Why do things float? Introduction Archimedes of Syracuse (ca. 287-212 B.C.), a physical scientist, is credited with understanding two basic principles: When describing the mechanical advantage gained by the use of a lever, Archimedes is claimed to have boasted: Give me somewhere to stand, and I will move the Earth. When having discovered, supposedly while in the midst of his bath, a method for determining whether the King s crown had been made of pure gold or a cheaper alloy, Archimedes is said to have run naked through the streets shouting: Eureka! The second of these two events led to Archimedes Principle which notes that a body immersed in a fluid is buoyed upward by a force equal in magnitude to the weight of the fluid displaced by the body. The upward force or buoyant force is a consequence of increasing pressure with increasing depth within a fluid in a gravitational field. One way to find a leak in a rubber inner tube is to immerse it in water. If you have ever tried this, you know that it is not easy to hold the whole tube under water. The buoyant force that the water exerts on the tube difficult to fight against. A hot air balloon is another example of Archimedes Principle. Theory To apply Archimedes Principle, we must determine the weight of the displaced fluid. To accomplish this, we need to know the density of the fluid and the volume displaced, which equals the volume of the submerged object. Recall we have defined density as the mass per unit volume of a substance. Density is what ultimately determines the buoyant force that a fluid exerts on an object. Remember the units of density are grams per cubic centimeter, or kilograms per cubic meter. The volume of an object depends on the object and how much of it is submerged.
Finding the volume of an object can be tricky, especially if it is not a nice even shape, like a sphere. One way to find an object s volume is to completely submerge it in water. The rise in the volume of water equals the volume of the object submerged. The beauty of this principle is that it does not depend on the shape of the object. Buoyant Force Measurement Suspend a film canister that contains several fishing weights from a spring balance scale as shown on the right side of the figure to the right. Record the weight. Weight suspended in air = Next place the suspended canister in a container with water as shown on the left side of the figure. Again record the weight. Weight suspended in water = Record the difference. Difference in the two measured weights = This difference in the two measured weights is the buoyancy force being applied to the canister by the water. Have you noticed that you can lift a person standing in a swimming pool much easier than when they are standing outside the pool? In fact have you noticed that as you lift them out of the water they seem to get heavier? This is because as you lift them out of the water they have less buoyancy force aiding you. In reality the air is also applying a buoyancy force to the canister. According to Archimedes Principle the buoyancy force is equal to the weight of the fluid being displaced by the object. For air, the weight of the displaced air is small. Continued Buoyant Force Measurements Fill a container to the brim with water. Be sure that the container is completely full. Carefully place the full container inside a larger container. Next, place several fishing weights in a film canister and slowly submerge it in a full container of water. Do not drop the canister into the water and splash water over the edge of the container. Be sure that all of the water that overflows
is collected in the Measure the volume of the water that overflowed using a graduated cylinder and record the value below. Volume of a canister = You now know the volume of the film canister. After drying the external container, repeat the experiment with 1/2-oz fishing weight in a canister. Again be careful not to splash water over the brim when placing the canister into the water. Capture and measure the water overflow as before. Also estimate the fractional amount of the canister is underwater. You may need to use the restaurant server s trick to aid in pouring water from a pitcher and coat the edge of the internal container with a light coat of butter to release the water s surface tension so the water will spill over the edge and not bulge above the containers edge. Volume of a water for a 1/2-oz weight = Next, repeat the measurements with a 3/4-oz weight in the canister. Again be sure to dry the Volume of a water for a 3/4-oz weight = Next, repeat the measurements with two 1/2-oz weights in the canister. Again be sure to dry the Volume of a water for the two 1/2-oz weights (Total 1 oz) = Next, repeat the measurements with 0.75-oz and 0.5-oz weights in the canister. Again be sure to dry the Volume of a water for the 0.75-oz and 0.5-oz weights (Total 1.25 oz) = Next, repeat the measurements with two 0.75-oz weights in the canister. Again be sure to dry the Volume of a water for two 0.75-oz weights (Total 1.5 oz) = Next, repeat the measurements with two 0.75-oz and 0.25 oz weights in the canister. Again be sure to dry the Volume of a water for two 0.75-oz and 0.25 oz weights (Total 1.75 oz) =
Repeat with a 2-oz weight. Again be sure to dry the Volume of a water for a 2-oz weight = What happens to the displaced volume of water as the weights change from 1/2 to 2 oz? What happens to the fractional amount of the canister that is underwater as the weights increase for 1/2 oz to 2 oz?? Calculate and record the density of the canister for each of the three experiments. The density of the canister can be calculated by dividing the canister s mass by its volume. The mass of the canister in each case can be determined by converting the canister s weight to its mass; on the Earth s surface, 1-lb weight is equivalent to 454-gm mass and 1 lb equals 16 oz. Note, we do not know the weight of the canister, we only know the weight of the fishing weights contained within the canister. For now, we will neglect the weight of the canister. Density of the canister with the 1/2-oz weight = gm per cubic centimeter Density of the canister with the 1-oz weight = gm per cubic centimeter Density of the canister with the 1.25-oz weight = gm per cubic centimeter Density of the canister with the 1.5-oz weight = gm per cubic centimeter Density of the canister with the 1.75-oz weight = gm per cubic centimeter Density of the canister with the 2-oz weight = gm per cubic centimeter Compare the density of the canisters with the density of water. Water s density is 1 gm per cubic centimeter. Predict the weight needed to exactly match the buoyant force of the completely submerged canister.
Climate and Global Change Recalling that ice cubes float, what does this imply about the density of ice as compared to water? What does this picture imply about the density of Coke Classic as compared to Diet Coke? http://www.physics.lsa.umich.edu/demolab /graphics2/2b40_u2.jpg Archimedes Principle applies to any fluid in hydrostatic equilibrium. Thus, it is equally valid for gases as for liquids. Therefore in the atmosphere, a parcel of air with density less than its surrounding environmental air will accelerate upward; an air parcel more dense than its surrounding air will accelerate downward. Combining the Ideal Gas Law which indicates that warm air is less dense than cold air if both are at equal pressure and Archimedes Principle, air warmer than its surrounding air rises while air colder than its surrounding air sinks. This is the basic driving force behind convection.