**Experiment 1 - Center of Pressure**

**Object**
To locate the center of pressure of an immersed rectangular surface and to compare this position with that predicted by theory.

**Apparatus**

The following equipment is required for this experiment:

- Center of Pressure Apparatus (Figure 1.1).
- Set of claibrated weights.

Figure 1.1.

**Theory**

The magnitude of the total force F, acting on the face of a torroid is given by:

### F = p g h_{c} A

where
A = area of the rectanglar face (2 7/8" x 4"),

p = density of water,

g = acceleration due to gravity, and

h_{c} = depth of centroid G below surface of water = d_{1} - ½(depth of rectangular section)

__Theoretical Determination of the Center of Pressure__ x_{p}:

Theory shows that

### x_{p} = h_{c} + (I_{G} ÷ A h_{c})

where
I_{G} = 2nd moment of area of the immersed surface about the horizontal axis passing
through the centroid

The equation for the second moment of area of the immersed surface about the horizontal axis is:

### I_{G} = (b d^{3}) ÷ 12

where
b = breadth of the rectangular section, and

d = depth of the rectangular section

__Experimental Determination of__ x_{p}:

For equilibrium of the apparatus, moments about the pivot give:

### F y = m g s

where
F = pressure force

y = distance from pivot to centre of pressure

m = mass added to hanger, and

s = distance from pivot to hanger.

Therefore

### y = (m g s) ÷ F

but

### y = x_{p} + d_{2}

where

### d_{2} = R_{1} - d_{1}

Therefore

### x_{p} = y - R_{1} + d_{1}

where
R_{1} = distance from pivot to bottom of rectangular surface, and

d_{1} = depth to water surface from bottom of rectangular surface

__Experimental Procedure__

The following procedure is to be performed during this experiment:

- With no water in the tank adjust balance arm until it is horizontal by using the rider and the
fine adjusting screw. This will be indicated by the spirit level situated at the top of the balance arm.
- Open valve 1 and allow water level to rise until it is level with the lower rectangular face of the
torroid. Close valve 1.
- Using the hook gauge, set this water level to zero on the vernier scale.
- Again open valve 1 until the tank is nearly full. Close valve 1.
- Using hook gauge find the increase in dept of water d
_{1}.
- Place weights on scale pan to restore balance arm to the horizontal position.
- Record mass (m) and depth of water (d
_{1}).
- Repeat for various values of d
_{1}. The face should be completely submerged.

**Report**

- Locate the center of pressure x
_{p} for various values of depth.
- Plot this relationship and compare it with the theoretical relationship.
- When doing your calculations remember to include the forces due to pressure on the curved surfaces.
- Discuss your results.

#### Table 1.1. Data Sheet: Center of Pressure