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CIEG-306 Fluid Mechanics Laboratory

1. Hydrostatic Pressure and Force on Submerged Surfaces

Objective

To determine the force that the water acts on the submerged part of the torroid and tocompare with the theory of hydrostatics.

Apparatus

The torroid device is sketched in Figure 1. The torroid is suspended at a point O and can be

rotated around point O. A scale (to the left in the figure) can be loaded to balance the forces

acting on the torroid.

Theory

Reference is made to the chapter in the CIEG305 textbook on Fluid Statics. We consider a

submerged surface in a stationary fluid. When a fluid is stationary, it has only normal stress,

which is called pressure, but it has no shear stress. Hence, any submerged surface in a

stationary fluid would experience hydrostatic force. Another characteristic of stationary fluid is

that its free surface is always perpendicular to the direction of gravitational acceleration. What

we need to determine here are the magnitude and location of the hydrostatic forces acting on

the submerged surface of the torroid (see figure 1). The hydrostatic pressure p below the

horizontal free surface is given by

p =y + patm (1)

where is the specific weight of the fluid, yis the vertical distance below the free surface, andpatm is the atmospheric pressure which can be taken to be zero. (Explain why this does not

change your results). The fluid pressure acts normal to the surface of an object and is positive in

the direction into the surface. Integration of the pressure over a submerged surface yields the

total hydrostatic pressure force acting on that surface. Similarly the resultant moment about a

suitable specified point can be obtained by integrating the moments from the pressure over the

body surface. Through total moment of momentum balance, the rotating part of the equipment

is balanced with the load W on the scale. During the lecture, we will go through the general

derivation of finding the force and location of force for a submerged plate and how the

problem shown in figure 1 can be solved efficiently. The derivations related to these steps are

required in your report.

Procedure

First, with the tank empty, adjust the balance arm until it is horizontal as indicated by the level

located on top of the arm. Allow the tap water to run for several minutes before using it to

safeguard against fluctuations in the temperature. Fill the tank with water until the water

surface is level with the torroid bottom. Record this depth. Then, the group should repeat the

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following procedure with 10 combinations of added weight W, in order to get 10

measurements.

1. Place weights on the scale pan.

2. Allow water to flow into or out of the tank until the balance arm is again level.

3. Record this depth of water and the weight required to restore equilibrium.

Also record the water temperature to determine the specific weight (each member of the

group takes at least one reading before the experiment and one after to establish a sample).

Data Reduction and Report

Derive the equations (outlined above) needed for your data analysis.

Is there hydrostatic pressure force acting on the circular surface of the torroid? How

does it contribute to the moment acting on the arm?

Using the measured data (i.e., the water depth), calculate the force and the position Y cp

of the pressure center on the vertical plane, and the resultant moment M f about the

pivot point O.

Explain why this moment should balance the moment resulting from the weight on the

scale pan, Mw.

Plot Mfversus Mw.

Perform a linear regression analysis between Mfand Mw. How do the results compare to

what would be theoretically expected?

Finally, compute a correlation coefficient r for the plot (M f , Mw). (See the handout

Simple statistical measures for data analysis). What does r tell you about your

measurements?