A simple pendulum undergoes damped oscillations in a vertical plane. The 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 oscillation number.
In one trial, six replicate measurements for the amplitude A4A_4A4 after 4 full oscillations are recorded in the table below:
0.312 m0.354 m0.318 m0.315 m0.309 m0.316 m\begin{array}{|c|c|c|c|c|c|} \hline 0.312\text{ m} & 0.354\text{ m} & 0.318\text{ m} & 0.315\text{ m} & 0.309\text{ m} & 0.316\text{ m} \\ \hline \end{array}0.312 m0.354 m0.318 m0.315 m0.309 m0.316 mIdentify any anomalous result, state your reasoned value for A4A_4A4, and calculate its percentage uncertainty.
The table below shows the amplitude AnA_nAn and the corresponding logarithmic values of ln(An/m)\ln(A_n / \text{m})ln(An/m) for various values of nnn:
nAn/mln(An/m)30.295−1.22160.264?90.236−1.444120.211−1.556\begin{array}{|c|c|c|} \hline n & A_n / \text{m} & \ln(A_n / \text{m}) \\ \hline 3 & 0.295 & -1.221 \\ \hline 6 & 0.264 & \text{?} \\ \hline 9 & 0.236 & -1.444 \\ \hline 12 & 0.211 & -1.556 \\ \hline \end{array}n36912An/m0.2950.2640.2360.211ln(An/m)−1.221?−1.444−1.556Calculate the missing value in the table for n=6n = 6n=6 to three decimal places.
The relationship between the amplitude and oscillation number is given by the relation:
An=A0γ−n A_n = A_0 \gamma^{-n} An=A0γ−nwhere A0A_0A0 is the initial release amplitude and γ\gammaγ is a constant known as the damping factor.
Explain how the value of the damping factor, γ\gammaγ, can be determined from the gradient, mmm, of a straight-line graph of ln(An/m)\ln(A_n / \text{m})ln(An/m) plotted against nnn. Units or numerical values are not required.
