in Problem 121 is overlooking the direction of the force and displacement vectors, which can lead to incorrect calculations. Another misconception is assuming that dW is equal to zero when no work is done, as it can still have a non-zero value depending on the system and its conditions.

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