Fluid Statics

Introduction:
In the study of fluids we may take into account the interactions of an object submersed, or partially submersed, in a fluid substance.  In this experiment we observe the upward force exerted on an object in such a scenario known as buoyancy.  We will consider three methods of obtaining a numerical value for the buoyant force for the purpose of comparing the three measurements and the uncertainties associated with each.  The three methods are named as follows: A) Underwater Weighing Method B) Displaced Fluid Method C) Volume of Object Method.  Fresh water will be used as our fluid substance and the density ρw = 1000 kg/m3 will be assumed.

Steps:


A) Underwater Weighing Method


In using this method, we first take a cylindrical mass and find the downward force due to gravity with the assistance of Logger Pro equipment.  We attach the mass with a string on the Dual-Range Force Sensor and measure the weight W of the cylinder in air(given by the tension force).  Next, we submerge our cylinder in water and measure the new tension Tsub . We know the buoyant force will account for the different in tension readings.  The submerged free body diagram is shown below.

The buoyant force experienced by the submerged cylinder will  contribute to the upward force to cancel with the downward weight W.

The Logger Pro Dual-Ranged Force Sensor will measure the new tension Tsub. 

We measure our weight and and tension to be
W = 1.09 ± .01N
Tsub = .71 ± .01N 

Thus using the free body diagram and rearrangement we may find our buoyant force.
ΣFy = T + B - W = 0N
B = W - T
B = .38    ± .02N 

B) Displaced Fluid Method

This method requires a measurement of the amount of fluid displacement  that occurs when the cylinder is submerged in our body of water.  In principle, the weight of this displaced fluid should be the buoyant force exerted on the cylinder.  We fill a graduated cylinder to a maximum level so that any object placed inside will cause that much water to be displaced onto a overflow beaker placed below.  Acquiring the mass of the beaker, rescaling it with the displaced water, and taking the difference of these two measurements will give us the mass of the displaced water.  Finally, calculating the weight of this water will yield the buoyant force.

The overflow of water is sent into an overflow beaker below to obtain the weight of displaced water.

 The measurements are as follows:
Beaker m: .141 ± .001kg
Beaker + water m: .180 ± .001kg
Water m = (Beaker + water m) - (Beaker m) = .039 ± .002kg
Wmax = (.039+.01) * 9.8
Wmin = (.039-.01)*9.8
Wf = B = Water m * g = .38 ± .02N

C) Volume of Object Method

This method uses Archimedes Principle that states the upward force (buoyant force) on an object is equal to the weight of the displaced fluid created by the object.  We will simply measure the dimensions of the cylinder using calipers and acquire the volume V of the cylinder. Using the density of water ρw we may calculate the mass of the displaced fluid and thus find the buoyant force by the the equation B = Vρwg.

A method of measuring the dimensions of the cylinder using calipers.

 
 The volume should be given by V = πr2h
Caliper measurements were
h = .077  ± .001m    D = .025 ± .001m
Bmax = π(.025/2+.001)2(.077+.001)ρwg = .438N
Bmin =  π(.025/2-.001)2(.077-.001)ρwg = .309N
Uncertainty B =  Bmax-Bmin/2 = .06N
Thus
B = ρwVg = .37 ± .06N

Questions/Conclusions:
Comparing the three values for the buoyant force, we observe that the values for methods A B C are .38N ± .02N, .38N ± .02N, .37 ± .06N respectively.  Clearly, all methods are agreeable within their uncertainties and what is especially notable is the complete overlap of method A and B.  The nature of each experiment contributes to the overall uncertainty of our answer, but our numerical uncertainty is derived from simple minimum and maximum values of our buoyant forces.  Multiple measurements taken with a caliper in method C account for the greater uncertainty obtained.
Of the three experiments,  method A seems most accurate, or at least more precise, based on the uncertainty obtained and the fact the the Logger Pro instrumentation was most reliable.  Although A and B both resulted in similar uncertainty, method B relies on the spillage of water which is affected by variables including surface tension adhesion and incomplete runoff from the graduated cylinder.

Considering if the cylinder had been touching the bottom of the water container:
According to our equation for buoyant force, B = W - Tsub , the buoyancy may be calculated with tension.  If the cylinder were touching the bottom of the container then the string would go slack, causing our Dual-Force sensor to measure 0N of tension.  It is clear this would increase our calculation for buoyant force which would make our value too high.