An automotive shock absorber piston undergoes damped harmonic oscillations during a quality-control test. The amplitude of successive oscillations is monitored, where N=1,2,3,4,5,…N = 1, 2, 3, 4, 5, \dotsN=1,2,3,4,5,… represents the cycle number.
In one test run, six replicate measurements for the amplitude X5X_5X5 after 5 completed cycles are recorded in the table below:
12.5 mm12.8 mm15.2 mm12.3 mm12.6 mm12.8 mm\begin{array}{|c|c|c|c|c|c|} \hline 12.5\text{ mm} & 12.8\text{ mm} & 15.2\text{ mm} & 12.3\text{ mm} & 12.6\text{ mm} & 12.8\text{ mm} \\ \hline \end{array}12.5 mm12.8 mm15.2 mm12.3 mm12.6 mm12.8 mmIdentify any anomalous result, state your reasoned value for X5X_5X5, and calculate its percentage uncertainty.
The table below shows the amplitude XNX_NXN and the corresponding logarithmic values of ln(XN/mm)\ln(X_N / \text{mm})ln(XN/mm) for various values of NNN:
NXN/mmln(XN/mm)412.342.513810.15?128.352.122166.871.927\begin{array}{|c|c|c|} \hline N & X_N / \text{mm} & \ln(X_N / \text{mm}) \\ \hline 4 & 12.34 & 2.513 \\ \hline 8 & 10.15 & \text{?} \\ \hline 12 & 8.35 & 2.122 \\ \hline 16 & 6.87 & 1.927 \\ \hline \end{array}N481216XN/mm12.3410.158.356.87ln(XN/mm)2.513?2.1221.927Calculate the missing value in the table for N=8N = 8N=8 to three decimal places.
The relationship between the amplitude and the cycle number is given by the relation:
XN=X0β−N X_N = X_0 \beta^{-N} XN=X0β−Nwhere X0X_0X0 is the initial amplitude and β\betaβ is a constant known as the damping constant.
Explain how the value of the damping constant, β\betaβ, can be determined from the gradient, mmm, of a straight-line graph of ln(XN/mm)\ln(X_N / \text{mm})ln(XN/mm) plotted against NNN. Units or numerical values are not required.
