# 4.5.1.1: Pressure Center

In the literature, pressure centers are commonly defined. These definitions are mathematical in nature and has physical meaning of equivalent force that will act through this center. The definition is derived or obtained from equation (145) and equation (153). The pressure center is the distance that will create the moment with the hydrostatic force on point "O." Thsu, the pressure center in the \(x\) direction is \[x_{p} = \frac{1}{F} \int_{A}xPdA \tag{159}\] In the same way, the pressure center in the \(y\) direction is defined as \[y_{p} = \frac{1}{F} \int_{A} y P dA \tag{160}\] To show relationship between the pressure center and the other properties, it can be found by setting the atmospheric pressure and \(ll_{0}\) to zero as following \[x_{p} = \frac{g \rho sin\beta I_{x'x'}}{A \rho g sin\beta x_{c}} \tag{161}\] Expanding \(I_{x'x'}\) according to Chapter on Mechanics results in \[x_{p} = \frac{I_{xx}}{x_{c}A} + x_{c} \tag{162}\] and in the same fashion in (\y\) direction \[y_{p} = \frac{I _{xy}}{y_{c}A} + y_{c} \tag{163}\] It has to emphasis that these definitions are useful only for case where the atmospheric pressure can be neglected or cancelled and where \(ll_{0}\) is zero. Thus, these limitations diminish the usefulness of pressure center definitions. In fact, the reader can find that direct calculations can sometimes simplify the problem.

### 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.