Fluid Dynamics and Wave Properties

Continuity Principle and Flow Rate

• The flow rate (volume of liquid per second) is consistent as liquid moves through pipes.

• The continuity principle is expressed as: $A<em>1v</em>1 = A<em>2v</em>2$, where:

  • $A<em>1$ and $A</em>2$ are the cross-sectional areas of the pipe at two different points.
  • $v<em>1$ and $v</em>2$ are the velocities of the liquid at those points.

Pressure in Open Pipes

• When a pipe is open to the air, the pressure exerted on the liquid is atmospheric pressure.

Bernoulli's Equation: Energy Conservation

• Kinetic energy of the liquid is expressed as $\frac{1}{2}mv^2$, which can be rewritten as $\frac{1}{2}\rho Vv^2$, where $\rho$ is density and $V$ is volume.

• General form of Bernoulli's equation:
  $P<em>1 + \frac{1}{2}\rho v</em>1^2 = P<em>2 + \frac{1}{2}\rho v</em>2^2$

  • $P<em>1$ and $P</em>2$ are the pressures at two different points in the fluid.
  • $\rho$ is the density of the fluid.
  • $v<em>1$ and $v</em>2$ are the velocities of the fluid at those points.

• This equation accounts for scenarios where pressures at the two ends are different (e.g., a motor pushing water).

• If height is a factor, the conservation of energy equation includes the term $mgh$, which is replaced by $\rho Vgh$.

• Bernoulli's equation including height:
  $P<em>1 + \frac{1}{2}\rho v</em>1^2 + \rho g h<em>1 = P</em>2 + \frac{1}{2}\rho v<em>2^2 + \rho g h</em>2$

  • $h<em>1$ and $h</em>2$ are the heights at the two points.

Applications of Flow Rate and Bernoulli's Equation

• Two equations are used to solve problems about liquids in pipes: flow rate conservation and Bernoulli's equation.

• With multiple exits, the sum of flow rates from outgoing pipes equals the flow rate into the incoming pipe.

Specific Example: Liquid Cooler with Nozzle

• Consider a cooler filled with liquid (e.g., lemonade) with a nozzle at a height $h$ below the liquid's surface.

• When the nozzle is opened, the liquid exits with a velocity, traveling a distance $x$ before landing (projectile motion).

• The horizontal distance $x$ is calculated as: $x = v \cdot t$, where $v$ is the exit velocity and $t$ is the time of flight.

• The time of flight $t$ depends on the height: $t = \sqrt{\frac{2h}{g}}$.

• As the liquid level decreases, the landing position moves closer to the cooler.

Mathematical Proof

• The pressure at the nozzle inside the liquid is: $P = P_0 + \rho g h$, where:

  • $P_0$ is atmospheric pressure.
  • $\rho$ is the density of the liquid.
  • $g$ is the acceleration due to gravity.
  • $h$ is the height of the liquid above the nozzle.

• Applying Bernoulli's equation:
  $P<em>0 + \rho g h + \frac{1}{2}\rho v</em>1^2 = P<em>0 + \frac{1}{2}\rho v</em>2^2$

  • $v_1$ is the velocity at the top of the liquid (negligible, ≈ 0).
  • $v_2$ is the velocity at the nozzle.

• Solving for $v<em>2$: $v</em>2 = \sqrt{2gh}$

• As $h$ decreases, $v_2$ decreases, causing the liquid to land closer to the cooler.

Key Concepts for Exam

• Continuity equation.
• Bernoulli's equation.
• Pascal's principle.
• Buoyant force and Archimedes' principle.

Wave Properties: Sound and Light

• Waves are oscillations that propagate energy.
• Sound waves are longitudinal waves, involving compression and rarefaction.
• Tuning forks produce specific frequencies (notes) based on their geometry.
• Vocal cords also produce frequencies based on their geometry.

Intensity and Power of Waves

• "Loudness" is an invented concept; the physical concept is intensity.
• Intensity is the power (energy per second) distributed over a surface area.
• Waves carry energy and have a fixed power unless absorbed by an obstacle.
• Waves propagate as spherical wave fronts, expanding outward.
• The distance between wave fronts is the wavelength $\lambda$.

Wave Speed, Frequency, and Wavelength

• Period (T) is the time for one full oscillation.

• Wave speed (v) is the distance traveled (wavelength $\lambda$) divided by the time (period T):
  $\frac{\lambda}{T} = v$

• Since frequency (f) is the inverse of period ($\frac{1}{T}$), the wave speed is also:
  $v = \lambda f$

• The speed of sound depends on the medium (air, solid, liquid) and temperature; it is approximately 340 m/s in air at 0°C.

• Mechanical waves (like sound) require a medium to propagate.

Frequency as Wave Identifier

• Frequency is the identifier of a wave; it is determined by the oscillator.
• Wavelength changes with the medium, while frequency remains constant.
• If speed changes (v goes down or up), wavelength changes accordingly ($\lambda$ goes down or up).

Perception of Color and Resonance

• Eyes perceive frequency, not wavelength.
• Resonance occurs when the external frequency matches the internal frequency of molecules.
• The ability to see certain colors (frequencies) depends on the composition of the material (e.g., water).