ArchimedesPrinciple

PHY221 – ARCHIMEDES’ PRINCIPLE (revised 11/28/22 J) REQUIRED SUPPLIES AND ANALYSIS TOOLS  Steel ball and plastic ball; large, medium and small nuts; spring scale for measuring masses; graduated cylinder, water, string; a small object that floats in water and is less than one inch across; anything else as needed to make better measurements.  Calculator for doing what calculators do. 1

LAB GOALS By the end of this lab students  Conduct an experiment to confidently verify Archimedes’ Principle and measure the density of an object that sinks and an object that floats..  Identify sources of statistical uncertainty, instrumental precision, and systematic errors.  Write a summary of all the decisions made during the experiment including justifications of those decisions.  Present data and calculations in a well-organized fashion including labeled tables and results that are referenced in the summary and/or conclusion.  Write a detailed conclusion that discusses the outcomes of the investigation specifically detailing results of revisions and iterations. INTRODUCTION Why do some objects float and other objects sink when placed in water? The same question can be asked of floating and sinking objects in air. The key to answering this question is to consider a free-body diagram of an object submerged in a gas or liquid. When the object is completely submerged, but not resting on a solid surface, the force of gravity pulls downward on the object. If you hold the object in place and then release it, the object will do one of three things. If the object rises then there must be a force that opposes gravity and is greater than the weight of the object. (In this lab we are using the term weight as being equivalent to the force of gravity.) If the object remains still then there must be a force that opposes gravity and equals the weight of the object. Finally, if the object sinks then there must be a force that opposes gravity and is less than the weight of the object (the force must be opposing gravity since the object sinks at a rate less than it would under the force of gravity alone). This force that opposes gravity is called the buoyant force . The buoyant force is essentially the result of the fact that as you descend into a gas or liquid the pressure increases. As a result, the object experiences a greater pressure at its bottom than at its top. This difference in pressure gives rise to an upward buoyant force. As an interesting aside, the buoyant force may not always be upward. For example, if the pressure in a gas or liquid increases to the right through the gas or liquid there will be a buoyant force to the left. An example of this is when you have a helium balloon in your car while you are driving down a road. If you accelerate you introduce a fictitious force toward the back of the car (you feel this “force” as it pushes you into your seat like a horizontal force of gravity). The air in your car feels this force as well and the air pressure increases in the back of the car and decreases in the front (let’s assume your windows are closed). The helium balloon, not being fixed to your car, feels a force toward the front of the car, opposite this fictitious force. The balloon thus moves forward as you accelerate forward even though you feel pressed back into your seat. 2

It can be shown that the buoyant force is equal in magnitude to the weight of volume of gas or liquid that the object displaces . (This is true as long as the pressure differences in the gas or liquid are due to differences in depth within the gas or liquid and it is gravity that is causing the pressure differences). This is known as Archimedes’ Principle . This can be understood by considering the differences in pressure on the top and bottom of the object along with the height of the object. You may have seen this in class or in your text, but if not you can find a discussion of this in just about any physics text. As an example, suppose you drop a rock, with a volume of 2 cm 3 , into water. Clearly the rock will sink, but as it sinks it pushes away, or displaces, a volume of water equal to its volume. The buoyant force on the rock will be equal to the weight of 2 cm 3 of water, and since the rock sinks we know that the buoyant force must be less than the weight of the rock. What does this example imply about the density of water compared to the density of the rock? The weight of an object (the rock) is equal to the object’s mass times g, w o = mg, but m = r o V o , where r o is the object’s density and V o is the object’s volume. Thus, w o = r o V o g. Similarly, the weight of the fluid displaced (the water) equals the density of the fluid times the volume of fluid displaced times g. Thus, w f = r f V f g. But, the weight of the fluid displaced equals the buoyant force, F b = r f V f g. (The subscript o is for object and the subscript f is for fluid .) If the object is completely submerged then V o = V f and we have: F b w o = ρ f ρ o or F b = ρ f ρ o w o (1) From this equation we see that if the density of the object is less than the density of the fluid then the buoyant force will be greater than the object’s weight and the object will accelerate upward. If the object’s density is greater than the fluid’s density then the object will sink. Finally, if the densities are equal the object will be in equilibrium and remain in one location, or move at a constant velocity up or down. If you hang a small metal block by a string and lower it into a beaker, or graduated cylinder of water you can determine the density of the metal block by measuring the change in tension of the string. When the block is hanging in the air the tension in the string is equal in magnitude to the weight, w o , of the block (T = w o ). When the block is completely submerged the tension in the string is a measure not of the object’s weight (a.k.a force of gravity), but of the object’s apparent weight, w’ o . Since the tension and the buoyant force are acting upwards while the object’s actual weight is acting downward we can write: w’ o + F b = w o . (2) 3