Buoyancy PDF

# Upthrust ## What is Upthrust? - The upward force which acts on a body by the fluid (liquid/gas) when the body is partially or completely submerged in it is called upthrust or buoyant force. - SI unit: Newton (N) ## In which direction does upthrust act? - Upthrust always acts in the upward dir...

# Upthrust ## What is Upthrust? - The upward force which acts on a body by the fluid (liquid/gas) when the body is partially or completely submerged in it is called upthrust or buoyant force. - SI unit: Newton (N) ## In which direction does upthrust act? - Upthrust always acts in the upward direction. - The point through which upthrust acts is the center of buoyancy. ## What is buoyancy? - The property of a fluid to exert an upthrust on a body immersed in it. ## Characteristics of Upthrust 1. Upthrust increases with the increase in the volume of the body submerged in the fluid. It is maximum when the object is completely immersed. 2. Upthrust increases with the increase in the density of the fluid in which the body is submerged. 3. Upthrust acts on a body away in an upward direction and is equal to the weight of the displaced liquid, i.e., the center of gravity of displaced liquid. ## A bunch of feathers and a pebble of the same mass are allowed to fall. Which will fall faster? 1. **In vacuum:** Both will fall at the same time due to the absence of upthrust. 2. **In air:** The pebble will fall faster. Since the volume of the bunch of feathers is more than a pebble of the same mass, it will experience more upthrust by air and will take more time to fall. ## A body experiences an upthrust F₁ in a river and F₂ in the sea. When dipped to the same level, which one is more and why? - F₂ > F₁ - This is because upthrust by a fluid increases with the increase in density of the fluid. So, the upthrust by seawater is more due to its density. ## A small block of wood is held completely immersed in a) Water b) Glycerine and then released. In which case will you observed submerged portion is more? - Density of glycerine is more than water. So, it will exert more upthrust on the block of wood. Thus, the submerged portion is more in water. ## A cylindrical body PQRS of cross-sectional area A immersed in a liquid of density ρ. - Let us take a cylindrical body PQRS of cross-sectional area A immersed in a liquid of density ρ. - At depth h₁, the liquid pressure on surface PQ = h₁ρg - Downward thrust on PQ, F₁ = h₁ρgA - At depth h₂, the liquid pressure on RS, F₂ = h₂ρgA - Upward thrust on RS = h₂ρgA - At various points on the vertical sides of the body, the horizontal thrust is balanced as liquid pressure is same at all points. - As h₂ > h₁, so F₂ > F₁ - The resultant upward thrust/ buoyancy force: - F_B = F₂ - F₁ = (h₂ - h₁)ρgA = hρgA (h = h₂ - h₁: height of cylinder) - hρgA = volume of submerged portion of body × density of liquid × g - = volume of liquid displaced × density of liquid × g - = weight of liquid displaced ## A body of volume V and density ρ is kept completely immersed in a liquid of density ρ_L. If g = acceleration due to gravity: #### Weight of the body: Vρg #### Upthrust on the body: Vρ_Lg #### Apparent weight of the body in liquid: Real weight – weight of displaced liquid = Vρg - Vρ_Lg = V(ρ - ρ_L)g #### What is the effect of upthrust on a body? - The effect of upthrust on a body submerged in a liquid is that the body appears to be of less weight than the actual weight. ## A body will weigh more in a vacuum/air? Why? - A body will weigh more in vacuum/air because it will appear to be of less weight than actual weight due to the upthrust of air. ## It is easier to lift a heavy stone underwater than in air, why? - The stone appears to be of less weight than its actual weight under water due to the upthrust, so it is easier to lift a heavy stone underwater than in air. # Archimedes’ Principle - It states that when a body is immersed partially or completely in a liquid, it experiences an upthrust, which is equal to the weight of the liquid displaced by the body. Due to this upthrust, there is an apparent loss in weight of the body = Upthrust on the body. ## Archimedes’ principle applicable to only liquids or gases as well? - The Archimedes’ principle is applicable to both liquids and gases. - It is also applicable to the position of a body submerged in a liquid. - Upthrust (Buoyant force) = V × ρ_L × g - V = volume of body submerged in the liquid - ρ_L = density of liquid - g = acceleration due to gravity. - = volume of liquid displaced × density of liquid × g - = mass of liquid displaced × g - = weight of liquid displaced by the boy. - Volume of liquid (fluid) displaced = Volume of the submerged portion of the body immersed. ## A body held immersed in a liquid, considering forces along the horizontal are balanced. What are the two forces acting along the vertical direction and how do these forces determine whether the body will sink, float or remain in liquid when released? - Considering forces along the horizontal are balanced, the two forces acting along the vertical direction are: - Downward force i.e., weight of the body (W) - Upward force i.e., upthrust (F_B) acting on the body. - Different cases can arise: - If the body sinks, W > F_B: Vρg > Vρ_Lg; ρ > ρ_L - Due to the net force (W-F_B) acting downwards, the body will sink. - When the body floats, W < F_B, the body is completely immersed but just below the surface of the liquid: Vρg < Vρ_Lg; ρ < ρ_L - If the body partially immersed such that W = F_B, the body floats: Vρg = Vρ_Lg; ρ = ρ_L - The amount of the submerged part of the body now, the upthrust will balance the weight of the body: W = F_B: Vρg = Vρ_Lg - This means the volume of submerged portion of the body, V = V_solid = V ## If the density of a substance = ρ_s immersed in a liquid of density ρ_L, state the relation between ρ_s and ρ_L for the body floating and sinking. - ρ_s > ρ_L: The body will sink. - ρ_s = ρ_L: The body will float completely immersed in a liquid. - ρ_s < ρ_L: The body will float partially immersed with part of its volume submerged in the liquid. ## State the principle of flotation - It states that the weight of a floating body is equal to the weight of the liquid displaced by the submerged part. - This apparent weight of a floating body = 0. ## A body with volume V and density ρ_s floats with volume V_s inside a liquid of density ρ_L. What is the relation between the volume of submerged part of floating body, density of liquid and the body? - Weight of the body, Vρ_sg - Weight of displaced liquid, V_sρ_Lg - According to the principle of flotation, Vρ_sg = V_sρ_Lg - Therefore, V/V_s = ρ_L/ρ_s # If the density of ice is 0.9 g cm^-3, what portion of an iceberg will remain below the surface of water in the sea? - Density of ice = 0.9 g cm^-3 - Density of water = 1 g cm^-3 - Fraction of the volume of the Iceberg that remains below the surface of sea water = 0.9/1 = 9/10th ## Why are icebergs floating in sea water dangerous for ships? - Icebergs are lighter than water. They float in the sea with the major portion inside the surface of seawater (almost 9/11th). The rest of the Iceberg remains visible outside. The displacement of sea water (almost 2/11th), thereby causing accidents. ## Why does an iron nail float on mercury but sink in water? - The density of iron is less than that of mercury. So, it floats in mercury but sinks in water as the density of iron is more than water. ## An iron nail sinks in water, iron ship floats in water? - An iron ship has more volume. Its density is less hence, it experiences more upthrust, whereas volume of an iron nail is less; hence floats in water with part of its volume submerged in water, while iron nail sinks more, its density is much more so, it sinks. ## A ship is sailing from river water to the sea water: - In which case will it experience more upthrust? - In sea water, the upthrust is more due to the more density of sea water. - In which case will the submerged portion of the ship be more? - The submerged portion of the ship will be more in river water. ## A man swimming in sea water and then river water. In which case will he find easier to swim? - A man swimming from sea water to river water will find easier to swim because sea water density decreases in a river, so upthrust becomes less. Thus, the submerged portion of his body becomes less. The upthrust will balance his weight and he can float. ## A balloon filled with hydrogen rises up to a certain height and then stops rising further. Why? - A balloon filled with hydrogen rises up to a height as long as the density of air is lesser than that of hydrogen. As soon as the density of the air is equal, the balloon provides upthrust, which is greater than the weight of the balloon. As the balloon moves up, the upthrust balances the weight of the balloon. It stops rising further and keeps floating. # Factors affecting the buoyancy - Buoyant force (upthrust) = weight of the displaced liquid = mg = Vpg - Thus, the magnitude of buoyant force depends on: - Buoyant force ∞ V - Buoyant force increases as the volume of the object immersed in the liquid increases. Buoyant force is maximum when the object is completely immersed in the liquid. - Buoyant force ∞ ρ - As the density of the liquid increases, the buoyant force exerted by the liquid on the object also increases. For example, it is easier to swim in seawater than in river water because the density of seawater is more than the river water. Thus, the weight of the displaced seawater by a swimmer is more than the weight of the displaced river water, i.e., the swimmer experiences a greater buoyant force in seawater than in river water. # Experimental Verification of Archimedes’ Principle - Objective: To establish the relation between the loss in weight of a solid, and the weight of water displaced when the solid is fully immersed in the liquid. - Experiment: - Take a jar and a beaker placed under the spout of the jar. - Pour water into the jar up to the level of the spout. - Suspend a solid by a thin thread from the hook of a spring balance and measure the weight of the body in the air. - Now carefully lower the body into the jar and measure its new weight when it is completely immersed in the water. - Collect the displaced water in the beaker, and measure its weight. - Result: It is found that the weight of the solid in the air – weight of the solid when immersed in the liquid = weight of the displaced water, - Or, the loss in weight of the body = weight of the displaced liquid. - Therefore, Archimedes’ principle is verified. # Practical Applications of Archimedes’ Principle - Archimedes’ principle has many practical applications. Some are given below: - Hydrometer: Used in determining the density of liquids, which is an instrument to measure the relative density of liquids. - Lactometers: used for measuring the purity of milk, which is based on the Archimedes’ principle. - Archimedes’ principle is used in designing ships and submarines. - Ship - A ship floats on the surface of the sea because the volume of water displaced by the ship is enough to have a weight equal to the weight of the ship. - A ship is constructed in a way so that the shape is hollow, to make the average density of the ship lesser than the sea water. - Therefore, the buoyant force acting on the ship is large enough to support its weight. - A ship submerge lower in freshwater as freshwater density is lesser than seawater. - Ships will float higher in cold water as cold water has a relatively higher density than warm water. - Submarine - A submarine has a large ballast tank, which is used to control its position and depth from the surface of the sea. - A submarine submerges by letting seawater into the ballast tank so that its weight becomes greater than the buoyant force (and vice versa). - It floats by reducing seawater in the ballast tank. Thus its weight is less than the buoyant force. - Hot-air balloon - The atmosphere is filled with air that exerts buoyant force on any object. - A hot-air balloon rises and floats due to the buoyant force which arises by the surrounding air. - It descends when the balloon weight is more than the buoyant force. - It becomes stationary when the weight = buoyant force. - The weight of the hot-air balloon can be controlled by varying the quantity of hot air in the balloon. # Why the objects sink or float in a liquid - When an object is put in a liquid, then two forces act on it - Weight of the object acting vertically downwards - Buoyant force (or upthrust) acting upwards - There are three cases: - If the buoyant force (or upthrust) exerted by the liquid is less than the weight of the body then, the object will sink in the liquid. - If the buoyant force (or upthrust) exerted by the liquid is equal to the weight of the body then, the object will float in the liquid. - If the buoyant force (or upthrust) exerted by the liquid is greater than the weight of the body then, the object will rise in the liquid and float. - All bodies experience a force of buoyancy when they are immersed in a liquid. Objects of density less than that of a liquid float on the liquid. The objects of density greater than that of a liquid sink in the liquid. - For example, the cork floats while the nail sinks. This happens because of the difference in their densities. The density of cork is less than the density of water so, the upthrust of water on the cork is greater than the weight of the cork. So, it floats. - The density of an iron nail is more than the density of water so, the upthrust of water on the iron nail is less than the weight of the nail. So, it sinks. # Law of floatation - According to this law, a body will float in a liquid if the weight of the body is equal to the weight of the liquid displaced by it. - A floating body may be partly or totally submerged in the liquid. - The liquid is displaced by that portion of the body which is submerged under the liquid. - If the weight of the immersed body is more than the weight of the water displaced, the body will sink. - In determining whether a given body will float in a given fluid both weight and volume must be considered; that is, the relative density, or weight per unit of volume, of the body compared to the fluid determines the buoyant force. - If the body is less dense than the fluid, it will float or, in the case of a balloon, it will rise. If the body is denser than the fluid, it will sink. - Relative density also determines the proportion of a floating body that will be submerged in a fluid. - If the body is two thirds as dense as the fluid, then two thirds of its volume will be submerged, displacing in the process a volume of fluid whose weight is equal to the entire weight of the body.