Learning Objectives: Recalling Understanding Applying Analyzing Evaluating Introduction: Buoyancy Statics Force Balance Buoyancy - Archimedes' Principle Hydrostatic Pressure Distribution Depth of water necessary to counter the buoyant force (FB) acting

on the cone(s) Experimental and Theoretical Results and Sources of Errors The principle of buoyancy was first discovered by the Greek scientist Archimedes (287-212 B.C.), which states that "when a body is placed in a static fluid, it is buoyed up by a force that is equal to the weight of the fluid that is displaced by the body.” (Hibbeler 2015). Understanding buoyancy is incredibly important when working in the fields of civil and environmental engineering since buoyant forces are always present when water is a part of the system; e.g. sediment transport in rivers, structural foundations, transportation vis shipping vessels, etc. In this experiment, we will demonstrate the nature of the buoyant force by a simple experiment of a submerge cone. Theory: An object in water, whether floating or submerged, experiences an upward buoyant force. This buoyant force is a result of the pressure difference that the object experiences acting downward on top of the body and upward on the bottom of the body. Recall that pressure is defined as the force acting normal to an area divided by the area. Therefore the average pressure is defined as B.1 P = hr The difference in pressure acting on the object in water is due to the hydrostatic pressure distribution within a fluid. For on incompressible fluid (such as water) the pressure increases as depth increases. The hydrostatic pressure (p) is defined as: p = yh B.2 Since the hydrostatic pressure increases with depth, the force acting on an object in a fluid by the fluid is greater at the bottom than at the top of the object causing a net upward force which is called the buoyant force (FB). F A To study the buoyant force, a simple experiment using a cone submerged in a cylinder of water can be conducted (see Figure B1 below). If the depth of water is decreased, then the pressure acting on the submerge cone decreases. Eventually the cone will become “un-seated” and begin to float upward (i.e. the buoyant force is greater than the weight forces). The moment this occurs the system is considered to be in equilibrium (i.e. the sum of the forces in the vertical direction are equal to zero). The depth of release (hr) can therefore be predicted using Statics: FB-W cone y Aseat 7 B.3 (a) Dcone Equipment and Materials: h Cylindrical acrylic (Plexiglas®) Apparatus o graduated scale (resolution = 0.01 m) O a retractable magnetic pole (b) Caliper (resolution = 0.01 mm) Thermometer (resolution = 0.01 °C) Ptop Wcone Wwater Dseat Figure B1. (a) Schematic of experimental set-up with the system defined by the dotted line and (b) Free Body Diagram of the system Pbottom Three (3) solid acrylic cones with similar dimensions but different masses. Container to transfer water Procedure: 1. Create data tables for data collection. Think about all of the independent and dependent variables that require measurements. 2. Measure the cone dimensions. 3. While holding the cone down using the retractable magnetic pole, fill the cylindrical acrylic (Plexiglas®) Apparatus with water. 4. Retract the magnetic pole and ensure that the cone remains submerged at the bottom. 5. Allow the tank to slowly drain until the cone becomes unseated. The unseating is a process. As the cone begins to unseat, water starts to flow out. This creates a low-pressure region at the bottom of the tank next to the cone causing it to be 'sucked' back into the hole. Be consistent in your determination as to when the cone has become buoyant. 6. Measure the depth when the cone becomes unseated (hå). 7. Repeat steps 4-7 a total of five (5) runs. 8. Repeat steps 4-8 for a total of three (3) cones. Calculations 1. Find Volsubmerged of each cone subjected to buoyant forces. 2. Calculate hr for each cone used in experiment. 3. Calculate descriptive statistics (mean and standard deviation) of the experimental depth. 4. Determine the percent (%) difference between hr's and he's.* *h is the theoretical depth; he is the experimental depth Analysis Show graphically that the buoyant force is a result of the hydrostatic pressure difference. Provide a written description and explanation. Assignment(s): Calculations and Data Analysis Derive the Eq. B.3 for the theoretical depth of release (ht) Answer the following discussion questions: 1. What assumptions were made and did they hold true? Explain using results. 2. What were the sources of error in this experiment? Were they systematic or random errors? How did each of these errors impact the overall results? 3. Explain why the greater the weight of cone, the depth of water at release was lower. 4. Theoretically, would the depth of water at release increase or decrease if the seat diameter increased? Show how mathematically and/or conceptually. (Note that the submerged volume will change.) 5. If the density of the fluid were less than water (e.g. vegetable oil), how would the depth of fluid at release be affected? Show how mathematically and/or conceptually. 9/n