Fluid Mechanics Quiz

Field of Basic Sciences, Lecture 6

• Course: Physics I (PHY111), Fluid Mechanics
• Essential textbooks:
  • Physics for Scientists and Engineers with Modern Physics, by Raymond A. Serway, John W. Jewett, Jr., 8th edition (2010)
  • Fundamentals of Physics, by Halliday, Resnick, Walker
• Instructor: Dr. Shehab E. Ali
• Date: 21/11/2023

Chapter 14: Fluid Mechanics

• Topics include:
  • Pressure
  • Variation of Pressure with Depth
  • Pressure Measurements
  • Buoyant Forces and Archimedes' Principle

States of Matter

• Solids: Have a definite volume and shape.
• Liquids: Have a definite volume but not a definite shape.
• Gases: Have neither a definite volume nor shape (unconfined).

States of Matter, continued

• The time it takes a substance to change shape in response to external force determines whether it's treated as a solid, liquid, or gas. The definitions in previous slides are somewhat artificial.

Fluids

• Fluids are collections of randomly arranged molecules held together by weak cohesive forces plus forces from the container walls.
• Both liquids and gases are fluids.

Statics and Dynamics with Fluids

• Fluid Statics: Describes fluids at rest.
• Fluid Dynamics: Describes fluids in motion.
• The same physical principles of statics and dynamics apply to fluids.

Forces in Fluids

• Fluids do not sustain shearing or tensile stresses.
• The only stress in a static fluid is one tending to compress an object from all directions.
• The force exerted by a static fluid on an object is always perpendicular to the surfaces of the object.

Pressure

• Pressure (P) is the force (F) per unit area (A): P = F/A
• The unit of pressure is the Pascal (Pa), where 1 Pa = 1 N/m².

Pressure, Continued

• Pressure is a scalar quantity, proportional to the force magnitude.
• If pressure varies over an area, calculate the force (dF) on a small area (dA) as dF = P dA.

Pressure vs. Force

• Pressure is a scalar, force is a vector.
• The direction of a force that produces pressure is perpendicular to the area of interest.

Measuring Pressure

• A calibrated spring measures the force exerted by a fluid, providing pressure readings.

Density Notes

• Density is mass per unit volume for a substance.
• Density values vary slightly with temperature because volume is temperature-dependent.
• The molecular spacing in a gas is much larger than in a solid or liquid.

Density Table (Table 14.1)

• Lists densities of common substances at standard temperature and pressure (0°C, atmospheric)

Variation of Pressure with Depth

• Fluid pressure varies with depth.
• If a fluid is at rest, all portions must be in static equilibrium.
• All points at the same depth are at the same pressure (otherwise, the fluid would not be in equilibrium).

Pressure and Depth

• Examining a sample of liquid within a cylinder:
  • Cross-sectional area (A)
  • Depth from d to (d+h) below the surface
  • External forces act on the liquid sample.

Pressure and Depth, Continued

• The liquid has a uniform density (p). This implies an incompressible liquid.
• External forces acting on the sample:
  • Downward force (PA) on the top.
  • Upward force (PA) on the bottom.
  • Downward gravity force (Mg).
• Mass (M) = p*V = ρAh

Pressure and Depth, Final

• ΣF = 0 (net force must be zero).
• P = P₀ + ρgh
• Pressure (P) at depth (h) is greater than pressure (P₀) at a shallower point by ρgh. (P₀ is atmospheric pressure).

Atmospheric Pressure

• When a liquid is open to the atmosphere (no lid or cover): P₀ is atmospheric pressure.
• 1.00 atmosphere (atm) is approximately equal to 1.013 x 10⁵ Pa.

Pascal's Law

• Pressure in a fluid depends on depth and P₀.
• Pressure increase at the fluid surface transmits undiminished to all other points in the fluid.

Pascal's Law, Continued

• Named after Blaise Pascal.
• A change in pressure applied to a fluid is transmitted undiminished to every point of the fluid and the container walls.
• P₁ = P₂
• F₁/ A₁ = F₂ / A₂

Pascal's Law, Example

• Hydraulic press diagram with input and output forces and areas.
• Large output force can be achieved using a small input force, due to ratios in pressure and areas.
• Volume of fluid displaced equals the volume of fluid displaced elsewhere in the closed system.

Pascal's Law, Example (Continued)

• Volume displacement on one side (ΔV₁) = volume displacement on other side (ΔV₂).
• Work₁ = Work₂

Pascal's Law: Other Applications

• Hydraulic brakes
• Car lifts
• Hydraulic jacks
• Forklifts

Pressure Measurements: Barometer

• Invented by Torricelli.
• A long, closed tube filled with mercury, inverted into a dish of mercury.
• Measures atmospheric pressure.
• P₀ = ρ_Hggh
• 1 atm = 0.760 m (of Hg).

Pressure Measurements: Manometer

• A device measuring gas pressure in a container.
• One end of a U-shaped tube is open to the atmosphere.
• The other end is connected to the container with the gas.
• Pressure at point B = P₀ + ρgh

Absolute vs. Gauge Pressure

• Absolute pressure (P) = P₀ + ρgh
• Gauge pressure = P - P₀

Buoyant Force

• The buoyant force is the upward force exerted by a fluid on any immersed object in the fluid.

Buoyant Force, Continued

• The magnitude of the buoyant force equals the downward gravitational force.
• The buoyant force is the resultant force due to all forces applied by the fluid to the submerged object.

Archimedes

• Greek mathematician, physicist, and engineer (c. 287 – 212 BC).
• Calculated ratio of circumference to diameter of a circle.
• Discovered the nature of buoyant force.
• Inventor (catapults, levers, screws, etc.).

Archimedes's Principle

• The magnitude of the buoyant force equals the weight of the fluid displaced by the object.
• Archimedes’s Principle does not assess the material properties of the object.

Archimedes's Principle, Continued

• Pressure at the top and bottom of a cube submerged in a fluid creates forces that balance the gravitational force.

  • B = (P_bot - P_top) A = ρ_fluid g V = Mg

Archimedes's Principle: Totally Submerged Object

• An object totally submerged in a fluid experiences an upward buoyant force equal to weight of the fluid displaced.
• The downward gravitational force is equal to the object's weight.
• Net force = (Density of fluid - Density of object) * g * Volume of object.

Archimedes's Principle: Totally Submerged Object, Continued

• If object density < fluid density, the object accelerates upward.
• If object density > fluid density, the object sinks.
• The direction of the motion of an object is based solely on the densities of the fluid and object.

Archimedes's Principle: Floating Object

• A floating object is in static equilibrium.

• The upward buoyant force equals the downward force of gravity.

• Volume of fluid displaced = volume of object below the fluid level:

  V_fluid = ρ_object/ρ_fluid * V_object

Archimedes's Principle: Floating Object, Continued

• The fraction of a floating object submerged is equal to the ratio of its density to the fluid density.

Archimedes's Principle, Crown Example

• Archimedes was supposedly asked if a crown was made of pure gold.
• Crown's weight in air = 7.84 N.
• Crown's weight in water = 6.84 N.
• Buoyant force = apparent weight loss.
• Difference in scale readings = buoyant force.

Archimedes's Principle, Crown Example (Continued)

• ∑ F = B + T₂ - Fg = 0
• B = Fg -T₂
• Archimedes's Principle: B = ρgV (weight (in air) minus apparent weight (in water) )
• Determining the volume and density of the crown to determine if it's pure gold—solving ρ_crown = m_crown / V

Archimedes's Principle, Iceberg Example

• What fraction of an iceberg is below water?
• The iceberg only submerges some, so Vs/Vi = ρi/ρs relation applies.
• The fraction below the water = ratio of the volumes of sections (V_s/V_i).

Archimedes's Principle, Iceberg Example, Continued

• V_ice = total volume of the iceberg
• V_water = volume of the displaced water V_water = volume of the iceberg submerged
• Approximately 89% of the iceberg is beneath the surface.

Test your knowledge on the fundamental concepts of fluid mechanics. This quiz covers topics such as the characteristics of solids and fluids, fluid dynamics, and pressure measurement. Perfect for students studying physics or engineering.