Fluid Statics

The objective of the experiment is to observe the buoyant force on a known cylindrical mass and determine its value using three methods.

A. The first method is to weigh it underwater. Using a string, attach the mass to a force probe. When placing the mass in water, the tension that the force probe measures is the mass' weight minus the buoyant force.

B. The second method is to measure how much fluid is displaced. After measuring the mass of a beaker, take a graduated cylinder and fill it completely with water. Place the graduated cylinder in the beaker, drop in the mass, and then measure the mass of the beaker plus the displaced water. The weight of the displaced water equals the buoyant force.

C. The third method is to calculate the volume of the cylinder (πr²h) which is equivalent to the volume of water displaced.  Afterward, calculate the weight of the displaced water (ρVg).

Data

Measured Values:

•  Method A- Mass: 0.11237 kg, Tension in Force Probe: 0.73 N
•  Method B- Mass of Beaker: 0.14718 kg, Mass of Beaker + Water: 0.17872 kg
• Method C- Diameter of Mass: 0.0253 m, Height of Mass: 0.0771 m 

Calculated Values: 

• Weight of Mass: 1.1023497 N
• Mass of Water (Method B): 0.03154 kg 
• Volume of Mass/Water: 3.876 *10^-5 m³

Buoyant Forces:

Method A	Method B	Method C
0.3723497 N	0.3094074 N	0.380236943 N

Error Analysis

Assumed Uncertainties:

• Mass: 0.11237 ± 0.00005 kg, Tension: 0.73 ± 0.05 N
• Mass of Beaker: 0.14718 ± 0.00005 kg, Mass of Beaker plus Water: 0.17872 ± 0.00005 kg
• Height: 0.0771 ± 0.0005m, Diameter: 0.0253 ± 0.0005 m

Adjusted Calculations:

• Weight of Mass: 1.1023497±0.0004905 N
• Buoyant Force (Method A): w - T = 1.1023497±0.0004905 N - 0.73 ± 0.05 N = 0.3723497 ± 0.0504905 N [0.37 ± 0.05 N]
• Mass of Water (Method B): 0.03154 ± 0.0001 kg
• Buoyant Force (Method B): mg= 0.3094074 ± 0.000981 N [0.309 ±0.001 N]
• Radius: 0.01265 ± 0.00025 m
• Volume:  3.8785*10^-5 ± 1.785 *10^-6m³
• Buoyant Force (Method C):  ρVg= 0.38048085 ± 0.01751085 N [0.380 ± 0.018 N]

Adjusted Buoyant Forces:

Method A	Method B	Method C
0.37 ± 0.05 N	0.309 ±0.001 N	0.380 ± 0.018 N

 Summary

The three experiments provided vastly different results, with little to no overlay. This is most likely due to uncertainties from instruments. For example, the force probe provides a rather imprecise value, and the analytical scales are very sensitive, so they could have factored in the weight of moving air. 

Method C probably provides the most precise values since the measurements were done using digital calipers.

In Method A, if the cylinder had come into contact with the base of the beaker, then it would also have a normal force acting on it in the upward direction. This would have reduced the value of the buoyant force.