Chapter 13: Liquids – Pressure, Buoyancy, and Fluid Principles

Liquids

Pressure

Pressure is a fundamental concept in fluid mechanics, describing the force exerted per unit area perpendicular to the surface of an object. It is crucial for understanding how fluids interact with surfaces and objects submerged within them.

• Definition: Pressure is the force per unit area exerted perpendicular to a surface.

• Formula: $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$

• Units: Pascal (Pa), where $1\,\text{Pa} = 1\,\text{N}/\text{m}^2$; also PSI (pounds per square inch).

• Area Dependence: The pressure increases as the area decreases for a given force.

• Example: A person lying on a bed of nails is unharmed because the force is distributed over many nails, resulting in low pressure on each nail.

Pressure in a Liquid

Liquid pressure is the force per unit area that a liquid exerts on an object. It is dependent on depth and the density of the liquid, but not on the volume or shape of the container.

• Depth Dependence: Pressure increases with depth; swimming deeper increases the pressure experienced.

• Directionality: Pressure acts equally in all directions at a given depth.

• Container Shape Independence: Pressure at a particular depth is the same regardless of the container's shape.

• Formula: $\text{Liquid Pressure} = \text{Weight Density} \times \text{Depth}$

• Example: Water pressure from a tower depends on the height (depth), not the amount of water.

• Effects: Liquid pressure acts perpendicular to surfaces; water spurts at right angles from holes in a container, with greater speed at greater depth.

Buoyancy in a Liquid

Buoyancy is the apparent loss of weight experienced by objects submerged in a fluid, resulting from the upward force exerted by the fluid.

• Displacement Rule: A completely submerged object displaces a volume of liquid equal to its own volume.

• Buoyant Force: The net upward force equals the weight of the fluid displaced.

• Forces on Submerged Objects: Two forces act: buoyancy (upward) and gravity (downward).

• Sink or Float:

  • Object sinks if its weight is greater than the buoyant force.

  • Object floats if its weight is less than the buoyant force.

  • Object remains at constant depth if its weight equals the buoyant force.

Archimedes’ Principle

Archimedes’ Principle relates buoyancy to the weight of the fluid displaced by an object. It applies to both liquids and gases.

• Statement: An immersed body is buoyed up by a force equal to the weight of the fluid it displaces.

• Apparent Weight: The apparent weight of a submerged object is its weight out of water minus the buoyant force.

• Density and Buoyancy: Objects with lower density displace more fluid and experience greater buoyant force.

• Applications:

  • Fish adjust their density by expanding or contracting their air bladders.

  • Submarines control their density by taking in or expelling water from ballast tanks.

Flotation

The principle of flotation states that a floating object displaces a weight of fluid equal to its own weight.

• Example: A solid iron block sinks, but the same mass shaped as a bowl floats by displacing more water.

• Salt Water vs. Fresh Water: Salt water is denser, so objects float higher and experience greater buoyant force.

• Life Preservers: Styrofoam-filled preservers float, while lead-filled ones sink despite greater buoyant force.

• Buoyancy in Air: The more air an object displaces, the greater the buoyant force. If the object displaces its weight, it hovers; if less, it descends.

What Makes an Object Float or Sink?

Whether an object floats or sinks depends on its density relative to the fluid and the volume of fluid displaced.

• Rules:

  1. An object more dense than the fluid sinks.

  2. An object less dense than the fluid floats.

  3. An object with equal density remains submerged at constant depth.

• Buoyant Force: Depends on the volume of fluid displaced, not the material of the object.

Pascal’s Principle

Pascal’s Principle states that a change in pressure at any point in an enclosed fluid is transmitted undiminished to all points in the fluid. This principle is foundational for hydraulic systems.

• Applications: Hydraulic presses, auto lifts, and construction equipment.

• Hydraulic Press Example: Pressure applied to a small piston is transmitted to a larger piston, allowing a small force to lift a much larger load.

• Formula: $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$ (applies to both pistons)

• Energy Conservation: Energy output cannot exceed energy input in hydraulic devices.

Surface Tension

Surface tension is the contractive tendency of the surface of liquids, caused by molecular attractions. It explains why drops and bubbles are spherical and why certain objects can float on water.

• Examples:

  • Wire submerged and pulled out stretches a spring.

  • Paintbrush hairs contract when removed from water.

  • Drops and bubbles form spheres due to surface tension.

• Molecular Attraction: Molecules at the surface are pulled inward, creating tension.

• Factors Affecting Surface Tension:

  • Type of liquid (water has higher surface tension than oil).

  • Mixtures (soapy water has lower surface tension).

  • Temperature (hot liquids have lower surface tension).

Capillarity

Capillarity is the rise of a liquid in a fine, hollow tube or narrow space, resulting from adhesion and surface tension. It is important in biological and physical systems.

• Mechanism: Adhesion draws liquid into the tube, surface tension contracts the film, and the liquid rises until balanced by its weight.

• Height of Rise: Depends on the weight of the liquid and the narrowness of the tube; lighter liquids and narrower tubes result in higher capillary rise.

• Examples: Oil rises in a wick, hair gets wet in a bathtub, insects struggle to escape water.