# Accuracy Analysis of the Measurement of Centre of Gravity and Moment of Inertia with a Swing
## 2 Materials and Methods
### Basic Equations  
Two different sets of equations are needed to calculate the centre of gravity (CG) as well as the moments of inertia (MI). The location of CG is based on steady measurements of an inclined swing due to added calibration weights. Knowing this value allows for the calculation of the MI based on the period or frequency of an oscillating system. The presented experimental set-up enabled measurements to be conducted with two different pivot points, but is limited to one direction at a time. For the calculation of the CG, the starting point is [PERSON]'s first law and the assumption that the system is in equilibrium and not moving. A zero acceleration results in a zero sum of forces in all directions. Furthermore, it can be assumed that the system can be reduced to a two-dimensional problem, hence all imbalances along the pivot axis are absorbed by the small changes in the reacting forces at the blade edges. Ideally, each side holds half of the sum of the mass of the swing \(F_{S}\), the additional model \(F_{M}\) as well as the additional calibration weights \(F_{W,j}\). The latter is applied to incline the swing with an angle of \(\alpha_{i}\). The purpose of the following equations is to connect the individual forces \(F\) and distances or levers \(d\) based on \(i\) measurements with different \(F_{W,i}\) at \(d_{W,j}\). To connect those, the moment balance of one example configuration is investigated in Figure 3. In this example a large angle \(\alpha\) of \(10^{\circ}\) is applied and the distances of the chosen to be easily identifiable. The sum of the moments/torques around the pivot point (red triangle in Figure 3) can be calculated based on the following equation:  
\[\sum M=0=F_{W}\;\;d_{W}\;\;\cos(\alpha)-(F_{S}\;\;d_{S}+F_{M}\;\;d_{M})\;\;\sin (\alpha) \tag{1}\]  
All forces in this equation are known by weighting the model as well as the swing and for the additional \(F_{W}\) specific calibration weights are used. The distances \(d_{S}\) and \(d_{M}\) describe the distance of the pivot centre to the centre of gravity of the swing or the model, respectively. The distance \(d_{W}\) is also constant and predefined. By using the solution with the rope, the added force always acts at the same point and with a fixed distance to the pivot point. Alternatively, the calibration weights could be placed on the swing but in this particular case, the specific location of the CG for the masses has to be calculated, which could also result in inaccuracies. Furthermore, the angle \(\alpha\) is the key output of the measurement and can be calculated based on the following equation:  
\[\tan(\alpha)=\frac{\sin(\alpha)}{\cos(\alpha)}=\frac{F_{W}\;\;d_{W}}{F_{S}\;\;d _{S}+F_{M}\;\;d_{M}} \tag{2}\]  
The distance \(d_{M}\) should be the only unknown for future applications, which can fitted to a point cloud of different additional masses \(F_{W}\) at difference distances \(d_{W}\). Therefore,the swing is analysed without an additional model (\(F_{W}=d_{W}=0\)) and the distance \(d_{S}\) representing the distance between the CG of the swing to the pivot point can be calculated based on the following equation:  
\[d_{S}=\frac{F_{W}\ d_{W}}{F_{S}\ \tan(\alpha)} \tag{3}\]  
This equation is utilised and further investigated in Section 3.1 based on measurements of a stable new equilibrium of the swing due the additional calibration weights. In a second step, the moment of inertia (MI) is evaluated based on the period of an oscillating swing based on the following equation for a free oscillating pendulum around a horizontal axis [1]:  
\[T=2\pi\sqrt{\frac{I_{P}}{m\ g\ R}}\to I_{P}=\frac{T^{2}}{4\pi^{2}}\ m\ g\ R \tag{4}\]  
The variable \(I_{P}\), representing the MI around the pivot axis, can be calculated based on the period of the oscillation \(T\), the mass of the rigid body \(m\), the constant gravity acceleration \(g\) and the distance \(R\) (length between the pivot point and the CG), which is equal to \(d_{S}\) from the previous CG calculation.