Hydrostatic Pressure Lab: Plate Submerged in Water. Purpose: Calculate the
center of pressure (yR) and resultant force (FR) of water acting on a wall.
Variables ...
Hydrostatic Pressure Lab: Plate Submerged in Water Purpose: Calculate the center of pressure (yR) and resultant force (FR) of water acting on a wall. Variables: ρ – Density of water g – Acceleration due to gravity γ – ρ*g h – depth a – Height of water touching the wall b – Width of water touching the wall A –Area of the wall α – Angle of the wall
Ixc, Iyc – Centroidal second moments of inertias [xc, yc] – Centroidal coordinates of the wall in a coordinate system where y runs along the wall and [0,0] is at the intersection of the wall and water surface (see figure 2). [xR, yR] – Center of pressure coordinates in same coordinate system as above. FR – Resultant force
Calculation of quantities: We know that pressure increases linearly with depth as: = (1) This pressure acting on a differential area results in a differential force: =
(2)
Since we will be changing the angle of the wall we want to express hc in terms of yc. Also, to find the force over the whole wall (resultant force) we will integrate over area. = = =
(3)
Although we have an intuition that this force should act at the centroid of the wall, it does not. Because pressure increases with depth the resultant force will always act at a location lower than the centroid. This location is known as the center of pressure. For a non-accelerating surface we can use a balance of moments to find the location of the center of pressure: = = (4) Since FR=γAyccos(α), we find = Finally we use the parallel axis theorem
= (5)
= (6) To put yR in terms of the centroid
= (7)
For a rectangular area Ixc is =
(8)
For more information on this derivation see section 2.8 of the textbook.
Figure 1: The Hydrostatic Pressure Apparatus
With the apparatus shown in Figure 1 we will be able to measure the resultant force acting on a tilted wall. The measurement process will involve balancing moments about a fulcrum (see Figure 2). The first moment is produced by a set of weights (Fmass) attached to a lever arm (Lmass). The opposing moment is produced by the resultant force (FR) acting at a lever arm that is determined by the center of pressure (LR). = → =
(9)
*Hint: The differences between fully and partially submerged plates are vital for your calculations. Pay extra attention to the meanings variables and their origins. For this lab you will be choosing Fmass and Lmass and measuring water levels to determine LR. Note that this process is not as straight forward as it seems. You will have to use some trigonometry along with equation (7) to arrive an answer. Since you will have to repeat this calculation for many values I suggest using software to do this. How does your measured FR compare to the theoretical calculation described by equation (3)? Figure 2: Schematic
Procedure: I. II. III. IV. V. VI. VII.
VIII. IX. X. XI.
Find the apparatus and make sure that you have a pitcher and short length of hose Level the whole apparatus using the tube level Remove all weights and tip the apparatus upside-down to remove any water. Select an angle using the detent (2), we will do 0, 28, and xx degrees Move the sliders (3) forwards or backwards until the bull’s eye level is centered and the apparatus is balanced Add the weight hanger to the rider (6) SLOWLY pour water into the water vessel (1) until the bull’s eye level is again centered. If you pour too much water use the length of hose to pipette small amounts of water out until balanced. Record all necessary information Add 0.5 N to the weight hanger Repeat VII-IX six times so that your data can be averaged. Change to the next angle and repeat II-X
