Experiment 1: Fluid Statics

The purpose of this lab is to measure the buoyant force of an object with three different methods. In this case, the object is a metal cylinder.

Procedure: For the first method, we attached the cylinder to a force gauge and recorded the string tension. Then we submerged the cylinder in water and again recorded the string tension. For the second method, we took a large graduated cylinder and filled it to the top with water. We then placed the graduated cylinder in an overflow container and put the metal cylinder into the graduated cylinder. We then measured the weight of the overflowed water. For the third method, we measured the radius and height of the metal cylinder, then calculated the weight of the displaced water.

Data: The difference in tensions when the cylinder was submerged in water and hanging in air is the buoyant force. When hanging in air, the tension read 1.09 Newtons. When submerged in water, the tension read 0.666 newtons. This means that the buoyant force is 0.42 Newtons.
Using the overflow method, the displaced water 0.39 Newtons. According to Archimedes principle, this measurement is equal to the buoyant force.
For the final method, the diameter of the cylinder measured in at 25.7mm, and the height measured in at 76.7mm. This means in meters the height measures 0.0767m and the diameter measures 0.0257m. Using pi x r x h, the volume equals 3.98 x 10^-5 cubic meters. The volume of the cylinder is the same volume of water that would be displaced if the cylinder was submerged. Using density of water as 1000 kilograms per cubic meter x 3.98 x 10^-5, you get 0.0398 kg. Multiply this number by 9.81 for gravity and you get 0.390 Newtons. This, again, would equal the buoyant force.

Questions:
1. For the force gauge example, the uncertainty lies in the meter. Since we didn't take multiple measurements for this lab, we compared numbers for data with other groups. Given this, a conservative guess for uncertainty would be plus or minus .05 Newtons. When applying this uncertainty, the maximum value for the buoyant force would equal 0.52 Newtons and the minimum would equal 0.32 Newtons. This would mean that the buoyant force for this method equals 0.42 Newtons plus or minus 0.1 Newtons.
For the overflow example, the uncertainty lies in the scale and the residual water that did not transfer containers. We measured the weight of water by applying the zero function to the scale while the beaker was on it and poured in the overflowed water. Factoring in the uncertainty of the scale and the water that remained in the first container, a conservative guess at the uncertainty of the mass of water would be plus or minus 5 grams. This would mean the maximum value of the weight would be 0.44 Newtons and the minimum would be 0.34, giving a buoyant force of 0.39 Newtons plus or minus 0.05 Newtons.
For the volume method, uncertainty lies in the actual measurement of the dimensions of the cylinder. If an uncertainty of plus or minus a millimeter is given for every dimension, then the maximum value for volume would equal 4.35 x 10^-5 and the minimum value would equal 3.63 x 10^-5. This means the maximum buoyant force would equal 0.427 Newtons and the minimum force would equal 0.356 Newtons, giving a buoyant of force of 0.392 Newtons plus or minus 0.036 Newtons.
2. Accuracy is difficult to qualify, given the fact that we do not know the true value of the buoyant force. However, we can use logic to determine which method would be most accurate. In the tension method, uncertainty lies in the gauge, but the gauge will read the same every time. This gives a precise measurement that is assumed to be accurate. With the overflow method, the same can be assumed of the scale as of the force gauge, however, the transfer of water from one container to the other gives further uncertainty. We can assume that this method will be less accurate that the first. The final method requires multiple measurements to give a proper answer. The uncertainty lies in two places, one being the markings of the ruler itself, and the other being the discretion of the reader. These circumstances lead to lack of precision, and it is not possible to have accuracy without precision. Considering these things, the first method seems to be the most accurate.
3. If the cylinder had touched the bottom of the container while submerged in water, the result would be an incorrect buoyant force. In this method, the tension in the cable plus the buoyant force equals acceleration due to gravity times the mass of the cylinder. If the cylinder had touched the bottom of the container that held the water, a normal force would arise that would lessen the tension. Therefore when subtracting the two tensions, the buoyant force would appear larger than it really is because the normal force would be included in this calculation.