Surface Forces
Objectives
In this laboratory you will measure surface forces acting on a round port in the sloping side of
a small tank. You will compare measured values of force with theoretical values. We will also
measure the difference between the centroid and the center of pressure.
Theory
Because pressure increases with depth the pressure acting on a submerged surface is function
of depth. The slant distance from the centroid to the line of action of the resulting force is
yR  
 g cos I xc
pc
A
1.1
For a circle we can substitute the following
I xc rs2

A
4
1.2
Substituting equation 1.2 into equation 1.1 and solving for the slant distance between the line
of action and the centroid we obtain
yR  
 g cos rs2
pc
4
1.3
The slant distance, d, between the hinge and the line of action is equal to the slant distance to
the centroid plus the distance between the centroid and the line of action
d  c  yR
1.4
Substituting equation 1.3 we obtain an equation for the lever arm, d, based on the geometry
and the depth of the water.
d c
 g cos rs2
pc
4
1.5
We will compare this calculation of the length of the lever arm with a measured value
based on the moment exerted by the port.
The force acting on a submerged surface is
FR  pc A
1.6
where pc is the pressure (relative to a reference or datum pressure) acting on the centroid
of the surface.
FR  pc rs2
1.7
The moment applied by the water in the enclosed tank on the lever arm is
dFR  dpc rs2
The counterbalance moment applied by the water in the cylindrical tank is
1.8
aFb  apb rb2
1.9
where Fb is the force applied by the mass of water in the cylindrical counterbalance tank.
Setting the two moments equal (equations 1.8 and 1.9)
dpc rs2  apb rb2
1.10
apb rb2
d
pc rs2
1.11
and solving for d we obtain
The length of the lever arm
calculated using equation 1.5 is
based on the moment equations of
statics and the depth of the water.
The length of the lever arm
calculated using equation 1.11 is
based on the calculated force acting
on the port and the measured
moment of the counterbalance. If the
equations are correct these two
methods of calculating the lever arm
should yield the same value within
experimental
error.
The
experimental error will be the result
of measurement errors for pc and pb.
Experimental Methods
and Analysis
Fb
Fzero
a
pb
rb
hc
b
c
d
hc

Front view
of sloping
surface
rs
e Side view Fr
Figure 1-1.
Schematic
drawing
of
the
experimental apparatus used to measure surface forces.
The
static
surface
forces
apparatus consists of an enclosed
tank with a circular port that is covered with a thin membrane (Figure 1-1). A matching circular
port mounted on a hinge supports the membrane. The membrane is not attached to the plug and so
the membrane can only transmit pressure from the water to the plug when the water pressure is
greater than atmospheric pressure. The membrane transmits the pressure to the port without
requiring a leak tight seal. Although the port is hinged its motion is constrained by the load cell
and by stops so that it always supports the membrane.
The hinged system is connected to the tank through a load
cell (Figure 1-2) that can be monitored to determine when
Figure 1-2.
Load cell
the moment applied by the membrane is matched by the
moment of the counterbalance.
used to balance moments.
Water can be added or removed from the port tank by
using centrifugal pump. The pump is plumbed so that switching four valves can reverse the
direction of the water. It is important that the centrifugal pump not be operated dry. The rotor
relies on water for lubrication and the rotor will be damaged in about 60 seconds. Water enters
the centrifugal pump at the center of the rotor and exits at the periphery of the rotor.
The measured values of the apparatus parameters are given in Table 1-1.
Procedure
1) Drain the water in
the tank to below
the level of the
port using the
centrifugal pump.
2) Monitor the load
cell using the Easy
Data
Software.
3) The water in the
counterbalance
tank should also be
close to empty.
Table 1-1.
Parameter
rs
rb
c
a
b

e
Apparatus measurements.
Description
Radius of the port
Radius of the cylindrical counterbalance tank
Slant distance between the hinge and the
center of the port.
Lever arm of the counterbalance tank
Vertical distance between the center of the
port and the outside top of the tank
Angle between the vertical and the port
surface
Vertical distance between the bench top and
the center of the port
Value
3.774 cm
5 cm
16.5 cm
12.7 cm
13.82 cm
30 deg
7.784 cm
4) Reduce the moment acting on the load cell so that the voltage output is less than 0.5 mV by
adding (or moving) counter weights (Fzero) to the left of the hinge.
5) Make sure the adjustable stop bolt and the lip of the plug are not touching the tank so the
hinged lever arm is free to move in either direction.
6) Zero the load cell output using the Easy Data software by clicking on
.
7) Measure the initial height of water in the counterbalance tank using a ruler. Note that this
measure can be from any convenient datum.
8) Add water to the port tank until the water level is at the top of the port.
9) Add water to the counterbalance tank until the load cell reads zero.
10) Accurately measure the vertical distance from the centroid of the port to the surface where the
pressure is equal to the reference pressure, hc, either with a pressure sensor or using what you
know about statics.
11) Measure the height of water in the counterbalance tank.
12) Pump air into the top of the tank using the peristaltic pump until the air pressure is equivalent
to 20 cm of water and repeat steps 9-11.
13) Make appropriate adjustments and measurements for the other cases listed in Table 1-2.
Table 1-2.
Suggested trials.
Water
level in
Air pressure
Case
tank
in tank
pc pb
Top of
1
Atmospheric
port
Top of
20 cm of
2
port
water gage
5 cm
15 cm of
3
above port water gage
5 cm
-5 cm of
4
above port water gage
Middle of
5
Atmospheric
port
d
rs2 cos  
4hc
c
ahb rb2
d
hc rs2
Develop your own equations
Data Analysis and Discussion
1) Calculate the distance between the center of pressure and the centroid for each case based on
each method of measuring d.
2) Explain how you measured pc.
3) Create a plot of measured pb vs pc. On the same graph draw the line representing the
relationship given by equation 1.10.
4) Explain why d changed between the first 2 cases even though the free surface didn’t move.
5) Show using equations 1.1 and 1.6 that although FR=0 when the port is submerged with gage
pressure equal to zero at the centroid that the moment yR*FR is greater than zero. What is the
magnitude of yR*FR (in Newton meters) when pc is zero?
6) Explain when the location of the free surface is important and when it is insufficient in
determining the force and line of action for this tank.
7) Develop equations for the length of the lever arm, d, for the case where only half of the port
was submerged. Why is this case different than when the entire port is submerged?
Lab Setup
1) Create a configuration file for each workstation for Easy Data. Setup the Channel and apply a
conversion “From File” so that the load cell output is equivalent to the number of mL of
water that need to be added to the counterbalance tank.
2) Install a peristaltic pump with #18 tubing to pump air into the tank.
3) Install a centrifugal pump and 4 valves and a reservoir so that the tank can either be filled or
emptied quickly.