Experiment 6: Archimedes' Principle OBJECTIVES Archimedes discovered that you can measure the volume of a geometrically irregular solid by measuring the displacement of a liquid in which the solid is completely submerged. When scientists quantified the force of buoyancy, they discovered that you can find the density of an irregular object by comparing the weight of the liquid displaced by a submerged object with the apparent loss in the weight of an object. In this experiment, you measure the displacement of liquids by submerged and floating objects and measure the buoyancy force of liquids on submerged objects. The objectives of this experiment are as follows: 1. To measure the liquid displaced by floating and submerged objects 2. To test Archimedes' law 3. To calculate the density of liquids, solid objects that sink, and solid objects that float THEORY Archimedes' law states that an object immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced by the object as shown in equation 6.1. Archimedes' Law Fb = PLVg (6.1) Here pi is the density of the liquid and Vis the volume of the object, so that pil is the mass of the displaced liquid. This law can measure the density of an irregular solid object pob; if the density of the liquid p, is known. Weigh the object in the standard way (without the liquid) determining its weight as a function of density as shown in equation 6.2. Weight (standard) W = mg = Pobj 8 (6.2) Then weigh the object while immersed in a liquid to determining its apparent weight as shown in equation 6.3. Apparent Weight (in liquid) Wapp = W - Fb = (Pobj - PL)Vg (6.3) Dividing equation 6.3 by equation 6.2 eliminates V, which might be difficult to measure directly, and results in the relation shown in equation 6.4. Ratio of Apparent Weight W app (6.4) to Standard Weight - = 1- 4 W objp Solving equation 6.4 for pobj yields a formula for the density of the object, as shown in equation 6.5. W Density of a solid Pobj = PL W - Wapp (6.5) Equation 6.5 can also calculate the density of the liquid, such as ethanol, once the density of the submerged object is known, as shown in equation 6.6. 42Density of a liquid W -Wapp PL = Pobj W (6.6) In this experiment, you weight the object in two different liquids, water and ethanol, thus it is convenient to label the symbols by the indices W and E. Then equation 6.5 becomes W Density of a solid, using water Pobj = PW W - Wapp, w (6.7 ) Substituting equation 6.7 in 6.6 gives a formula for the density of ethanol. Density of Ethanol W - W app, E PE = PW W - Wapp, W (6.8) An object which floats in a liquid displaces a weight of the liquid equal to its own weight. If the object is elongated with length L and constant cross-sectional area S, its volume is V = LS and its mass is pob; V. If the object is floating in a vertical position and the length of its submerged part is Lsubs the volume of the displaced liquid is Vsub = LsubS and the mass of displaced liquid is PL sub. So for a floating object PobjV = PL sub which algebraically yields equation 6.10. Density of a solid, floating L sub Pobj = PL L (6.10) In the equations above, it is convenient to measure all the weights in grams instead of Newtons because the acceleration of gravity g cancels in all cases. ACCEPTED VALUES The accepted value when testing Archimedes' Law in Part A is the weight of the water displaced by the submerged solid object. The accepted values for the density of the liquids and solids analyzed in this lab at 20'C and in an atmosphere of 1 bar of pressure are as follows: Water: PL = 998.21 kg/m3 . Ethanol: PL = 789.3 kg/m3 Aluminum: Pobj = 2698.9 kg/m3 APPARATUS metal object overflow can . thread (attached to . platform balance metal can metal object) (supported above ethanol . meter stick the table by a stand) wooden dowel graduated cylinder beakerPobyg Fb - PL x V x g density density Volume 64 of object liquid liquid that gets Overflow Can displaced Solid Object Poby = W - wapp (immersed) PL W PW = 1000 kg/ m 3 Liquid Density of water Beaker Figure 6.1 An overflow can directs displaced liquid into a beaker PROCEDURE Table 6.1 is for steps 1-5 (for the metal object), while Table 6.2 is for steps 6-10 (for the wooden block) 1. Hang the object from the hook under the left pan of the balance using the thread. Measure the object's weight and record the result as the object weight, W (g) on Table 6.1. 2. Arrange the overflow can and the beaker so that water can flow from the spout of the overflow can into the beaker. Pour water into the can until it overflows. When the water has stopped dripping from the spout, weigh the beaker with the water in it and record the result as the initial weight of the beaker, Wb (8). 3. Place the beaker and its contents back under the spout. While keeping the object hung from the balance, lower the object by a thread into the water in the overflow can until it is completely immersed. When all the water displaced by the object has flowed into the beaker, weigh the beaker with the water in it and record the result as the final weight of the beaker, Wb, (8) 4. Adjust the apparatus so that the object isn't touching the sides or bottom of the overflow can. Measure the apparent weight of the object when immersed in water and record the result on Table 6.1 under Wapp, w (8). 5. Set the overflow can aside and dry the object. Under the instructor's supervision, fill the metal can with ethanol. Measure the apparent weight of the object when completely immersed in ethanol, Wapp, E and record your result under Wapp, E (g). Under the instructor's supervision (once again), pour the ethanol back into the ethanol bottle and close the bottle. 6. Measure the length of the wood block and record the result on Table 6.2 as the total length of the block, Liotal (cm) . 7. Measure and record the width W (cm) and the height H (cm) of the wood block. 8. Measure and record the mass m (g) of the wood block. 9. Fill the graduated cylinder approximately to the halfway mark with water. Lower the wooden dowel into the water in the graduated cylinder until it floats. 10. Measure the length of the wood that is below the level of the water and record the result as the submerged length Lab (cm). Dispose of any remaining water to end the experiment. 44 Thus, a plot of F KAXPart A: Testing Archimedes' Law W (g) W app, w (8) W b. (g) 159. 1 W b,f (g) 102.9 205.9 8 259-9 Part B: Calculating the density of a solid object and ethanol W (g) W app, w (8) app,E (8) 159, 1 HETT 102.9 11 6 .1 Part C: Calculating the density of a floating object Ltotal (cm ) W (cm) H (cm) m (g) Ljub (cm ) 3.8 3. 6 20 1 9 259.5-205. 26 CALCULATION AND ANALYSIS = 53, 6 1. Calculate the weight of the water displaced by the immersed object, Why Wb,;, using the data from Part A. 2. Calculate the apparent loss of weight of the object when completely immersed in water, W-Wapp,w, using the data from Part A. 159 , 1 - |6 2, q - 56 . 29 3 . Archimedes' Law predicts that the weight of the displaced water equals the apparent loss of es weight of the object. Do your results support Archimedes' Law? 4. Calculate the density of the object Pob; using equation 8.7 and the data from Part B. 6998 27 15 Calculate the % error in your result for Pobj. (159 Calculate the density of the ethanol PE using equation 8.8 and the data from Part B. 2 1 826 - . Calculate the % error in your result for PE. 8. Calculate the density of the wood block Pwood, float using equation 8.10 and the data from Part C. 9. Calculate the volume of the wood block using your data from Part C. (Remember volume = length X width X height.) 10, Calculate the density of the wood block Pwood, actual m volume using your data from Part C. 11. Using Pwood, actual as the accepted value, calculate the % error in your result for Pwood, float V 789, 3 /000 = 0, 7893 45 - PY platto. Thus, a plot or considered " simple is TUS ratus