Lab 9. Archimedes Principle and Applications. Upon successful completion of this exercise you will have...

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1 Lab 9 Archimedes Principle and Applications Objectives: Upon successful completion of this exercise you will have utilized Archimedes principle to determine the density and specific gravity of a variety of substances utilized Archimedes principle to determine the density and volume of an irregularly shaped object and then through extrapolation methods determine the amount of mass removed to create the irregular shaped object determined the mass density of an unknown liquid. Theory: The average mass density, ρ m, of an object is defined as the mass, M, of the object divided by its volume, V, or ρ m = M/V [1] The average weight density, ρ w, is defined as the weight, W (equals M*g), of an object divided by its volume, or ρ W = W/V = Mg/V = ρ m g, [2] where g is the acceleration of gravity. The specific gravity, s.g., of a material is defined as the ratio of the density of the material to the density of water. Thus, s.g. = density of material/density of water Note that specific gravity is a unitless quantity which depends only on the material. The density of water is approximately 1.00x10 3 kg/m 3 in the SI system of units. Archimedes principle states that an object partially or wholly immersed in a fluid will be buoyed up by a force equal to the weight of the fluid displaced by the object. From equation [2] the weight of the object is W = (ρ m g)v [3] Thus, according to Archimedes principle, the buoyant force, F B, on an object submerged in a liquid F B = weight of liquid displaced = (ρ m ) liq gv, [4]

2 where (ρ m ) liq is the mass density of the liquid and V is the volume of the liquid displaced. First consider an object suspended at rest by a string in air as shown in Figure 9-1(a), the tension in the string, T, is equal to the weight of the object, W. Consider, then a submerged object, suspended by a string, as shown in Figure 9-1 (b). m T F B T W W m (a) Figure 9-1 (b) If T is the tension in the string, W is the weight of the object (equals mg) and F B is the buoyant force on the object, then in equilibrium, T = W - F B [5] T is commonly (though incorrectly) called the weight of the object when submerged, since it is the downward pull of the string on the balance. In this experiment the measurements made using the force sensor are in Newton s. The weight, W, of the object is first made when the object is freely suspended in air as shown in Figure 9-1(a). Later when the object is submersed into the liquid the measurement will represent the tension in the string, T. The difference between these two measurements results in the buoyant force supplied by the liquid, F B. Consider that the volume of the liquid displaced is equal to the volume of the submersed object. By examining equations [3] and [4] and solving each for the volume, V, and setting them equal to one another. W/(ρ m g )=F B /(ρ mliq g ) Factor out gravity, g and also solving for F B in equation [5] then substituting it into the equation yields

3 W W T = ρ m ρ mliq [6] The density of the submersed material, ρ m, can then be determined from equation [6] Where ρ mliq is the density of water 1.00x10 3 kg/m 3, W is the weight of the object in air and T is the tension in the string when the object is submersed. The density of a liquid, ρ mliq, can likewise be determined if the density of the material, ρ m, is known by manipulating equation [6] to solve for ρ mliq. Substance Mass density ρ (kg/m 3 ) Aluminum x 10 3 Brass x 10 3 Copper 8.9 x 10 3 Gold 19.3 x 10 3 Silver 10.5 x 10 3 Steel x 10 3 Water 1.00 x 10 3

4 Procedure: Density of on object: 1. Start the computer. Run the Data Studio program and then open the Archimedes file. Measurements 2. Choose one of the cylinders and measure its height and diameter using the vernier caliper. Convert the mm measurement to meters. Measure its mass, convert the gram measurement into kg and record the information into the Data Table. Setting up the equipment 3. Fill a cup to about a ¼ inch from the top with water. Suspend the cylinder from the force sensor. Adjust the force sensor so that the cup with the water can be placed beneath the cylinder, such that the cylinder is just above the brim of the cup. Getting Data Weight in Air 4. Remove the cylinder from the force sensor. Press the TARE button located on the side of the force sensor. This zeros the force sensor and compensates for any electronic drift that may occur between measurements Suspend the cylinder from the force sensor. Select Start from the Experimental ToolBar. Data will be gathered for 20 seconds then stop automatically. The value will be a negative number, you are only interested in the magnitude not the sign. Record the magnitude of the mean value as the Weight, W, of the cylinder. Apparent Weight in Liquid 5. Lower the force sensor so that the cylinder is completely submersed into the water. Start another data run. Record the mean value as the Tension, T, in the string, due to the submersed cylinder. Calculations of measured Mass Density 6. Calculate the density of the cylinder use equation [6] and solve for ρ m. ρ mliq = density of water = 1.00x10 3 kg/m 3 From Table-1 determine the substance of the cylinder. Calculation of theoretical mass density With a regular shaped object such as a cylinder whose volume can be easily determine, another method can be used to determine the density of the object. 7. Calculate the volume of the cylinder (Volume of cylinder V = πr 2 H). Use equation [1] calculate the Theoretical density of the cylinder. 8. Calculate the Specific gravity of the material. s.g = density of material / density of water. 9. Repeat for each cylinder.

5 Volume of an irregular shape object: 1. Use the experimental methods from the Getting Data section and determine the measured density of the irregular shape object. 2. Measure the mass, M, of the irregular shaped object. 3. Use equation [1] to determine the volume of the irregular shaped object. Pour the water into the sink if available or return it to the container. Density of an unknown liquid. 1. Fill a cup to about a ¼ inch from the top with the unknown liquid. Choose the cylinder whose experimental value in density matched closest to the theoretical value of density. Record the experimental value as your material density. 2. Use the experimental methods from the Getting Data section in part 1 obtain the weight, tension and buoyant force measurements of the object for the unknown liquid. 3. Solve equation [6] for the mass density of the liquid, ρ mliq, then calculate the density of the unknown liquid. 4. Calculate the liquids specific gravity. 5. Return the liquid to the appropriate container do not pour into the sink. Dry off all objects and wipe up any spilled liquid.

6 Density of an Object Data Page I Weight W (N) Tension when Submersed T (N) Buoyant Force F B (N) Measured Mass Density Height of Cylinder H (m) Diameter of Cylinder D(m) Radius of Cylinder R(m) Volume of Cylinder V (m 3 ) Mass of Cylinder M (kg) Theoretical Mass Density ρ = M/V (kg/m 3 ) Specific Gravity Postulated Substance Cylinder Blue Cylinder Orange Cylinder Black

7 Data Page II Volume of an Irregular Shaped object: Weight W (N) Tension when submersed T(N) Bouyant Force F B (N) Mass M (kg) Measured Mass Density Volume of object (m 3 ) Density of an Unknown liquid Weight W (N) Tension when submersed T(N) Bouyant Force F B (N) Measured Mass Density Calculated Mass Density of Liquid Specific Gravity

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