Physics 181- Summer 2011 - Experiment #8 1 Experiment #8, Measurement of Density and Archimedes' Principle 1 Purpose 1. To determine the density of a fluid, such as water, by measurement of its mass when confined in a known volume; e.g., a 25ml pycnometer (glass bottle). 2. To determine the density of a cubic-shaped solid, such as aluminum, by measuring its mass and volume. 3. To become familiar with some basic properties of fluids, such as density, pressure, and buoyant forces, and to verify Archimedes' principle. The physics of such phenomena is referred to as hydrostatics. 4. To utilize Archimedes' principle to determine the density of water. 2 Introduction 2.1 Definition of mass density Many physical properties of solid objects or fluids depend not on their total mass, but on their mass density. The mass density of a material, designated by the Greek letter ρ (rho), is defined as the amount of mass per unit volume. Its unit is g/cm 3 in cgs, or kg/m 3 in SI. For a solid object of mass m, and volume V, we have: m ρ = (1) V The density of a homogeneous fluid is similarly defined as the mass (m) of a sample of the fluid divided by the volume (V) it occupies. Normally, a fluid is confined in a container and therefore its volume is related to the volume of the container. Caution: Do not confuse the mass of an object with its weight. The SI units of mass (m) and weight, W, are the kilogram (kg) and the Newton (N), respectively. The pound (lb) is a unit of weight and not of mass. The weight and the mass of an object are related by W = mg where g is the acceleration of gravity (980.35 cm/s 2 ). 2.2 Fluid pressure At any depth h in a fluid, there is a pressure due to the weight of the fluid above this depth. The deeper we descend, the greater is the pressure due to the overlying liquid. In fact, the pressure increases linearly with the depth of a fluid if its density remains constant. This can be deduced easily by considering a column of fluid shown as shaded in Fig. 1, which depicts a container filled with a fluid (f) of density, ρ f. The pressure, P, across any surface of area A in the fluid is defined in terms of the normal force, F, exerted by the fluid one the side of the surface: P =F/A (force per unit area), i.e, F=PA. The volume (V), in Fig 1, of the shaded cylindrical column of height h, and the base area A, is given by V=Ah, so that the mass of fluid contained in this volume is m = ρv = ρ f Ah. Therefore, the weight of the column is given by W = mg = ρ f Ahg.
2 Figure 1: Forces on a column of liquid Figure 2: Pressure P vs. depth h As the column is stationary, it is in static equilibrium, i.e., the sum of vertical forces is zero. Therefore, its weight, W, which acts in the downward direction on the base area A, is balanced by the difference between an upward force F - pushed up from below by the fluid - and the downward force F o - due to the atmospheric pressure. This net upward force acting on the column is called the buoyant force B, and its magnitude is equal to the weight (W) of the column (fluid). At equilibrium, the pressure P that the fluid exerts upwards on the bottom of this column must equal the atmospheric pressure P o, plus the weight (W) of the column divided by its area (A), i.e., the weight of the fluid column is supported by the difference in pressure forces pushing up from the bottom and down from the top of the fluid column. P = F/A, P o = F o /A, W/A = ρ f gh (2) P = P o + ρ f gh (3) Please note that from equations (2) and (3) we deduce the following: The fluid pressure increases linearly with depth in a liquid and also linearly with fluid density. The pressure increases with depth, P - P o, is directly proportional to the depth below the surface of the fluid as indicated by the linear relationship in Fig. 2. The pressure is propagated uniformly in all directions and across all cross sections in the fluid. This result is general and applies at any depth since the fluid below that depth does not contribute to the weight supported by the area A. Pressure P changes if the atmospheric pressure changes, in fact, it changes by the same amount as the change in the atmospheric pressure. 2.3 Archimedes' principle and its application Fig. 3 shows the outline of an irregular object immersed in a fluid and acted on by two forces, one due to its weight and the other due to the pressure exerted on its surface by the fluid. As shown in section 2.2 above, the pressure in a fluid increases with depth, causing the force from the fluid pushing up on the bottom of the object to be greater than that pushing down from the top of the object. The net effect is for the fluid to exert an upward resultant force on the object.
Physics 181- Summer 2011 - Experiment #8 3 2.3.1 Archimedes' principle Figure 3. Forces on an irregular object immersed in a fluid. To calculate the net buoyant force on an irregularly shaped object, such as shown in Fig. 3, is difficult. However, the ancient Greek scientist and mathematician Archimedes (287C212 B.C.) managed to deduce the correct size of this buoyant force, independent of the shape of the object. The basic ideas in his reasoning are as follows. Archimedes considered the force exerted on a small portion of the fluid in a container. This small portion is itself in static equilibrium, otherwise it would rise or sink through the surrounding fluid. Because the small portion is in static equilibrium, the net force on it must be zero. Hence, the upward buoyant force, B, must equal the downward gravitational force, W = mg, i.e., the buoyant force on a volume of fluid is equal to the weight of the fluid. Archimedes further reasoned that if the liquid in this volume were removed and replaced by an object of exactly the same size and shape as this liquid portion, none of the liquid pressure forces acting on its surface would change. Because the object is exactly the same shape and volume as the fluid removed, it would fit exactly into the previous volume without compressing the surrounding fluid. Therefore, he arrived at the conclusion known as Archimedes' principle, that the net buoyant force B upward on any object immersed in a fluid is equal to the weight of the fluid displaced. Using the subscript f for fluid, we have: B = W f = m f g = (ρ f V obj )g (4) where the weight of the fluid has been expressed in terms of its density, ρ f, and its volume, which is equal to the volume of the object, V obj. This principle applies to any object immersed in any gas or fluid, such as a hot-air or helium balloon or a submarine in water. The balloon (plus any weight it carries) is totally supported by the buoyant force of the air it displaces. 2.3.2 Application of Archimedes' principle Using Archimedes' principle we can easily determine whether an object (obj) will sink or float in a given fluid. This clearly depends on whether the buoyant force acting upward on the object, totally immersed in the fluid, is greater or less than the weight of the object. Using Archimedes' principle, as expressed by equation (4) to give the buoyant force in terms of the fluid density ρ f, we can calculate the net downward force as:
4 F net = W B F net = mg - W f F net = g(ρ obj V obj ) g(ρ f V obj ) F net = g(ρ obj - ρ f )V obj (5) Therefore, if the density of the object, ρ obj, is greater than the density of the fluid ρ f, the net force on the object is downward and the object sinks, but if the density of the object is less than that of the liquid, the net force is upward and the object floats. For example, because ice floats in water we conclude that the density of ice is less than the density of water. 3 Experimental Apparatus and Procedure 3.1 Theory of the balance The balance shown in Fig. 4 is used in this experiment to measure unknown masses and forces. An unknown mass, m, placed on the left-hand pan can be determined by finding a combination of standard masses, m s, which when placed on the right-hand pan, have a weight W s = m s g that is equal to the weight of the unknown mass W = mg. In this case, the forces and torques exerted on both sides of the balance are equal and there is no net deflection or rotation of the indicator relative to the scale. The balance is sensitive to the gravitational forces (weight) on the unknown and standard objects and not on their masses. One is able to calibrate the balance in units of mass (grams) only because the acceleration of gravity is constant. (mg = m s g, where g is constant) Note that m = m s when W = W s. The balance used for this experiment has mass riders on two horizontal scales on the front of the balance (as shown in Fig. 4) which are used as the standard mass. Figure 4: The trip balance. 3.1.1 Use of the balance to measure the buoyant force A method for using the balance to measure buoyant forces on objects denser than the fluid is shown in the three diagrams in Fig. 5. The object is first balanced by placing an equal standard mass on the right-hand pan, as shown in the center diagram. The object, suspended on a string attached to the pan, is immersed in the liquid, as shown in the last diagram. The buoyant force is calculated from the amount of mass Δm that must be removed from the right-hand pan to obtain balance in the presence of the buoyant force acting on the object. Figure 5: Method to determine the buoyant force using Archimedes' principle.
The buoyant force is given by: Physics 181- Summer 2011 - Experiment #8 5 B = (Δm)g (6) The density of the liquid may then be found by equating this buoyant force to the expression previously derived in terms of the fluid density and volume of the object: We solve this for the density of the fluid: B = ρ f V obj g (7) ρ f Δm = (8) Vobj We will use this method in the last part of the experiment to determine the density of water, by using an object of known volume and Archimedes' principle. This expression for the unknown fluid density has the dimensions of mass per unit volume. g/cm 3 = g/ml [1 milliliter of volume equals 1 cubic centimeter (cc) or 1 ml = 1 cm 3 ]. 3.2 Measurements 3.2.1 Determination of the density of water This method to determine the density of water consists of measuring the mass of a known volume of water using the balance and a 25-ml pycnometer, shown in Fig. 6. The pycnometer is also known as a volumetric flask, or specific gravity bottle. Its glass stopper has a capillary opening. It should be filled until the fluid level reaches the top of the capillary; only then will the volume of the fluid in the pycnometer equal the number stamped on it. Fill the pycnometer bottle to the top and then, insert the stopper. Wipe off the excess fluid that overflows from the stopper. Method: 1. Record the volume of the pycnometer (not exactly 25.00 ml, be careful!) 2. First, balance the empty pycnometer and its stopper and record its mass. Figure 6: The pycnometer. 3. Fill the pycnometer with water until it is full, and insert stopper. Water will overflow the capillary opening. Wipe off any excess water on the outside of the pycnometer. 4. Measure and record the mass of the filled flask and stopper. 5. Empty the pycnometer when finished.
3.2.2 Measurement of the density of aluminum 6 Method: 1. Use a caliper to measure the three sides, (L a )(L b )(L c ), of the aluminum block. Record your results to the nearest tenth of a millimeter. 2. Measure and record the aluminum block's mass. A paper clip should be on the right pan so that the mass of the paper clip can be ignored. The mass of the thread is negligible. 3.2.3 Verification of Archimedes' principle Carry out the verification of Archimedes' principle using a device consisting of a pair of finely machined concentric cylinders, the cross section of which is shown in Fig. 7. The outer hollow cylinder is closed at one end. The matching solid cylinder fits inside the outer cylinder. Therefore, the volume of the solid cylinder 'precisely' equals the inner volume of the hollow cylinder. Determine whether Archimedes' principle holds by observing the volume of water needed to re balance the scale when the cylinder is immersed in water. Keep in mind that the volume in the hollow cylinder equals the volume of the solid cylinder. Figure 7. Concentric cylinders. Method: 1. Do not place the solid cylinder into the hollow one as the string may break when pulled. 2. Hang the solid cylinder (by use of the paper clip) over the side of the left pan and into an empty glass beaker. 3. Place the hollow cylinder on top of the left pan and balance with the sliding standard masses. Record the value of this combined mass as m 1. 4. Pour water into the glass beaker, making sure that the solid cylinder is totally immersed. 5. Record your observations as to what happened to the balance as you poured in the water. Which side of the pan balance is lifted as you poured in the water? Was the solid cylinder forced up by the buoyant force of water? 6. Remove any air bubbles which may have formed under the solid cylinder. 7. Rebalance the scale, and record this value as m 2. 8. Return the sliding masses so that the scale reads m 1 from step 3. The solid cylinder should be left immersed in the water, so the scale at this point will not be balanced! 9. Add water to the hollow cylinder until the scale is balanced. Be careful not to spill water onto the pan. (Use the eye dropper when the cylinder is about full.) 10. What happened to the left pan as you added water to the hollow cylinder in step 9? What can you
Physics 181- Summer 2011 - Experiment #8 7 say about the volume of water you added to the hollow cylinder and the volume of the water (in the beaker) displaced by the solid cylinder? According to Archimedes's principle, should these volumes be the same? 11. Leave the cylinders separated when finished. Dry the hollow cylinder. 3.2.4 Measurement of the volume of the aluminum cube by Archimedes' principle Method: 1. Hang the aluminum cube (using string and a hook made from a paper clip) into an empty glass beaker. 2. Balance and record its mass, m 1. 3. Fill the beaker with water to totally immerse the cube. Record your observation. Which pan goes up? What causes the unbalance? 4. Remove any air bubbles which may have formed under the cube. 5. Rebalance the scale and record the new mass as indicated by the balance, m 2. 4 Calculations and Analysis of the Data 4.1 Calculation of the density of water 1. Compute the mass of water in the pycnometer by calculating the difference in standard mass used to balance the pycnometer when empty and when full of water. 2. Compute the density of water (g/ml) by dividing this mass in grams by its volume indicated on the side of the pycnometer. 3. Compare your result with the accepted value of the density of water at 20EC (page 9). 4.2 Calculation of the volume of the aluminum cube and its density 1. Calculate the volume of the cube, where V cube,1 = (L a )(L b )(L c ). 2. From its mass and volume, calculate the density of aluminum. Compare it with the accepted handbook value given in the Appendix I on page 9. Calculate the percent difference between the accepted and experimental values for the density of the aluminum block. 4.3 Calculations of buoyant force 1. Using Δm B, calculate the value of the buoyant force on the solid cylinder when water is added to the glass beaker in step 4 of 3.2.3. 2. Note comments in step 5 of 3.2.3, and answer them in this part of your lab report. 3. Give your response to the questions in step 10 of 3.2.3.
4.4 Calculation of the volume of the aluminum cube. 8 As explained in section 2, the density of a liquid is given by: ( Δm) ρ f = V obj where V obj is the volume of the aluminum cube, V cube, - in this case - and where Δm is the amount of mass that must be removed from the right-hand side of the balance to rebalance the aluminum cube when it is immersed in water. 1. Use m 1 and m 2 to calculate Δm for water. In addition, answer part 3 of 3.2.4. 2. Use the handbook value of the density of water at 20EC, given in the Appendix I on page 9, and the corresponding Δm in water, to obtain the volume of the aluminum cube. Compare this value to the volume calculated in step 1 of 4.2 above. Use 4.2 as the accepted value. 5 Questions 1. In a short table, compare your experimental results with the accepted densities. This is a simple repeat of 4.1.3 and 4.2.2. Calculate and list the percent difference between corresponding accepted and experimental values of the densities. 2. Discuss qualitatively the errors (random and/or systematic) in your various measurements of densities. 3. Calculate the buoyant force on the aluminum cube exerted by water using your data from 4.4.1. Use cgs units. 4. 500 ml of glycerin is poured into a liter container. The aluminum cube used in this experiment is dropped into the same container. Would the cube sink? Show your calculations. See Appendix I on page 9. 5. Determine the volume of a hydrogen balloon just large enough to lift a 75 kg person in a 50 kg basket (includes mass of balloon) off the ground. Here, assume that F net = 0.00 N when the balloon starts to lift, and that the weight of the person, basket and inflated balloon, is equal to the weight of the air displaced. A balloon (plus any weight it carries) is totally supported by the buoyant force of the air it displaces. See Appendix I on page 9. 6 Conclusion State your conclusion. What did you learn? How could the experiment be improved?
Physics 181- Summer 2011 - Experiment #8 9 The Measurement of Density and Archimedes' Principle Appendix I Densities of some common materials. (The densities of the gases listed are given for 20EC and 1 atmospheric pressure.) Substance Density (g/cm 3 ) Air 1.293 x10!3 Aluminum Glycerin 2.70 at T = 20EC 1.26 at T = 20EC Hydrogen 0.0899 x10!3 Water 0.99823 at T = 20EC (g/ml = grams/cm 3 = g/cc) Data Sheet 3.2.1 Determination of the density of Water Measurements: 1. Volume of the pycnometer ml 2. Mass of the empty pycnometer and its stopper g 3. Mass of the full pycnometer and its stopper g 4. Temperature of the water o C 5. Error in reading thermometer 6. Error in reading scale balance ± δt = o C ± δm = g Computations: 1. Compute the mass of the water in the pycnometer g 2. Compute the density of the water g/ml 3. Handbook value of the density of the water (page 9) g/ml 4. Percent difference % 3.2.2 Measurement of the density of aluminum Measurements: 1. Length of each side of the aluminum cube: L a = cm, L b = cm, L c = cm 2. Mass of the aluminum cube: g 3. Error in reading caliper ± δl = cm Computations: 1. Volume of the aluminum cube, V cube,1 cc (Note: 1cc = 1cm 3 = 1ml) 2. Density of the aluminum cube g/cc 3. Handbook value of the density of aluminum g/cc 4. Percent difference % 3.2.3 Verification of the Archimedes' principle Measurements: Step 3: Combined mass of solid and hollow cylinders, m 1 g
10 Step 7: Recorded mass when scale is rebalanced after solid cylinder is immersed in water, m 2 g Change in mass due to immersion in water, Δm B = m 1 - m 2 g Calculations: 1. Compute the value of the buoyant force on the solid cylinder when immersed in water in step 4 of 3.2.3. B = (Δm B )g g =980.35 cm/s 2 B = dyne (1 dyne = 1 g-cm/s 2 ) 3.2.4 Second determination of the volume of the cube: Measurements: 1. Mass of the aluminum cube in air g 2. Mass of the aluminum cube immersed in water g Computations: 1. Calculate Δm for water Δm water g 2. Use the handbook value of the density of water at 20EC and the corresponding Δm in water to obtain the volume of the aluminum cube. Compare this value to the volume calculated in 4.2.1. V cube,2 = Δm water / ρ f = cc Percent difference in volumes %