# 4.6.1.2: Metacentric Height, \(\overline{GM}\), Measurement

The metacentric height can be measured by finding the change in the angle when a weight is moved on the floating body. Moving the weight, \(T\) a distance, \(d\) then the moment created is \[M_{weight} = Td\tag{56}\] This moment is balanced by \[M_{righting} = W_{total}\overline{GM_{new}}\theta\tag{57}\] Where, \(W_{total}\), is the total weight of the floating body including measuring weight. The angle, \(\theta\), is measured as the difference in the orientation of the floating body. The metacentric height is \[\overline{GM_{new}} = \frac{Td}{W_{total}\theta}\tag{58}\] If the change int he \(\overline{GM}\) can be neglected, equation 58 provides the solution. The calculation of \(overline{GM}\) can be improved by taking into account the effect of the measuring weight. The change in height of \(G\) is \[\not{g} m_{total} G_{new} = \not{g} m_{ship} G_{actual} + \not{g} T h \tag{59}\] Combinging equation 59 with equation 58 results in \[\overline{GM_{actual}} = \overline{GM_{new}} \frac{m_{total}}{m_{ship}} - h\frac{T}{m_{ship}}\tag{60}\] The weight of the ship is obtained from looking at the ship depth.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.