A mass-spring system undergoes damped oscillations while submerged in a viscous oil. The maximum displacement (amplitude) of successive oscillations is recorded, where n=1,2,3,4,5,…n = 1, 2, 3, 4, 5, \dotsn=1,2,3,4,5,… represents the complete oscillation cycle number.
In one trial, six replicate measurements for the amplitude X5X_5X5 after 5 full oscillations are recorded in the table below:
4.22 cm4.28 cm4.68 cm4.25 cm4.21 cm4.29 cm\begin{array}{|c|c|c|c|c|c|} \hline 4.22\text{ cm} & 4.28\text{ cm} & 4.68\text{ cm} & 4.25\text{ cm} & 4.21\text{ cm} & 4.29\text{ cm} \\ \hline \end{array}4.22 cm4.28 cm4.68 cm4.25 cm4.21 cm4.29 cmIdentify the anomalous result, state your reasoned mean 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/cm)\ln(X_n / \text{cm})ln(Xn/cm) for various values of nnn:
nXn/cmln(Xn/cm)25.251.65844.54?63.931.36983.401.224\begin{array}{|c|c|c|} \hline n & X_n / \text{cm} & \ln(X_n / \text{cm}) \\ \hline 2 & 5.25 & 1.658 \\ \hline 4 & 4.54 & \text{?} \\ \hline 6 & 3.93 & 1.369 \\ \hline 8 & 3.40 & 1.224 \\ \hline \end{array}n2468Xn/cm5.254.543.933.40ln(Xn/cm)1.658?1.3691.224Calculate the missing value in the table for n=4n = 4n=4 to three decimal places.
The relationship between the amplitude and oscillation number is given by:
Xn=X0δ−n X_n = X_0 \delta^{-n} Xn=X0δ−nwhere X0X_0X0 is the initial displacement and δ\deltaδ is the damping ratio.
Explain how the value of the damping ratio, δ\deltaδ, can be determined from the gradient, mmm, of a straight-line graph of ln(Xn/cm)\ln(X_n / \text{cm})ln(Xn/cm) plotted against nnn. Units or numerical values are not required.
