Switching between distance variables in a simple seismograph

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Discussion Overview

The discussion revolves around the substitution of distance variables in the context of a seismograph problem presented in a textbook. Participants are exploring the reasoning behind replacing the y variable with x, as illustrated in specific figures from the textbook, and the implications of these substitutions on the equations governing motion.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses confusion regarding the substitution of the y variable with x as per figure 3.6.3.
  • Another participant suggests that figure 3.2.5 may clarify the substitution process.
  • A third participant reiterates the suggestion about figure 3.2.5, indicating ongoing confusion.
  • A later reply explains the substitution of variables in detail, referencing Newton's 2nd law and providing equations to illustrate the simplification process.
  • The explanation includes a step-by-step derivation of the equations, concluding that the resulting equation (Eq. 3) is simpler than the original (Eq. 1).
  • The participant assumes that the textbook contains all necessary information to understand these substitutions.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the clarity of the figures or the substitutions, as confusion persists despite attempts to explain the process. Multiple viewpoints regarding the effectiveness of the figures in clarifying the problem remain evident.

Contextual Notes

Some participants may be missing assumptions or context provided in the textbook, which could affect their understanding of the substitutions and the equations involved. The discussion does not resolve the confusion expressed by the initial poster.

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Below is an example problem from my textbook. I don't fathom the highlighted portion where they replace the y variable with x in accordance with fig. 3.6.3?

Please help

Thanks

image.jpg
image.jpg
 
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Doesn't figure 3.2.5 explain it?
 
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paisiello2 said:
Doesn't figure 3.2.5 explain it?

Here is the other Figure. Unfortunately, I still don't fathom??
image.jpg
 
In figure 3.2.5 they are simply substituting in X'e to simplify the equation. From Newton's 2nd law of motion:
∑F = m⋅d2X/dt2
m⋅d2X/dt2 = mg - k(X - Xe) ...(Eq. 1)

At equilibrium we know that ∑F = 0, therefore:
mg - k(X - Xe) = 0
→ X - Xe = mg / k ...(Eq. 2)

Making the following two substitutions into (Eq. 1):
Xe = X'e - mg/k
X = X'e + x
→ m⋅d2(X'e + x)/dt2 = mg - k((X'e + x) - (X'e - mg/k))
→ m⋅d2(x)/dt2 = mg - kx + mg
→ m⋅d2x/dt2 = kx ...(Eq. 3)

Now Eq.3 is simpler than Eq. 1, agreed?

I assume your textbook already has all this.
 
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