Density and Archimedes Principle Objectives: To understand the concept of density and its relationship to various materials. To understand and use Archimedes Principle. Equipment: Dial calipers, Graduated cylinders, Solid aluminum, brass, steel cubes, cup size water containers, variety of irregular shapes, Electronic scales (200 g, 1 g resolution), 1- Electronic scale (2000 g, 1 g resolution), 2 lbs table salt, sugar Must be in a room with a sink. Background: Density (ρ) is defined to be the mass of a material contained in a unit volume of the material: ρ = m/v, where m is the total mass of the object and V is its volume. The usual units are g/cm 3 or kg/m 3. The importance of density is that for different objects made of the same material the mass and volume will vary from object to object but the density remains the same it only depends on the material used. Table 1 gives the density of some common materials. Table 1 - Density of Various Materials material ρ (g/cm 3 ) material ρ (g/cm 3 ) material ρ (g/cm 3 ) aluminum 2.70 gold 19.3 platinum 21.4 brass 8.56 iron 7.86 water 1.00 copper 8.93 lead 11.3 zinc 6.92 One of the activities described in Part II, is to study how the density of water changes as you dissolve a material in it. Salt is highly ionic and dissociates into Na + and Cl - ions. Sugar, on the other hand, does not dissociate when it dissolves. Measuring the changes in volume gives you information about the nature of how the salt ions and sugar dissolve in water. Archimedes Principle states that a body, immersed in a fluid, feels a buoyant force such that: if the body sinks, the buoyant force is just equal to the volume of the body times the density of the fluid. Once the object lies on the bottom of the container, three forces act on it: the buoyant force B, the normal force N (not drawn), and the weight of the submerged object mg. These three forces of course cancel each other because the object stays at rest. Density -- 1
Β = ρ g V mg if the body floats, the buoyant force equals the volume of displaced fluid times the density of the fluid. In this case only two forces act on the floating body: the buoyant force and its weight. Therefore the buoyant force alone must cancel the weight m g. B = mg = ρ g V* V* = volume of water displaced mg Note that in both cases the buoyant force equals the weight of the displaced fluid. PART I Activity 1. There are rectangular blocks made of several different materials available. Measure the dimensions and mass of two different metal blocks and calculate their densities. Identify the material and compare with your calculated density. To measure the dimensions of the shapes, you will need to use a dial caliper. Handle the caliper with care. Note the units used on the caliper. Only a limited number of calipers may be available. Activity 2. Determine the density of a cube of metal (aluminum or brass or steel). Weigh the metal on the electronic scale. Also weigh it using the computer interfaced force probe. Put a beaker of water on the electronic scale and record the weight. Suspend (with steady hands, elbows resting on the table) the unknown metal from the force probe and slowly lower it into the water. Record the changes in the scale and force probe readings for several depths of immersion and when it is completely immersed. Explain why the readings change as they do. (Note that water exerts a buoyant force on Density -- 2
the metal cube. Therefore by Newton's third law, the cube exerts a force on the water in the opposite direction.) Using your data for complete immersion, calculate the density of the metal. Compare with the value given in Table 1. NOTE: For using the computer interfaced force probe, do the following. 1. Be sure that the probe is connected to Port 1 of the computer interface. 2. Open Archimedes.xmbl file in the physics 161 course folder. 3. Click "Experiment" menu and select "calibrate". This procedure performs the calibration of the "zero" of the force probe and the scale factor between the applied force and the reading. 4.The Logger Pro program is set to measure the force for 5 minutes. This time interval can be changed if desired. If you do a linear fit of a constant force versus time measurement over a short time, you can improve the accuracy of the force in question. Activity 3. Now repeat the experiment that Archimedes did. Determine the density of 2 pieces of irregular shaped unknown metals (ring, slab, sphere, trapezoid). Do this by measuring the weight of the object, then again measuring the weight after complete immersion in water. The second measurement allows to determine the buoyant force. Since the buoyant force equals the weight of the displaced water, this allows for finding the density of the object compared to the density of water. So is your object made of pure gold like in Archimedes assignment? Finally, determine the density of an irregular object that will float in water. Use your imagination and any available scales. Density -- 3
PART II Activity 4. Investigate how the density of water changes when you dissolve salt in the water. Fill a graduated cylinder with about 200 cc of water and carefully measure the volume. Weigh out 20-30 g of salt (sodium chloride, density 2.16 g/cm 3 ), add it to the water, and measure the new volume. Make sure all salt has dissolved!! What can you conclude from the volume measurement about the sizes of Na and Cl ions? Activity 5. Repeat Activity 4 for sugar (sucrose C 12 H 22 O 11, density 1.58 g/cm 3 ). [As an aside, note that a typical 240 cc soft drink contains ~30 g of sugar.] Compare the results with activity 4. Density -- 4
Density and Archimedes Principle (preliminary questions) Name: Section: Date: 1. A beaker, filled to the very top with water, weighs 15 N. A 300 g weight is carefully dropped into the beaker. The water that overflows is wiped away and the beaker reweighed. It weighs 16.9 N. What is the density of the weight? [Use g=9.8 m/s 2.] Hint: The net force on the bottom of the beaker after the weight is lowered to the bottom is (the weight of the reduced volume of water) + (300 g weight). 2. The beaker is refilled to the top with water and a 100 g piece of wood of density 0.8 that of water is carefully floated on the water. What volume of water overflows? NOTE(1.): This is one possible way, Archimedes may have solved his assignment of finding if a golden crown was made of pure gold. NOTE(2.): The claim is that melting of arctic sea-ice, including large icebergs, does NOT increase sea-level, but that melting of antarctic glaciers does increase sea-level. 3. Are both of the claims in NOTE(2.) correct? Density -- 5
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Report --- Density and Archimedes Principle (Part I) Name Partner PART I (BE ACCURATE TO 3 SIGNIFICANT FIGURES) Activity 1. Density of different materials. materials dimensions (units) mass (units) measured density (units) 1 2 Expected density from Table 1 Show work V(1) = V(2) = Activity 2. Density of metal cube. Weight on electronic scale: Weight with force probe: Scales register mass (kg). Convert this to a force (N). Depth of immersion Scale reading Force probe reading Scale + Force probe zero ~1/4 ~1/3 ~1/2 ~3/4 Full immersion, still attached to probe Re-check the readings for zero immersion. If they are not the same, you spilled water, and you have to repeat all measurements. Density -- 7
Draw force diagram for the metal cube for complete immersion. How many forces act on the cube? Which of these forces was measured in above table? Draw force diagram for the beaker with the water for complete immersion of the cube. How many forces act on the system of beaker-water? Which of these forces was measured in above table? Explain why the scale reading, the force probe reading, and the sum of both readings change as they do, as you vary the depth of immersion. Why should the sum remain constant?. Calculate the buoyant force for each immersion and add a fifth column to the table above with the results. (3 significant figures) Using your numbers, calculate the density of metal in the cube. Show your work. Density -- 8
Activity 3. For 2 different non-cubic metal shapes find the density of each metal using the method of complete immersion in water. Describe which quantities you need to measure to be able to calculate the density. EXTRA CREDIT Try to find the density for 1 floating irregular object. How was your method modified? Density -- 9
Report --- Density and Archimedes Principle (Part II) Name Partner Partner Partner PART II Activity 4. Salt Water: Investigate how the density of water changes when you dissolve salt in the water. Fill a graduated cylinder with about 200 cc of water and carefully measure the volume. Weigh 20-30 g of salt (sodium chloride, density 2.16 g/cm 3 ), add it to the water, and measure the new volume. Make sure all salt has dissolved!! Initial volume of water: Weight of salt: Final volume of water: Density of salt water: Show your work. Activity 5. Repeat Activity 6 for sugar (sucrose C 12 H 22 O 11, density 1.58 g/cm 3 ), again using 20-30 g. Make sure all sugar has dissolved!! Initial volume of water: Weight of sugar: Final volume of sugar water solution: Density of sugar water: Show your work. Compare how much volume Na and Cl ions take up in the water with how much sugar take up? Why do you think they are different? Density -- 10