Lecture 13: Hydrostatics

Learning Outcomes

• Distinguish between the concepts of ‘fluid’, ‘liquid’, and ‘gas’

• Calculate density

• Understand how atmospheric pressure exerts forces on objects

• Distinguish between atmospheric and gauge pressures

• Work with the hydrostatic equation to calculate pressures at different depths in a liquid

• Use the hydrostatic equation to understand pressure measuring devices

• Compute buoyancy forces on objects

Terminology

• Fluids: Substances that can flow / have free moving particles, including liquids and gases.

  • Conforms to the shape of a container.

Density

• Density, denoted as $\rho$, relates volume of a solid or fluid to its mass.

  • Formula: $\rho = \frac{m}{V}$

    • Rearranged:

  • $m = \rho V$

  • $V = \frac{m}{\rho}$

• Unit of measurement: $kg/m^3$

Pressure

• Definition: Pressure is the force applied per unit area, represented by the formula: $P = \frac{F}{A}$

  • Unit: N/m² or Pascal (Pa)

• Example: Calculate the pressure exerted by a student with a mass of 83 kg and a shoe surface area of 0.0290 m².

  • To find the pressure, we first need to calculate the force (weight) exerted by the student, which is given by the formula: $F = m g$ where (g) is the acceleration due to gravity (approximately 9.81 m/s²). Therefore, the force is:

  $F = 83 { kg} \times 9.81 { m/s}^2 = 814.23 { N}$

  Substituting the force into the pressure formula:

  $P = \frac{814.23 { N}}{0.0290 { m}^2} \approx 28,073.79 { Pa}$

• Standard atmospheric pressure:
  $Po = 1 \text{ atm} = 101,325 \text{ Pa}$

• This is calculated using the pressure formula.

The Hydrostatic Equation

• Principle: According to Pascal’s Principle, any pressure applied to a fluid is transmitted undiminished throughout the fluid and to the walls of its container.

• Hydrostatic equation: To find the difference in pressure at a depth, use:

  $\Delta P = \rho g \Delta h$

  Where:

  • $\rho$ (density of the fluid)

  • $g$ (acceleration due to gravity)

  • $\Delta h$ (change in height/depth)

Thus pressure depends on height

• Example:

  

  P = P₀+ $\rho g \Delta h$ (where $P0$ is the atmospheric pressure, $\rho$ is the fluid density)

• P-P₀=$\Delta P = \rho g \Delta h$

Measuring Pressure

• Absolute Pressure: Measured by a barometer.

  • Historical invention by Torricelli.

  • Involves mercury and atmospheric conditions.

• Calculation: $P_{mercury} = 13546 \, kg/m^3$

  • Absolute pressure can also be calculated using:
    $AP = p g A h$

  • Where pressure is P=0$ at the reference point.

Gauge Pressure

• Definition: The pressure relative to atmospheric pressure.

  • Formula:
    𝑃_gauge = 𝑃 − 𝑃_atm

Sphygmomanometer

• Used to measure blood pressure.

  • Cuff should be at the same height as the brachial artery for accurate measurements.

Buoyancy

• Archimedes Principle:

  • The (magnitude of the) buoyancy force, 𝐹_𝐵, on a submerged object is equal to the weight of the fluid displaced

    • 𝐹_𝐵 = 𝜌𝑔$V$

  • Where:

    • $\rho$ = Density of the fluid

    • $V$ = Volume of the displaced fluid

Will the Object Sink or Float?

• To determine if an object will sink or float, compare:

  • Density of the object ($\rho{obj}$) to the density of the fluid ( $\rho{fluid}$)

Example: Iceberg Proportion Above Water

• Density of ice: $\rho_{ice} = 920 \, kg/m^3$

• Density of sea water: $\rho_{water} = 1030 \, kg/m^3$

• Calculate the proportion of the iceberg that is submerged based on these densities.