Processes of Science: Physics of Matter. Introduction to Buoyancy


 Anabel Jennings
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1 Processes of Science: Physics of Matter Introduction to Buoyancy Introduction: You already know that it's easier to lift something heavy when it's underwater. Why is this? Because the water helps support the weight of the object. Clearly, when you swim on the surface of the water, the water is supporting 100% of your weight. Scuba divers wear weights on a belt to help them sink under the surface of the water. Why does a fluid (it doesn't have to be water) provide buoyancy? Buoyance is a fluid's capacity to support some or all of the weight of an object submerged in it. How does buoyancy work? You already know that an object that is submerged in a liquid displaces some of that liquid. The level of the liquid rises. It turns out that a submerged object displaces its total volume of the liquid. If an object is only partially submerged, it will displace liquid equal to the submerged volume. In this series of pictures, notice how the level of the water rises as the object sinks. As more of the object goes beneath the surface, more water is displaced.
2 This displaced liquid has mass. The mass depends only on the volume and the density of the liquid. Let's do a couple of short calculations: Let's say that the yellow mass in the pictures above is made of lead. Lead's specific gravity is How much liquid is displaced when the lead object is submerged in alcohol (sg = 0.78), fresh water (sg = 1.00) and glycerin (sg = 1.73)? First of all, we know that the object will sink to the bottom of the container because its specific gravity is greater than the specific gravities of all three of the liquids. So, let's see what happens. Lead object in alcohol: The volume of the object is 7.0 cm 3. You know this because of the rise of the level of alcohol in the picture on the previous page. This means that the displaced alcohol also has a volume of 7.0 cm 3. How do we know this? Again, the level of the alcohol rose 7.0 ml, which is equal to 7.0 cm 3. How much mass does this displaced alcohol have? You know that density includes mass and volume. This makes density a good tool for finding the mass of the alcohol. ρ alcohol = m / V m = ρ alcohol V Do we know the density of alcohol? Not exactly. But we do know the SG alcohol = (ρ alcohol ) / (ρ water ) ρ alcohol = (SG alcohol )( ρ water ) m = ρ alcohol V = (SG alcohol )( ρ water )(V) m = (0.78) (1.00 g/cm 3 ) (7.0 cm 3 ) m = 5.46 g
3 So, 5.46 g of alcohol were displaced. Lead object in fresh water: The volume of the object is 7.0 cm 3. You know this because of the rise of the level of water in the picture on the previous page. This means that the displaced water also has a volume of 7.0 cm 3. How do we know this? Again, the level of the water rose 7.0 ml, which is equal to 7.0 cm 3. How much mass does this displaced water have? You know that density includes mass and volume. This makes density a good tool for finding the mass of the water. ρ water = m / V m = ρ water V Do we know the density of water? Not exactly. But we do know the SG water = (ρ water ) / (ρ water ) ρ water = (SG water )(ρ water ) m =ρ water V = (SG water )(ρ water )(V) m = (1.00) (1.00 g/cm 3 ) (7.0 cm 3 ) m = 7.0 g So, 7.0 g of water were displaced. Lead object in glycerine: The volume of the object is 7.0 cm 3. You know this because of the rise of the level of glycerin in the picture on the previous page. This means that the displaced glycerin also has a volume of 7.0 cm 3. How do we know this? Again, the level of the glycerin rose 7.0 ml, which is equal to 7.0 cm 3.
4 How much mass does this displaced glycerin have? You know that density includes mass and volume. This makes density a good tool for finding the mass of the glycerin. ρ glycerin = m / V m = ρ glycerin V Do we know the density of glycerin? Not exactly. But we do know the SG glycerin = (ρ glycerin ) / (ρ water ) ρ glycerin = (SG glycerin )(ρ water ) m = ρ glycerin V = (SG glycerin )( ρ water )(V) m = (1.73) (1.00 g/cm 3 ) (7.0 cm 3 ) m = g So, g of glycerin were displaced. Let's take a look at a summary of these results. Specific Gravity Volume of Displaced Alcohol cm g Water cm g Glycerin cm g Mass of Displaced The object displaced the same volume in all three liquids, but it displaced a different mass of liquid in each case. Partial Submersion What happens when an object goes partly below the surface, and then floats there? Let's suppose that we have the same size object that we had before, with a volume of 7.0 cm 3. But now, let's reduce the mass a lot. Let's reduce the mass so much that the specific gravity of the object is now This is much less than the very large specific gravity for lead. What happens when we drop this new object (which is
5 now red!) into the same three liquids? Clearly, each of the liquids has a specific gravity that is larger than 0.55, so the object will float in each one. What will it look like? What about the displaced liquids in these cases? Before, with the yellow object, the volume of displaced liquid was the same for all three cases. This is no longer true. However, the buoyant force on the object (the arrow in the pictures) is always equal to the mass of the object. How do we know that? Because the object is floating. The liquid must be supporting the entire weight of the object if it is floating. It would sink otherwise. What is the mass of the object? Again, we will use density and volume to calculate it. ρ object = m / V m =ρ object V m = ρ object V = (SG object )( ρ water )(V) m = (0.55) (1.00 g/cm 3 ) (7.0 cm 3 ) m = 3.85 g So, the mass of displaced liquid is 3.85 g for all three cases. How much liquid is displaced for each case? Light object in alcohol:
6 The volume of the object is 7.0 cm 3. The displaced alcohol has a volume of less than 7.0 cm 3. How much mass does this displaced alcohol have? The mass must be equal to the mass of the object, 3.85 g. What is the volume of this amount of alcohol? ρ alcohol = m / V V = m / ρ alcohol Do we know the density of alcohol? Not exactly. But we do know the SG alcohol = (ρ alcohol ) / (ρ water ) ρ alcohol = (SG alcohol )(ρ water ) V = m /r alcohol = m / [(SG alcohol )( r water )] V = (3.85 g) / [(0.78 ) (1.0 g/cm 3 )] V = 4.94 cm 3 So, 4.94 cm 3 of alcohol were displaced. Light object in water: The volume of the object is 7.0 cm 3. The displaced water has a volume of less than 7.0 cm 3. How much mass does this displaced water have? The mass must be equal to the mass of the object, 3.85 g. What is the volume of this amount of water? ρ water = m / V V = m / ρ water Do we know the density of water? Not exactly. But we do know the SG water = (ρ water ) / (ρ water ) ρ water = (SG water )( ρ water )
7 V = m / ρ water = m / [(SG water )( ρ water )] V = (3.85 g) / [(1.00 ) (1.0 g/cm 3 )] V = 3.85 cm 3 So, 3.85 cm 3 of water were displaced. Light object in glycerin : The volume of the object is 7.0 cm 3. The displaced glycerin also has a volume of less than 7.0 cm 3. How much mass does this displaced glycerin have? The mass must be equal to the mass of the object, 3.85 g. What is the volume of this amount of glycerin? ρ glycerin = m / V V = m / ρ glycerin Do we know the density of glycerin? Not exactly. But we do know the SG glycerin = (ρ glycerin ) / (ρ water ) ρ glycerin = (SG glycerin )(ρ water ) V = m / ρ glycerin = m / [(SG glycerin )(ρ water )] V = (3.85 g) / [(1.73 ) (1.0 g/cm 3 )] V = 2.23 cm 3 So, 2.23 cm 3 of glycerin were displaced.
8 Specific Gravity Volume of Displaced Alcohol cm g Water cm g Glycerin cm g Mass of Displaced Before, when the object sank to the bottom, the same volume but different masses of liquid were displaced. Now, we see that when an object floats, the same mass of liquid is displaced, but different volumes. Principle Concepts: A sunken object displaces its own volume of liquid. A floating object displaces its own mass of liquid. Buoyant force is equal to the weight of the displaced liquid, whether the object is submerged or floating. You should make a special effort to memorize these concepts, and use them.
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