Hydrostatic Force - Slide/Image
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Title
Introduction and
Objectives
Audio/Text
Music starts
Text: This video is intended to provide guidance for carrying out a laboratory experiment to measure the hydrostatic force on a submerged surface.
You would be well advised to have paper and pencil handy when you're viewing this program so that you can make relevant notes on some of the key points, dimensions and other figures that will be given to you.
Although the theoretical procedure for calculating resultant pressure on a submerged surface can be applied to any arbitrary shape, we're going to deal here with a relatively simple plane rectangular surface.
The objective then is to investigate the validity of theoretical procedures intended for calculating the resultant pressure forces acting on a submerged plane rectangular surface.
Equipment Features
-set-up
-adjusting
-calibrating
The most important feature of this experiment is a balance arm which you can see here on which is mounted an accurately machined quadrant that is one quarter of a toroid. When in use the orange shaped quadrant is partially immersed in water and is positioned relative to the knife edge shown by the blue pencil such that the hydrostatic force on the curved surfaces produces no moment about the knife edge.
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The balance arm should first be set in position with the knife edges in the appropriate sockets or V-notches. Water is then added to the tank until the level is as close as possible to the bottom edge of the quadrant.
The correct level can be gauged quite accurately by looking at the rectangular edge of the quadrant with the eye almost at the water level. The reflection of the two fortyfive degree lines should form a perfect cross when the water level is coincident with the lower edge of the quadrant. This technique avoids errors due to the meniscus, which is produced by the surface tension of the water.
If not already level the balancing arm should be adjusted by means of the weight, indicated by the blue pencil, which operates on a screwed rod.
The weight should be adjusted until the bubble of the circular spirit level is centered in the red circle.
When balancing the arm, make sure that the weight pan is correctly attached and free of any loose weights.
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The depth gauge should now be calibrated to obtain a zero reading for the initial water level. Notice that this need not be set to a zero reading.
In fact it is much better practice to set the pointer to the water level and record the vernier reading which can then be applied as a correction to subsequent readings.
The gauge is set using a hook point that is adjusted to touch the underside of the water surface.
Once again, the reflection is used as a guide to avoid the problem of surface tension meniscus.
First experiment The first experiment may now be carried out by adding water to the tank to increase the water level by approximately one centimeter. It will be noted that this immediately puts the balance arm out of balance.
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The situation may be visualized with the aid of this diagram in which the depth of water producing a resultant pressure F on the end of the quadrant is indicated by the dimension Y. The pressure acting on this face produces a counter-clockwise moment unbalancing the arm.
Balance can be restored by adding an appropriate mass M to the weighing pan at the right hand end of the balance arm.
-Measurements &
Dimensions
The required to perform the moment calculations are:
1. the lever arm of the mass and the weight pan measured from the knife edge to the point of application of the hanger carrying the weight, X;
2. the inner radius and outer radius of the quadrant, R and
R+ D respectively; and
3. the breadth B of the rectangular face
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-Constants
-Using & storing weights
In order to complete the calculations you are required to know certain physical constants relevant to the system. For example, the specific weight of water may be taken as
9802 newtons per cubic meter and in the SI system of units, gravitational acceleration is 9.802 metres per second squared.
Obviously the key to successful measurements is careful weighing of the force necessary to restore the balance.
You are provided with a simple set of metric weights as illustrated here and you should take a little time to examine the way in which these are stored in the box and also the particular values of the small weights less than one gram.
It is a good practice to get into the habit of weighing by adding weights. to the pan in descending order then total the amount by looking at the box to see which weights are missing. The calculation can be checked by noting and totaling the weights in the pan to make sure that there has been no error.
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Calculating and
Plotting Results
Finally the weights are restored to the correct positions in the box and a third total of the weight may be made mentally in returning each weight one at a time.
Since the slight movement of the quadrant in restoring the balance arm to a horizontal position may produce a very slight difference in the water level it's good practice to measure the water surface elevation and thus the depth Y after the balance has been made.
Let's turn our attention now to the calculation of results and the way in which these experiments may to use to cheek the theory on which this is based.
As illustrated in this slide, the pressure distribution is in the form of a triangle as long as the depth of immersion Y is less than the surface height D.
The centroid of the immersed surface is located at a depth
Y bar below the water surface and the magnitude of the resultant pressure force can be calculated as F=gamma
(specific weight) x Y bar x the area A. In this case both Y bar and A are simple functions of the depth Y.
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-Formula #1
The centre of pressure is located at a depth YP below the water surface where Ycp is given by the general form shown here.
By substituting the appropriate expressions for the simple rectangular shape used here, we obtain an expression for
YCP as a function of Y.
Here Y.P is two thirds of Y.
For convenience the mass may be divided by Y
2 so that the results may be displayed on a linear graph.
You should perhaps take a moment to copy down formula number one on this slide before proceeding to the next section.
Music (pause)
We see here that the quantity M/y
2 can be expressed as a linear function of y and that the slope and intercept of the line can be calculated in terms of the quantities and dimensions that were measured earlier.
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The objective of the first part of the experiment may be carried out by plotting experimental points against this theoretical straight line to see what kind of agreement is obtained.
These slide shows a typical result of a number of experiments and you will probably find that for very small values of y a significant error is introduced into the experiment by the presence of surface tension acting on the outer curved surface of the quadrant.
For this reason it is sometimes advisable to add a few drops of detergent or wetting agent to the water in the tank so as to reduce the surface tension and thus minimize the error at small values of y.
When the depth of immersion y is greater than D the pressure diagram is no longer triangular and the calculation must take a different form.
The centroidal depth y bar is now given as y-D/2 and the resultant pressure force F is calculated as before.
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Rotation Axis
Relationship
Instructions to students
The centre of pressure may be obtained by the same general formula as used previously and you should take a few minutes to satisfy yourself as to how this expression was obtained.
Taking moments about the axis of rotation leads to the relation shown here. Note that the centroidal depth y bar and not the actual depth y is used in developing this expression. In presenting your report you should demonstrate how this formula is obtained.
The relation shown in the previous slide is illustrated graphically here by using the reciprocal of the centroidal depth as abscissa a linear graph is obtained as shown here.
Points corresponding to five or six experimental observations may be plotted on a diagram such as this and compared with the theoretical line.
You should now be in a position to carry out the experiment. If any point is not clear view this tape-slide program again. If you are still unsure of the procedure contact the course instructor or a teaching assistant.
Obtain results for the two cases using about five or six depths in each.
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If you follow the procedure illustrated here and carry out the weighing accurately, you should obtain reasonable results.
Please leave the apparatus in a neat and tidy condition and make sure that all the weights have been returned to the box.
Report Instructions Each group must submit a laboratory report. The report should follow the format provided in the course outline handout.
Closing Credits Closing Credits
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