Comprehensive Study Guide on Atmospheric Pressure, Upthrust, and Archimedes' Principle

Demonstration of Pressure Variation using a U-Tube

Experimental Setup and Procedure: * Initial State: Water is poured into a U-tube. The water levels in the two arms, designated as arm X and arm Y, naturally become equal. This state is illustrated in Figure 15.10(a). * Preparation: An air-filled balloon is prepared with its opening tied by a piece of thread that can be easily untied. * Connection: The balloon is connected to arm X of the U-tube and secured with another piece of thread, as shown in Figure 15.14(b). * Activation: The knot on the balloon is slowly undone. * Observations: After removing the knot, the water level in arm X descends, while the water level in arm Y ascends, as shown in Figure 15.10(c).
Scientific Principles behind the Experiment: * Pressure Equality: In a liquid, the pressure at all points on the same horizontal level is identical. * Interpretation of Initial State: The equal liquid levels before the balloon is connected indicate that the pressures acting above the water surfaces in both arm X and arm Y are equal. * Concept of Air Compression: Filling a balloon with air restricts a large amount of air within a limited volume, thereby compressing the air. * Resultant Pressure Change: When the compressed air from the balloon is released into arm X, the water levels shift. The fact that the water level in arm Y is higher than in arm X proves that the pressure at the liquid surface in arm X is now higher than the pressure at the liquid surface in arm Y. * Conclusion: The reason for this higher pressure in arm X is the additional pressure exerted on the liquid by the compressed air originating from the balloon.

Atmospheric Pressure

Definition: Earth's atmosphere extends hundreds of kilometers above the surface. Just as water in a container exerts pressure at any point due to the weight of the water above it, the atmosphere exerts pressure at any point due to the weight of the air above that specific point. This is known as atmospheric pressure.
Historical Measurement (Torricelli's Experiment): * The Italian scientist Torricelli was the first to measure atmospheric pressure. * The Mercury Barometer Instrument: * The device consists of a glass tube approximately $1\,m$ long with one end closed. * The tube is filled completely with mercury ( $Hg$ ). * It is then inverted and immersed into a container of mercury, ensuring no air enters the tube. * When mounted upright, the mercury column in the tube drops by several centimeters, leaving a space at the top. * Standard Measurement: The vertical height of the mercury column remaining inside the tube is approximately $76\,cm$ .
Explanation of the Barometer Mechanism: * Torricelli understood that the mercury does not flow entirely out of the tube because the atmospheric pressure pushes down on the exposed mercury surface in the container. * The atmospheric pressure is strong enough to balance the pressure exerted by a mercury column of $76\,cm$ in height. * The empty space created above the mercury column is a vacuum because no air can enter that space.
Calculations and Variations: * Formula: Pressure at a point inside the tube at the same level as the outside surface corresponds to atmospheric pressure. This can be calculated using the formula: $P = h \rho g$ . * Measurement Units: The height of the mercury column itself is often used as a convenient unit for measuring pressure. * Sea Level baseline: At sea level, the standard height is $76\,cm\,Hg$ . * Experimental Behaviors: * If the tube is immersed deeper into the mercury container, the column height remains at $76\,cm$ , though the vacuum space above it is reduced. * If the tube is inclined (tilted), the physical length of the mercury within the tube increases, but the vertical height remains constant at $76\,cm$ .
Altitude and Pressure Relation: * As one moves upward from sea level, the height of the air column above decreases, which in turn decreases the atmospheric pressure. * Example: At the summit of Mount Everest, the atmospheric pressure is approximately $25\,cm\,Hg$ . * Pressure also fluctuates according to weather conditions.

Alternative Measurement: The Aneroid Barometer

Characteristics: Unlike the mercury barometer, an aneroid barometer does not contain any liquid.
Mechanism: * The device contains a cavity bounded by thin metallic walls from which air has been evacuated. * When outside atmospheric pressure varies, the shape of these thin metallic walls changes (contracting or expanding). * An indicator is attached to the walls; as the walls shift shape, the indicator rotates. * The pressure reading is taken from an attached scale.

Real-World Applications of Atmospheric Pressure

(I) Drinking with a Straw: * Sucking on the straw removes the air inside it, which enters the mouth. * This action reduces the pressure inside the straw. * Because the liquid surface outside the straw is under a higher atmospheric pressure, that pressure pushes the liquid up into the tube and into the mouth.
(II) The Siphon Method: * This method is used to draw water from a higher tank (Tank A) to a lower tank (Tank B). * Procedure: The tube must initially be filled with water. One end is blocked with a finger to prevent flow while it is lowered into Tank A. When the finger is removed at the lower end, flow begins. * Pressure Dynamics: Pressure at the tube end in Tank A equals the sum of the water column pressure above the end plus the atmospheric pressure. The tube end in Tank B is exposed only to atmospheric pressure. The resulting pressure differential pushes water toward Tank B.
(III) The Rubber Sucker: * When pressed onto a glass surface, most of the air between the sucker and the glass is forced out, leaving very little air inside. * The pressure inside the sucker becomes significantly less than the atmospheric pressure outside. * The atmospheric pressure holds the sucker firmly against the surface. * Condition for Success: The sucker only works if there is no air flow between the edge of the sucker and the glass surface.

Worked Example: Pressure Calculations

Data Provided: * Atmospheric pressure at sea level: $76\,cm\,Hg$ * Density of mercury ( $\rho$ ): $13600\,kg\,m^{-3}$ * Acceleration due to gravity ( $g$ ): $10\,m\,s^{-2}$ * Density of water: $1000\,kg\,m^{-3}$
(i) Find atmospheric pressure in Pascals ( $Pa$ ): * Formula: $P = h \rho g$ * Calculation: $P = (76/100\,m) \times (13600\,kg\,m^{-3}) \times (10\,m\,s^{-2})$ * Result: $103360\,Pa$
(ii) Find the height of a water column balanced by this pressure: * Setup: $h \rho g = 103360$ * Calculation: $h \times 1000\,kg\,m^{-3} \times 10\,m\,s^{-2} = 103360$ * Calculation: $h = \frac{103360}{10000}$ * Result: $10.3360\,m$

Upthrust and Archimedes' Principle

Concept of Upthrust: * Upthrust is an upward force exerted by a fluid (liquid or gas) on any object immersed or floating in it. * Floating objects like wood experience this force; even objects that sink weigh less when submerged in water than they do in air.
Experimental Demonstration of Upthrust (Activity 15.3): * A piece of metal is suspended from a spring balance to measure its weight in air. * Applying a downward force increases the balance reading. * Applying an upward force decreases the balance reading. * When the metal is immersed in water, the reading decreases, confirming that water exerts an upward force (upthrust).
Verification of Archimedes' Principle (Activity 15.4): * Experiment: A cubic metal piece is submerged in water at varying levels (partially to fully submerged). Readings from the spring balance and the weight of the displaced water are recorded. * Data Summary (Pupil's Observations): * Stage (a): Metal near surface (not submerged): Spring balance: $1.2\,N$ ; Weight of beaker + water: $1.3\,N$ . Resultant Upthrust: $0\,N$ . Displaced water: $0\,N$ . * Stage (b): Half submerged: Spring balance: $0.9\,N$ ; Weight of beaker + water: $1.6\,N$ . Resultant Upthrust: $0.3\,N$ . Displaced water: $0.3\,N$ . * Stage (c): Fully immersed near surface: Spring balance: $0.6\,N$ ; Weight of beaker + water: $1.9\,N$ . Resultant Upthrust: $0.6\,N$ . Displaced water: $0.6\,N$ . * Stage (d): Fully immersed far from surface: Spring balance: $0.6\,N$ ; Weight of beaker + water: $1.9\,N$ . Resultant Upthrust: $0.6\,N$ . Displaced water: $0.6\,N$ .
Archimedes' Principle (Verbatim Definition): "When an object is partially or completely submerged in a fluid, the upthrust acting on it is equal to the weight of the fluid displaced by the object."

Principles of Floatation

Analysis of Three Objects (Activity 15.5): * Object A: Partially submerged and floating. Upthrust ( $1.1\,N$ ) = Weight ( $1.1\,N$ ). Apparent weight = $0\,N$ . * Object B: Fully submerged and floating. Upthrust ( $1.8\,N$ ) = Weight ( $1.8\,N$ ). Apparent weight = $0\,N$ . * Object C: Sunk at the bottom. Weight ( $2.4\,N$ ) > Upthrust ( $1.9\,N$ ). Apparent weight = $0.5\,N$ .
Summary of Conditions for Floatation and Sinking: * (a) Sinking: If the upthrust on a fully immersed object is less than its weight, the object sinks. * (b) Neutral Buoyancy: If the upthrust equals the weight of the fully submerged object, it floats while remaining completely submerged. * (c) Partial Floatation: If the potential upthrust at full immersion is greater than the object's weight, it will float while partially submerged. It settles at a position where the upthrust of the submerged portion exactly equals the object's weight. If pushed down, it experiences a resultant upward force and returns to this equilibrium position.

The Hydrometer

Purpose: An instrument used to measure the density of liquids and solutions.
Design: Made of glass, featuring a cylindrical stem and a bulb. The bulb contains mercury or lead shots to ensure the device floats vertically.
Operation: * The hydrometer is placed in a liquid; the density is read directly from an integrated scale on the stem. * It operates based on Archimedes' principle: the hydrometer immerses until the weight of the liquid displaced equals its own weight. * Density Correlation: * High Density Liquid: Displaces a smaller volume of liquid to match the hydrometer's weight, so it sinks to a small depth. * Low Density Liquid: Requires a larger volume of liquid to be displaced to provide the necessary upthrust, so the hydrometer sinks deeper.