Switching between distance variables in a simple seismograph

[shanepitts]

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

 

[paisiello2]

Doesn't figure 3.2.5 explain it?

 

[shanepitts]

Doesn't figure 3.2.5 explain it?

Here is the other Figure. Unfortunately, I still don't fathom??

 

[paisiello2]

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 - X_e) ...(Eq. 1)

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

Making the following two substitutions into (Eq. 1):
X_e = 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.