Lecture on Fluid Mechanics

Topics Covered

• Density
• Buoyant Force
• Archimedes' Principle
• Pascal's Law
• Hydraulic Lifts
• Bernoulli's Equation

Density

• Definition: Density = Mass / Volume
• Symbol: (\rho) (rho)
• Water Density: 1 g/cm³ = 1000 kg/m³
• Specific Gravity: Density of a substance compared to water
  • Example: Aluminum's specific gravity = 2.7

Buoyant Force

• Weight = Mass (\times) Gravity
• Mass = Density (\times) Volume
• Buoyant Force = Density (\times) Volume (\times) Gravitational Acceleration

Archimedes' Principle

• Buoyant force = Weight of the fluid displaced by the object
• Heavy objects sink; Light objects float
• Example: Ice floats in water because its density is less than water

Behavior of Gases

• Helium rises; Carbon Dioxide sinks
• Air composition: 78% Nitrogen, 21% Oxygen
• Hot air is less dense than cold air

Pressure

• Definition: Pressure = Force / Area
• Unit: Pascal (Pa)
• 1 atm = 101,300 Pascals = 760 mmHg
• Pressure decreases with increased velocity (Bernoulli's Principle)

Pascal's Law

• Pressure applied to a confined fluid is transmitted equally throughout
• Hydraulic Lift: (F_2 = F_1 \times \frac{A_2}{A_1})
  • Example: Mechanical advantage

Hydraulic Lift Problems

• Output force increased by a factor
• Work input = Work output ( (F_1 \times d_1 = F_2 \times d_2))

Mercury Barometer

• Measures atmospheric pressure
• Calculating height difference using fluid density and gravitational acceleration

Fluid Flow Rate

• Mass Flow Rate: (\dot{m} = \rho \times A \times v)
• Volume Flow Rate: (Q = A \times v)
• Continuity Equation: (A_1v_1 = A_2v_2)

Bernoulli's Equation

• Describes energy conservation in fluid flow
• Equation: (P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2)
• Applications: Explains lift in airplane wings, pressure changes in pipes

Practical Application Examples

• Calculating fluid velocity exiting a tank (Torricelli's Law)
• Force exerted by wind on structures
• Pressure differences due to varying heights and velocities

Key Takeaways

• Fluids include liquids and gases due to their ability to flow.
• Density and pressure are crucial in determining fluid behavior.
• Buoyant force is a direct application of Archimedes' Principle, crucial in understanding floating and sinking objects.
• Pressure and velocity are inversely related in fluid systems, as described by Bernoulli's Principle.
• Hydraulic systems leverage Pascal's Law to multiply force.

Important Formulas

• Density: (\rho = \frac{m}{V})
• Pressure: (P = \frac{F}{A})
• Buoyant Force: (F_b = \rho Vg)
• Bernoulli's Equation: (P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2)
• Continuity Equation: (A_1v_1 = A_2v_2)