(ta initials) first name (print) last name (print) brock id (ab17cd) (lab date) Experiment 1 Archimedes principle Prelab preparation Print a copy of this experiment to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report. Compile these printouts to create a lab book for the course. Perform the following tasks as indicated. Then answer the following questions and submit them in your Archimedes Principle Prelab assignment, due at Turnitin the day before you perform your experiment in the lab. Turnitin will not accept submissions after the due date. Unsubmitted prelab reports are assigned a grade of zero. The Archimedes Prelab assignment template is found at the Lab Documents page at the course website: http://www.physics.brocku.ca/courses/1p92_dagostino/lab-manual/ 1. Read the relevant sections of the textbook and/or the lecture notes to review the following concepts: density pressure buoyant force specific gravity Archimedes principle 2. For a cartoon introduction to this lab experiment, and the famous story of Archimedes discovery, read http://archimedespalimpsest.org/images/kaltoon/ 3. To get a feel for the concepts underlying this experiment, play with the following simulation: https://phet.colorado.edu/en/simulation/buoyancy The section labelled Intro provides a basic introduction and the section labelled Buoyancy Playground allows you a lot more freedom to play. 4. Read through the rest of the lab instructions for this experiment in this document.! Important! Be sure to have every page of this printout signed by a TA before you leave at the end of the lab session. All your work needs to be kept for review by the instructor, if so requested. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 1
2 EXPERIMENT 1. ARCHIMEDES PRINCIPLE Archimedes principle Everybody knows the story s punch line: A man is so excited by the idea that came to him in a bathtub that he runs naked to the emperor s palace, shouting Eureka! But what exactly was the idea that made Archimedes forget the dress code? Survey your friends who are not taking this physics course, and most are likely to respond with some description of buoyancy: A body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid. That s important enough, and Archimedes did state a very concise formulation of the buoyancy principle. However, boats floated long before Archimedes came along. Why would a better formulation of an old idea excite him so? The full story of Eureka! is somewhat more subtle. Archimedes figured out a way to use the buoyant force to solve a very important practical problem of catching the crooks who were defrauding the treasury by passing off gold-silver alloy coins as pure gold ones. Before you continue reading the next paragraph, spend a few minutes trying to think of a solution to this challenge. It s not an easy one! Archimedes solution (which brought him both the satisfaction of resolving a intellectual challenge and a considerable monetary reward) could be implemented quickly and easily and required only the simplest of tools: A balance scale, weights made of pure silver and pure gold, and a tub of water. First, you had to use the weights made of gold balance out the unknown material. Then you would submerge both sides of the balance in water. If the two arms remained balanced, then the unknown material was also gold since the same mass of the material displaced the same volume of water on both sides and thus both sides experienced the same buoyant force equal to the weight of that water. If, however, the material was not really gold, its density was slightly different from that of the pure gold, and the same mass would displace a different volume of water. The buoyant force would be slightly different on the two sides of the balance scale, and the submerged balance would tilt. In fact, by replacing the pure gold weights with a mix of gold and silver weights and adjusting their ratio until the balance scale remained level in and out of water, one could measure the exact make-up of the alloy and catch the crooks! Inthisexperimentyouwill determinetheunknownratioofcopper(cu)toaluminum(al)inasimulated alloy of the two metals, resolving essentially the same conundrum. The density rho of a material is defined as the ratio of its mass m to its volume V, ρ = m V and has units such as kg/m 3 or g/cm 3. The specific gravity of a material is defined as the dimensionless (unitless) ratio of the density of the material to that of water: S x = ρ x, ρ H2 O Suppose that you submerge the object in question in water. If you multiply the numerator and denominator ofthepreviousequationbythevolumeofwaterdisplaced, thenumeratorwillbecomethemassoftheobject, because the volume of the object is equal to the volume of water displaced. The denominator will become the weight of the water displaced. The denominator is equal to the apparent weight loss of the object when it is submerged in water (sketch a free-body diagram to convince yourself that this is true), and this provides us with a practical expression for the object s specific gravity: S x = The density of an alloy of Al and Cu is weight of object in air apparent weight loss when submerged in water (1.1) ρ alloy = m Al + V Al +V Cu. (1.2)
3 Equation 1.2 can be re-arranged to compare a theoretical result, the mass ratio of the two metals, to that obtained from the experimentally determined specific gravities of the various materials: m Al = 1 S alloy/s Cu 1 S alloy /S Al. (1.3) Procedure The experimental apparatus consists of a precise digital weight scale, a volumetric flask, a pipette, distilled water, a long bar of Cu, a long bar of Al, and a simulated Al/Cu alloy made up of two short bars of Al and Cu. For each of the three metals (Al, Cu and the Al/Cu alloy ) you will need to perform three separate measurements, as summarized in Fig. 1.1: Figure 1.1: The three steps must be repeated for each metal (a) weight w a of the piece of metal; (b) weight w b of the piece of metal and of the volumetric task filled with distilled water to the exact mark on the neck of the flask; (c) weight w c of the piece of metal submerged in the volumetric task filled with distilled water to the exact same mark. You will need to withdraw some water from the volumetric flask between steps (b) and (c). This is best achieved by simply pouring off some water and then adding the required amount back, drop-by-drop from the pipette when the level gets close to the target mark. These weight measurements rely heavily on your experimental technique. To avoid introducing errors, be careful not to have any stray water droplets on the piece of metal, on the body of the flask or on the scale platform itself. To get a feeling for how these experimental errors affect your results, you will repeat steps (a) (c) several times and perform a statistical analysis of the data to determine the sample average w and the standard deviation σ(w), which is a measure of how closely your data is distributed around w.
4 EXPERIMENT 1. ARCHIMEDES PRINCIPLE Part 1: Aluminum bar? A major source of random error in this experiment is the precision with which you can reproduce the exact same water level time after time. How much does the water level rise after a drop is added? Starting with the aluminum bar, fill in Table 1. Proceed column by column. Close any open Physicalab windows, then start a new Physicalab session by clicking on the desktop icon and login with your Brock student ID. You will be emailing yourself all the graphs that you create for later inclusion in your lab report. Al 1 2 3 4 5 6 7 8 9 w σ(w) Metal bar, w a Metal & flask, w b Metal in flask, w c Table 1.1: Experimental data for aluminum bar In Physicalab, click File, Newto clear the data entry window, then enter in a column thedata points from step (a) for the aluminum metal bar. Click Options, Insert X index to insert a column of indices to your data points, then select scatter plot and click Draw to view the variation in your data. Select bellcurve and click bargraph to view the distribution of your data. Physicalab has calculated the average w and standard deviation σ(w) of your data set and these values are shown in the graph window as w ±σ(w). Enter these two values in the w and σ(w) columns of Table 1. Repeat the above steps for the Metal & flask data set and the Metal in flask data set. You can now determine the specific gravity S Al of aluminum with the aid of Equation 1.1 and your results from Table 1: S Al (S Al ) Part 2: Copper bar Proceed to acquire data and fill in Table 1 for the copper bar. S Cu (S Cu )
5 Cu 1 2 3 4 5 6 7 8 9 w σ(w) Metal bar, w a Metal & flask, w b Metal in flask, w c Table 1.2: Experimental data for copper bar Part 3: Alloy bar Complete Table 1 for the Al/Cu alloy bar composed of the two short Al, Cu bars. Alloy 1 2 3 4 5 6 7 8 9 w σ(w) Metal bar, w a Metal & flask, w b Metal in flask, w c Table 1.3: Experimental data for Al/Cu alloy bar S alloy (S alloy ) Part 4: Results Measure the weight m Al of the short bar of aluminum and the weight of the short bar of copper that make up the alloy bar. Include the error in these measurements. These are the theoretical values that will be used to compare to your experimental results based on the previously calculated specific gravities of the various materials. m Al =...±... =...±... From these measurements, determine the theoretical mass ratio of your alloy and the magnitude of uncertainty, or error, in this result: ( ) mal m Al
6 EXPERIMENT 1. ARCHIMEDES PRINCIPLE Use Equation 1.3 to calculate the experimental mass ratio and the associated error for your alloy : ( ) mal m Al IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Lab report Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab report submission deadline, at 11:00pm six days following your scheduled lab session. Remember that you must include a discussion of your results and your estimate of the validity of your results, including error estimates, as well as answering the questions embedded here. Turnitin will not accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.
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