Switching between distance variables in a simple seismograph

shanepitts · May 1, 2015

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

Textbook page showing Example 3.6.1: a seismograph modeled as a mass suspended by springs and a dashpot from a platform attached to the Earth, with the problem statement and the start of the solution.

Diagram of a seismograph model: a platform hangs a spring and a damping pot above a mass m, with labelled distances z and y measured from a dashed reference line.

paisiello2 · May 1, 2015

Doesn't figure 3.2.5 explain it?

shanepitts · May 1, 2015

paisiello2 said:

Doesn't figure 3.2.5 explain it?

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

Diagram of a mass m hanging from a ceiling by a vertical spring, showing natural length Xe, stretched length X, new equilibrium position X'e = Xe + mg/k, and displacement x below it.

paisiello2 · May 2, 2015

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.

Switching between distance variables in a simple seismograph

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