ARCHIMEDES PRINCIPLE AND THE COMPUTATION OF BUOYANT FORCES. Alexis Rodriguez-Carlson

Transcription

1 ARCHIMEDES PRINCIPLE AND THE COMPUTATION OF BUOYANT FORCES Alexis Rodriguez-Carlson September 20, 2006

2 Purpose: The purpose of this experiment is to show that the buoyant force acting on an object submerged in a liquid is equal to the weight of the liquid displaced. Equipment: This experiment required the use of: Cylindrical Samples of aluminum, brass, and copper. A triple beam balance. A vernier caliper. 2 beakers. One containing water, the other containing antifreeze. Discussion: Archimedes Principle states that the buoyant force acting on an object submerged in a liquid is equal to the weight of the liquid displaced. This means that if a 1 m 3 block of any material is placed into any liquid that the buoyant force will be equal to the weight of 1 m 3 of that liquid. F Buoyant = W Liquid Displaced F Buoyant = ρ liquid * V Displaced Liquid * g (equation 1) (ρ liquid = density of the liquid) The buoyant force can also be defined as the actual weight of the submerged object (meaning, the weight in a vacuum or, for our purposes, the

3 weight in air at 1 atmosphere of pressure) less the apparent weight of the submerged object (how much the object weighs once submerged in the liquid). F Buoyant = W object (actual) W object (apparent) (equation 2) Combining equations 1 and 2 yields: ρ liquid * V Displaced Liquid * g = W object (actual) W object (apparent) (equation 3) The point of this experiment is to prove that equation 3 is true. Procedure: The first step in our experiment was to collect data about the samples of metal which we would be submerging into the two liquids. For each of the samples we took the following data using the vernier calipers and the triple beam balance: Mass in grams. Diameter in cm. Height in cm. Diameter of the hole through which the string ran in each cylinder. and volume. With these values we were able to calculate each samples actual weight

4 Next, we tied one sample at a time to the triple beam balance and let the sample hang into the fluid. The apparent mass was then recorded. This was repeated for each of the three samples in both water and antifreeze. With that data collected, all that remained to be done was to calculate the buoyant forces of each liquid with each sample and to see how they compared. Data: Aluminum Brass Copper Actual Mass of Sample (g) Diameter of Sample (cm) Sample Height (cm) Void Diameter (mm) Density (ρ) H 2 O (g/cm³ ) Density (ρ) Antifreeze (g/cm³ ) Apparent Mass in H 2 O (g) Apparent Mass in Antifreeze (g) Calculations: Aluminum Brass Copper Actual Weight of Sample (dynes) (Actual M Sample * cm/s²) Density of Sample (g/cm 3 ) Area of Sample (cm²) (Π (D Sample ²)/4) Volume of Cylinder (cm³) (A Sample * H Sample ) Area of Void (mm²) (Π (D Void ²)/4) Volume of Void (mm³) (A Void * D Sample ) Volume of Void (cm³ ) Sample Volume = Displaced Liquid Volume (V Cylinder - V Void ) Mass of Displaced H 2 O (g) (V Sample * ρ H 2 O) Mass of Displaced H 2 O (g) (V Sample * ρ Antifreeze)

5 Weight of Displaced H2O (dynes) (M Displaced H2O * cm/s²) Weight of Displaced Antifreeze (dynes) (M Displaced Antifreeze * cm/s²) Apparent Weight in H 2 O (dynes) (M Apparent in H2O * cm/s²) Apparent Weight in Antifreeze (dynes) (M Apparent in Antifreeze * cm/s²) F Bouyant 1 of H 2 O (dynes) (ρ H 2 O * V Displaced H2O * cm/s²) F Bouyant 1 of Antifreeze (dynes) (ρ Antifreeze * V Displaced Antifreeze * cm/s²) F Buoyant 2 of H 2 O (dynes) ( Actual Sample Weight - Apparent W Sample in H2O ) F Buoyant 2 of Antifreeze (dynes) ( Actual W Sample - Apparent W Sample in Antifreeze ) F Buoyant Avg of H 2 O (dynes) ((F Buoyant 1 + F Buoyant 2) / 2) F Buoyant Avg of Anbtifreeze (dynes) ((F Buoyant 1 + F Buoyant 2) / 2) Percent Difference in Bouyant Force of H 2 O Calculations ( ((F Bouyant 1 - F Bouyant 2)/F Bouyant Avg) * 100 ) 1.94% 1.73% 2.32% Percent Difference in Bouyant Force of Antifreeze Calculations ( ((F Bouyant 1 - F Bouyant 2)/F Bouyant Avg) * 100 ) 4.19% 3.09% 3.94% Error Analysis: In this experiment, our goal was to prove that the buoyant force on an object submerged in a liquid was equal to the weight of the displaced liquid. By our calculations the two values were within 4.2 % of one another. This is a relatively small difference which could be accounted for by the fact the actual measurements were not taken in a vacuum and that the density of the water was estimated. The thing that is interesting about it is that the values for the antifreeze were so much more inaccurate than those in water. Since the antifreeze

6 measurements were taken before the water measurements, it is possible that we became more comfortable with the reading of the scale. It is also possible that the value for the density of the antifreeze was not accurate, but since this value was measured in the lab at the time of the experiment, this is unlikely. Without performing the experiment again, we will probably never know where the discrepancy came from or why it is so much larger in the calculations involving the antifreeze. Conclusions: The buoyant forces calculated for each of the samples in each of the liquids were sufficiently close enough to confirm Archimedes Principle that the buoyant force acting on an object submerged in a liquid will equal the weight of the liquid displaced by that object. After this experiment we can be confident is using either of the above methods to calculate the buoyant force acting up an object.