Download Archimedes Principle: Investigating Density through Buoyancy and more Schemes and Mind Maps Physics in PDF only on Docsity! NMU-Physics Archimedes Principle [1] Archimedes Principle Materials: Mass balance in lab stand configuration, two metal masses, one wooden object, a lead or other metal sinker, one bag of pellets, large metal cans and bench stands, string, and micrometer 1 Purpose The goal of this laboratory is to investigate the density of different materials using Archimedes’ buoyancy principle. 2 Introduction The Greek mathematician, physicist, and inventor Archimedes (287 BC - 212 BC) developed his buoyancy principle while supposedly taking a bath. The true root of this story is irrelevant. What matters here is that an object submersed in water experiences a buoyant force, B, equal (in magnitude) to the weight of the fluid which is displaced, Wdisplaced. B = Wdisplaced (1) This buoyant force can then be used to determine the density or volume of the object which is immersed in water (depending on the final goal). This allows the investigator to determine the volume and density of irregularly shaped objects without damaging the object. Here is how it works. Mass Balance Object Water line Figure 1: Schematic of the Archimedes setup for an object more dense than water. If an object is suspended from a mass balance so that the balance is at zero, then the tension in the string is equal to the weight of the object. Now let the object be submerged in a tub of water (Fig. 1). The free body diagrams for this scenario are shown in Fig. 2. In both cases the system is in equilibrium. Therefore, summing the forces for Fig. 2A yields ΣF = T ′ +B −Wdry = 0 (2) where B is the buoyant force and Wdry is the weight of the object when measured in dry air. Looking at Fig. 2B the sum of all forces yields ΣF = T ′ −Wwet = 0 (3) where Wwet is the weight of the object when suspended in water. Solving Eq. 3 for T ′, substituting into Eq. 2, and solving for B yields B = (mdry −mwet)g. (4) NMU-Physics Archimedes Principle [2] T’ B WW T’ (A) (B) Dry Wet Figure 2: (A) Free Body Diagram for an object suspend in water indicating the buoyant force. (B) Free Body Diagram for an object suspend in water indicating the apparent weight (wet weight). Archimedes principle states that the buoyant force is equal to the weight of the fluid displaced by the object (Eq. 1). Setting Eq. 4 equal to Eq. 1 yields mfluid = (mdry −mwet). (5) By substituting for the mass of the fluid the fluid’s density times volume and solving for the volume we get V = (mdry −mwet) ρfluid (6) where the fluid in this case is water. The volume of the fluid displaced is the same as the volume of the object as long as the object is completely submerged. This works well for the case in which the object is more dense than the fluid and will sink on its own. For objects which are less dense than the fluid and float in the fluid, a sinker is necessary to completely immerse the object in the fluid. Figure 3 shows the object in the “dry mass” setup and Fig. 4 shows the object in the “wet mass” setup. Mass Balance Object Sinker Water line Figure 3: Schematic with the floating object suspended in dry air with a sinker attached. Keeping the sinker submerged at all times is important to this experiment. Looking at the FBD’s for the sinker/object system as a whole (see Figure 5, it can easily be seen that the weight of the sinker is in both diagrams and is the same in both cases. By keeping the sinker submerged for both measurements, the weight of the sinker cancels in the force analysis. This can easily be shown by first summing the forces on the two diagrams to get the following two equations. ΣF = T ′ +B −Wdry −Wsinker = 0 (7)
