# Accuracy Analysis of the Measurement of Centre of Gravity and Moment of Inertia with a Swing
## 2 Materials and Methods  
### Construction and Theoretical Values of the Swing  
The newly constructed swing is split into two parts: (1) support structure and (2) the swing itself. Figure 1 presents an overview of the experimental set-up. The support structure is made of two identical steel frames providing a stable support due to its mass and the extended footprint. It can be moved by two persons and allows the use of two different pivot axes of the swing. This oscillating part is made of aluminium to reduce the total mass of the moving section. It has an inner spacing of 1.3 by 1.3 m and is open at the top to enable crane access. The blade edges are constructed of a quadratic stainless steel profile, which is machined to ensure a sharp edge by mounting it with an angle of 45\({}^{\circ}\) onto the aluminium profiles. Those parts are coloured green in Figure 2, which presents the dimensions of the swing.  
The values presented in Table 1 are theoretical values based on the CAD model [30] constructed with the software Rhinoceros. The only exception is the mass of the inclinometer (Section 2.2), which is measured with a typical length of cable. The sum of the theoretical masses is 41.913 kg, which is slightly lower than the measured value of 41.980 kg (measurement conducted without the approximately 0.3 kg for the inclinometer). This difference may be caused by the weld bead at the bottom of the structure, which is neglected in this theoretical approximation. Based on these values, the total centre of gravity (CG) is 72.2 mm (Table 1) under the pivot axis P1 and 1272.2 mm relative to P2 (Figure 2). A value of 64.4 mm is found based on the theoretical mass of the swing (neglecting the inclinometer). The measured mass of the swing gives a value 0.067 kg greater than the theoretical value. This calculation is based on the assumption of a point mass located a half profile under P0 and half inclinometer over P0. The comparison with the measurement of the CG is presented in Sections 3.1 and 3.3 focuses on the evaluation of the oscillation period, which leads to the calculation of the MI.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Name** & **Volume** & **Density** & **Mass** & **CG Part** & **I\({}_{\text{P1}}\)** & **I\({}_{\text{P2}}\)** \\  & **(m3)** & **(kg/m3)** & **(kg)** & **(m)** & **(kg/m2)** & **(kg/m2)** \\ \hline frame & 0.013544 & 2700 & 36.568 & 0.165 & 35.423 & 102.540 \\
4 pivot blocks & 0.000681 & 7850 & 5.345 & \(-\)0.624 & 4.007 & 3.698 \\ mass difference & & & 0.067 & 0.963 & 0.062 & 0.314 \\ inlinometer & & & 0.300 & 0.917 & 0.252 & 1.344 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of the two parts of the swing—Frame includes the red and blue elements shown in Figure 2 and the four pivot blocks are shown in green—the reference for the distance to the centre of gravity (CG) is the height P1. Moment of inertia \(I\) is based on theoretical calculations.  
Figure 1: Overview of the swing with support structure (1), swing (2), inclinometer (3) and masses, including the rope highlighted with a red line (4) as well as the two different pivot axis (P1 and P2): (**a**) Balanced swing at the pivot axis P1. (**b**,**c**) Swing at the pivot axis P2 with full calibration weights. (**d**) Support structure moved to remove the swing.