Hello, and welcome to today's physics lesson. Today, we explore upthrust in fluids, Archimedes' principle, and floatation. We will discover why objects feel lighter in water, why ships made of iron can float, and how submarines dive and surface at will. Let us begin our journey into the fascinating world of fluids and forces.
When you push a ball into a swimming pool, you feel something pushing back. This upward force that fluids exert on immersed objects is called upthrust, also known as buoyant force. We represent it by the symbol F_B.
Upthrust is the upward force exerted by a fluid on a body that is partially or completely submerged in it. The property of fluids to exert this upward force is called buoyancy.
Imagine pushing an empty can into water. At first, it floats with most of it above the surface. As you push deeper, you feel increasing resistance. When fully submerged, the upward push is at its maximum. If you release the can, it bounces back to the surface. This happens because two forces compete: gravity pulls down with the weight of the can, while upthrust pushes up. When upthrust exceeds weight, the can rises.
Let us examine the three key properties of upthrust. First, the larger the volume submerged, the greater the upthrust. Second, for the same submerged volume, the denser the fluid, the greater the upthrust. This is why you feel more buoyant in the salty Dead Sea than in a freshwater lake. Third, upthrust always acts upward through the centre of buoyancy, which is the centre of gravity of the displaced fluid.
Why does upthrust exist at all? The answer lies in fluid pressure. Pressure in a fluid increases with depth. When a block is immersed, its bottom surface sits deeper than its top surface. Therefore, the upward pressure on the bottom exceeds the downward pressure on the top. This pressure difference creates a net upward force. The sideways pressures on the block's vertical faces cancel each other out.
We can prove mathematically that upthrust equals the weight of displaced fluid. Consider a cylindrical body with cross-sectional area A, with its top at depth h₁ and bottom at depth h₂. The downward force on top is pressure times area, or h₁ρgA. The upward force on bottom is h₂ρgA. The net upward force, or upthrust, equals A(h₂ − h₁)ρg. Since A(h₂ − h₁) equals the submerged volume V, this simplifies to Vρg.
Since Vρ is the mass of displaced fluid, Vρg is its weight. Therefore, upthrust equals the weight of the displaced fluid.
This brings us to one of the most important principles in physics. Archimedes' principle states that when a body is immersed partially or completely in a fluid, it experiences an upthrust equal to the weight of the fluid displaced by it.
This principle explains why objects seem lighter in water. The apparent loss in weight exactly equals the upthrust. When you lift a bucket from a well, it feels lighter while underwater. Only when it breaks the surface does its full weight become apparent.
Whether an object sinks or floats depends on the balance between its weight and the maximum possible upthrust when fully immersed. If maximum upthrust exceeds weight, the object floats. If maximum upthrust equals weight, the object floats fully submerged. If maximum upthrust is less than weight, the object sinks.
Since maximum upthrust equals Vρ_L g and weight equals Vρg, we can compare densities directly. An object sinks if its density exceeds the liquid's density, floats fully submerged if densities are equal, and floats partially if its density is less. Here, V is volume in cubic metres, ρ_L is liquid density in kg m⁻³, ρ is object density in kg m⁻³, and g is acceleration due to gravity in m s⁻².
Now let us discuss relative density. Density is mass per unit volume, with S.I. unit kg m⁻³ and C.G.S. unit g cm⁻³. The conversion is simple: 1 g cm⁻³ = 1000 kg m⁻³.
Relative density compares a substance's density to that of water at 4°C. It is a pure ratio with no units. In C.G.S. units, relative density simply equals the numerical value of density in g cm⁻³, since water's density is 1 g cm⁻³. In S.I. units, relative density equals density in kg m⁻³ divided by one thousand.
Archimedes' principle gives us a practical way to measure relative density. For a solid denser than water, relative density equals weight in air divided by loss of weight in water. If the solid dissolves in water, we use another liquid of known relative density instead. For liquids, we compare the loss of weight of a standard solid in the liquid versus in water.
Let us now explore the principle of floatation. When a body floats, its weight exactly balances the upthrust from the submerged portion. This means the weight of the floating body equals the weight of fluid displaced by its submerged part.
The apparent weight of a floating object is zero.
For a floating body, we can derive a beautiful relationship. If total volume is V, submerged volume is v, body density is ρ_s, and liquid density is ρ_L, then equating weight and upthrust gives Vρ_s g = vρ_L g. This simplifies to v/V = ρ_s/ρ_L. Here, V and v are volumes in m³, ρ_s and ρ_L are densities in kg m⁻³, and g is acceleration due to gravity. The fraction submerged equals the ratio of densities. A cork with density 0.24 g cm⁻³ floats in water with twenty-four percent of its volume submerged. Ice with density 0.92 g cm⁻³ floats with ninety-two percent of its volume below water.
These principles explain many everyday phenomena. An iron nail sinks because iron's density exceeds water's, but a ship floats because its hollow structure gives it an average density less than water. Loaded ships sit lower in water, displacing more fluid to balance their greater weight. Ships sink deeper when moving from seawater to river water because freshwater is less dense. The Plimsoll line painted on ships marks the safe loading limit for standard density water.
Submarines use ballast tanks to control their average density. Filling tanks with water increases density, causing descent. Pumping water out and replacing it with compressed air decreases density, allowing ascent.
Icebergs float with about ninety percent hidden underwater, making them dangerous navigational hazards. When floating ice melts in a container, the water level remains unchanged because the melted ice exactly fills the volume that was previously displaced by the submerged portion.
Fish regulate their depth using swim bladders. Adding gas increases volume, decreasing density and causing ascent. Releasing gas has the opposite effect. Balloons rise when filled with light gases like hydrogen or helium, stopping only when the decreasing air density at altitude reduces upthrust to match their weight.
Let us work through an example. Imagine a wooden block of volume twenty-five cubic centimetres floating with twenty cubic centimetres submerged. Using our relationship, density of wood equals submerged fraction times water density: 20/25 times 1 g cm⁻³ equals 0.8 g cm⁻³. The block's weight equals the weight of displaced water: twenty cubic centimetres times 1 g cm⁻³ times g, which is twenty gram-force.
Consider another example: an iceberg with eight hundred cubic centimetres above water. With ice density 920 kg m⁻³ and seawater density 1025 kg m⁻³, the submerged fraction is 920/1025 or about ninety percent. If ten percent equals eight hundred cubic centimetres, the total volume is eight thousand cubic centimetres. Of this, seven thousand two hundred cubic centimetres remain submerged beneath the surface.
Let us recap the key insights from today's lesson. First, upthrust is the upward buoyant force exerted by fluids on immersed objects. Second, Archimedes' principle tells us that upthrust equals the weight of displaced fluid. Third, objects sink or float based on the comparison between their density and the fluid's density. Fourth, relative density provides a convenient unitless comparison to water. Fifth, floating bodies displace fluid whose weight equals their own weight. Sixth, the fraction of a floating body that is submerged equals the ratio of its density to the fluid's density.
These principles govern everything from swimming pools to ocean liners, from weather balloons to deep-sea exploration. Understanding upthrust and floatation opens your eyes to the hidden forces shaping our interaction with the world of fluids.
Thank you for your attention today. Keep questioning, keep exploring, and remember that physics explains the wonders we encounter every day. Until next time, stay curious.