Archimedes' Principle Lab Report

Abstract: In this experiment Archimedes Principle was demonstrated by concept and magnitude. The mass, true weight, and apparent weight of a brass cylinder, silver cylinder, and brass block were measured. When measuring the apparent weight of each object, the volume displaced of the water in the cylinder was also measured. The Buoyant force was also calculated which allowed us to calculate for the percent deviation of the buoyant forces resulting in percent deviations below 6%. Introduction: Archimedes' principle states that the buoyant force on an object which is fully submerged in a fluid is equal to the weight of the fluid displaced by the object. Fb = mf x g = 𝜌fVg Where 𝜌f is the density of the fluid, V equals the volume of the object submerged, and g is acceleration due to gravity (9.8 m/s^2). Buoyant pressure is understood as force per unit area on an object due to the weight of an imaginary column of fluid which extends from the bottom of an object to the fluid’s surface. Static equilibrium will determine how far down the object will sink and will be determined by the following conditions: buoyant force (Fb) which pushes upward from the object while the downward force is the object’s weight (W): W= V 𝜌 g, The object will reach equilibrium at the depth where the forces balance out. Therefore, if buoyant force is less than the object's weight when the object is fully submerged, per Newton’s 2nd Law, the object will sink. The magnitude of buoyant force is determined by weighing an object both in and out of the fluid. Difference of dry weight (true weight) and submerged weight (true weight minus the buoyant force) is equal to the buoyant force. Fb= W- Ws Theory: In the experiment the three objects were measured to find the mass and true weight by hanging and balancing the objects on the triple beam balance. The true weight was calculated by multiplying the mass times the acceleration of gravity (9.8 m/s^2). The apparent weight was found by placing the object on the hanger and balancing it on the triple beam balance, while it was fully submerged in water. The tube of water was held by the stand ensuring that the object was not touching the tube at all. To find the volume displaced the initial water level was recorded when there was no object in the water and subtracted from the water level when the object was submerged and balanced in the water. The buoyant force was found by using rho times the volume times the acceleration of gravity (9.8 m/s^2); Fb= 𝜌Vg. The percent deviation of the buoyant forces was calculated by subtracting the buoyant force of the apparent weight from the buoyant force of the true weight divided by the buoyant force of the true weight and multiplied that by 100%. Experimental Procedure: During this experiment we used a triple beam balance, a graduated cylinder, a light string, a rod stand, a clamp, a brass cylinder, a silver cylinder, and a brass block. First we measured the mass of the brass cylinder, the silver cylinder, and the brass block using the triple beam balance. After finding the mass, we determined the true weight by multiplying the mass by gravity. Once we found that, we filled the graduate cylinder up to the 75mL mark to measure the volume of each of the items. We then used the test tube clamp to hold the graduated cylinder off of the bottom of the triple beam balance and measured the mass of the items submerged in the water, then multiplied by gravity to give us the apparent weight. We used the weight and the apparent weight to find the Buoyant force. Results: Table 1 Sample Brass Cylinder Mass m (kg) True Weight (W = mg) Volume displaced (m3) X 10-6 m3 Buoyant Force (Archimedes) 𝑭𝑩 = 𝝆𝑽𝒈 0.20819 kg 2.0403 25 0.245 0.06714 kg 0.6579 25 0.245 0.0698 kg 0.68404 7.958 0.07799 Silver Cylinder Brass Block Table 2 Sample True Weight (W = mg) Apparent Weight (W’ = m’g) Buoyant Force 𝑭𝑩 = 𝑾 − 𝑾′ Percent deviation Brass Cylinder 2.0403 1.8011 N 0.2392 2.37% Silver Cylinder 0.6579 0.42581 N 0.23209 5.27% Brass Block 0.68404 0.60417 N 0.07987 2.41% Table 3 Sample True Weight (W = mg) Apparent Weight (W’ = m’g) Specific gravity 𝑾 𝒔. 𝒈. = ′ 𝑾−𝑾 Density of sample (kg/m3) Brass Cylinder 2.0403 1.8011 N 8.533 8327.6 Silver Cylinder 0.6579 0.42581 N 2.835 2685.6 Brass Block 0.68404 0.60417 N 8.564 8771 Discussion: Our results gave us two values for the buoyant force, for each of the three sample objects we measured. Using Archimedes' Principle, the buoyant force for each sample is shown in table 1. Using the equation we derived to calculate the buoyant force, the values for each sample are shown in table 2. Because of the accuracy of our calculations, we were able to obtain percent deviations for the buoyant force of all three samples below 6%. The percent deviation for the brass cylinder came out to be 2.37%. The percent deviation for the silver cylinder came out to be 5.27%, and the percent deviation for the brass block was 2.41%. Lastly, we calculated the density of each sample, as shown in table 3. During the lab we did encounter a systematic error. The scale we were using to take measurements was faulty and could not accurately take the mass of the samples. We were able to notice this error early and switch our scale with one that functioned properly so as not to offset our data. Conclusion: We were able to properly demonstrate Archimedes Principle using our three samples; a brass cylinder, a silver cylinder, and a brass block. Using Archimedes Principle, we were able to find the buoyant force of each sample with the buoyant force of the brass cylinder being 0.245 N, the buoyant force of the silver cylinder being 0.245 N, and the buoyant force of the brass block being 0.07799 N. Using the equation we derived to calculate the buoyant force, we got 0.2392 N for the brass cylinder, 0.23209 N for the silver cylinder, and 0.07987 N for the brass block. The percent deviation for the brass cylinder came out to be 2.37%. The percent deviation for the silver cylinder came out to be 5.27%, and the percent deviation for the brass block was 2.41%. Lastly, we calculated the density of each sample. We found the density of the brass cylinder to be 8327.6 kg/m3. The density of the silver cylinder was 2685.6 kg/m3, and the density of the brass block was 8771 kg/m3. Questions: 1. Derive an equation for specific gravity in terms of the dry sample weight and the apparently lighter submerged weight (which is the real weight minus the buoyant force). You will use no other quantities in the equation. Calculate the specific gravity of each sample. density ρ = mass of sample / volume of sample weight = mg m = ρV W = ρ Vg s.g. = (ρ sampleVsampleg) / (ρ waterVsampleg) = W / FB = W / W - W’ FB = ρ waterVsampleg s.g. = W / W - W’ 2. Calculate the density of each sample using Eq. 2. Calculate the specific gravity of each sample from Eq.4 and the percent deviation between this value and the first. Is this value consistent with the total relative error (expressed as a percent)? Explain. density specific gravity brass cylinder 8327.6 8.533 silver cylinder 2685.6 2.835 brass block 8771 8.564 percent deviation 2.37% 5.27% 2.41% This value is consistent with the total relative error. 3. How might you measure the specific gravity of a substance that is less dense than water (without using a different fluid)? s.g. = density of material / density of water at 4 degrees C 4. If Archimedes used his principle to verify the purity of a block of silver, would the method be foolproof? Explain your answer. No, the method would not be foolproof because an alloy of two different metals such as copper and mercury would have an average density equal to that of silver. The possibility of different combinations of different metals would make it difficult to verify the purity.