For the first part, the weight of the cylinder was determined. We measured the mass of the cylinder to be 0.1128 +/- .0002 kg. This makes the weight of the cylinder to be:
Wcyl = mg = 0.1128 +/- .0002 * 9.81 = 1.107 +/- 0.0020 N.
With the help of the force probe, we could get the tension force when the cylinder got submerged inside a beaker.
T = 0.7530 +/- 0.051
Looking at the freebody diagram, we have:
Statics principles then suggest the following expression:
T + B = mg
or,
B = mg - T = 1.107 +/- 0.0020 - 0.7530 +/- 0.051 = 0.35 +/- 0.051 N
For part B, we measured the buoyant force based on how much water leaked from the container when we slowly dropped the weight.
We measured the mass of the beaker to be Beaker m = 0.04032 +/- .00003 kg.
After dropping the mass in the water, the displaced water that was poured in the beaker was:
Beaker + Water m = 0.04031 +/- .00002 kg, but having known the mass of the empty beaker we got the mass of the water to be:
Water m = (Beaker + Water m) - Beaker m = (0.07639 +/- .00002) - ( 0.04032 +/- .00003) = 0.03607 +/- 0.000036 kg
The buoyant force will be the mass of the water displaced times gravity to yield:
B = (water m)(g) = ( 0.03607 +/- 0.000036)(9.81) = 0.3538 +/- 0.00035 N.
In the last part of the experiment, the buoyant force was measured by involving the volume of a cylinder ( V = πr2 h).
The measured height and diameter were:
h = 0.076 +/- 0.0005 m d = 0.0253 +/- 0.0005 m, with r = 0.0127±0.00025
The calculated volume was then:
V = πr2 h = π (0.0127±0.00025)^2(0.076± 0.0005) = 0.000038±0.0000011 m^3
Having the volume, and the value for the density of water = 1000 kg/m^3 , we can express the mass as:
density = m/v, density * volume = m
m = ( 1000)(0.000038±0.0000011) = 0.038±0.0011 kg
We now can express the weight of the water as
weight water = m water * g = (0.038±0.0011)*(9.81) = 0.37 ± 0.011 N
The buoyant force, according to Archimede's principle, will be
weight of the water = B = 0.37 ± 0.011 N.
When comparing the three buoyant forces, we can see that the first two methods are very close to one another. It is the last one that is off by about 0.02 N.
I think the best way to come up with buoyant force was the second method since the masses were weighed in the same scale. In the first method, the force probe is not very reliable because the probe itself is made out of other measuring devices which could greatly affect the results of the experiment. In the third case, the density of water varies with pressure and temperature, and also it had two more possible sources of error due to miscalibration: the scale and the caliper.
If in part A, the cylinder had been touching the bottom of the container, then the tension would have been compromised: there would be less tension or even none at all. This would have yielded a smaller T and larger B, or T would have been zero, modifying the static equilibrium equation to B = (mass of cylinder)(g).
