Unit notes

Density is mass per volume. Pressure is force per area and acts normal to a surface. Gauge pressure is measured relative to atmospheric pressure; absolute pressure includes atmospheric pressure.

Pressure describes normal force per unit area. In a fluid at rest, pressure at a point acts in all directions. Hydrostatic pressure increases with depth because lower fluid supports the weight above it. In a connected uniform fluid at rest, points at the same height have the same pressure under the usual conditions.

Distinguish absolute pressure from gauge pressure. Atmospheric pressure is included in absolute pressure; gauge pressure measures excess above a reference atmosphere. The container’s shape does not directly enter the simple depth relation, although it affects total forces on surfaces.

A submerged object experiences greater pressure on lower surfaces than on upper surfaces, producing an upward resultant force. Archimedes’ principle equates buoyant force to the weight of displaced fluid. Use displaced volume, which may be less than the object’s full volume when it floats partially submerged.

For a floating object at rest with no additional vertical forces, buoyant force equals weight. A fully submerged object need not be in equilibrium: compare buoyancy and weight to determine acceleration. The fluid’s density belongs in the buoyancy expression, not the object’s density.

For steady incompressible flow through a pipe without leaks, volume flow rate Av is constant. A narrower section therefore has a greater average speed. This is an accounting statement about fluid passing through sections per unit time, not a claim that fluid is created or stored in the narrow region.

If density varies appreciably, use mass flow rather than the simplified equal-volume-flow relationship. Turbulence, leakage, or unsteady accumulation can invalidate a simple two-section model. State what is held constant before predicting a speed change.

Along a streamline in the ideal steady, incompressible, nonviscous model, pressure energy per volume, kinetic energy per volume, and gravitational energy per volume sum to a constant. If height is unchanged and speed rises, pressure falls within this model.

Do not apply “faster fluid means lower pressure” without checking height, pumps, losses, and model assumptions. A pump can add energy, and viscosity can dissipate mechanical energy. The pressure difference required to drive real flow through a resistive pipe is not captured by loss-free Bernoulli alone.

A pressure sensor, water column, or force measurement can test hydrostatic predictions. Vary depth while holding fluid density fixed; a pressure-versus-depth graph should have slope ρg under the model. The intercept depends on the pressure reference.

Buoyancy can be investigated by comparing an object’s scale reading in air and while immersed. Account for support forces, contact with the container, trapped air, and whether the object is fully submerged. A lighter apparent weight is not a reduction in the object’s gravitational mass.

For a fluid of constant density at rest, pressure increases by ρgΔh when moving downward. Points at the same depth in a connected resting fluid have the same pressure under the usual assumptions. A wide container and a narrow container can therefore have equal pressure at equal depth even though they contain different total masses of liquid.

Pressure is not the same as total force: multiplying pressure by the relevant surface area gives force when pressure is uniform over that surface. In a hydraulic system, an applied pressure change is transmitted through an ideal confined fluid, so a larger piston area can produce a larger force. The larger force is accompanied by a smaller displacement for the same displaced volume; energy is not multiplied for free.

For steady incompressible flow through a pipe, volume flow rate Av is constant. A narrower section therefore has greater speed, provided there are no branches, leaks, or accumulation. Continuity expresses conservation of mass, not conservation of pressure.

Bernoulli’s equation connects pressure, kinetic-energy density, and gravitational potential-energy density along a streamline for the ideal assumptions. At equal height, greater speed corresponds to lower pressure within that model. A pump, viscosity, turbulence, or substantial energy loss can require additional terms or invalidate the simple application. State the assumptions rather than applying the equation to every fluid situation.

Pressure is force per area and acts normally on a surface in a fluid at rest. The same pressure on a larger area produces a larger force. In a static fluid of approximately uniform density, pressure increases with depth by ρgΔh. Absolute pressure includes the pressure at the reference surface; gauge pressure measures the difference from a chosen reference, often atmospheric pressure.

Pressure at a depth does not depend simply on the total amount of water in the container. Container shape can change wall forces and total weight while the hydrostatic pressure relation remains the same. Compare depths below the same surface under the same fluid and gravity conditions.

In a hydraulic idealization, an applied pressure change is transmitted through the confined fluid. A larger output area can produce a larger force, but the corresponding displacement is smaller under volume conservation. Force multiplication does not create energy.

Buoyant force equals the weight of displaced fluid. For a fully submerged object in a uniform-density liquid, the displaced volume is its submerged volume, not automatically the volume of the entire container. Buoyancy arises from the pressure difference between lower and upper surfaces, with lateral contributions balancing in the simple case.

A floating object in equilibrium displaces enough fluid to balance its weight. The submerged fraction depends on the ratio of average object density to fluid density under the ideal assumptions. An object can float even when some of its materials are denser than the fluid if its overall average density is sufficiently low.

A fully submerged object held by a string may have buoyancy, weight and tension all acting. Do not set buoyant force equal to weight unless equilibrium and the absence of other relevant vertical forces justify it. An apparent-weight measurement can reveal buoyancy through the difference in supporting force.

For steady incompressible flow through a single unbranched passage, volume flow rate Av is constant. A narrower section has greater average speed under that model. This does not imply that every narrow pipe in every real network carries the same flow rate; boundary conditions and branching matter.

Bernoulli’s relation combines pressure, kinetic and gravitational terms along a streamline under suitable ideal conditions. Faster flow at the same height can correspond to lower pressure when those assumptions hold. Pumps, viscosity, turbulence and energy losses require additional accounting rather than blind use of the ideal relation.

Distinguish volume flow rate from speed: one measures volume per time, the other distance per time. State whether a pressure is static, a height is relative to a datum, and the fluid density is approximately constant. These choices determine whether the simplified equations answer the question.

Hydrostatic pressure applies to fluid at rest under gravity. Pressure at a depth depends on the surface pressure, density, and depth, not simply the container’s total volume. Pressure is a scalar; the force it exerts on a surface is normal to that surface and depends on area.

Continuity relates flow through cross-sections when the modeled fluid is incompressible and flow is steady: Av is constant along the flow path. A narrower section then has greater average speed. It does not automatically have a larger volume flow rate, because the same flow passes through each section.

Bernoulli’s relation combines pressure, kinetic, and gravitational terms along an appropriate streamline under ideal steady conditions without relevant dissipative losses or added pump work. Faster flow can correspond to lower pressure when height is unchanged, but do not apply that shortcut when height or energy input differs.

The buoyant force equals the weight of displaced fluid. The displaced volume is the submerged volume, which equals the object’s full volume only when fully submerged. For a floating object at rest without other vertical forces, buoyancy equals weight and the submerged fraction is object density divided by fluid density.

A fully submerged object can rise, sink, or remain in equilibrium depending on its weight and other forces. A scale supporting an object underwater can read an apparent weight smaller than its true gravitational weight because buoyancy supplies part of the support. The object’s mass has not decreased.

For a sealed rigid object in an approximately incompressible fluid, buoyant force is nearly independent of depth because displaced volume and fluid density remain nearly constant. Pressure rises with depth on all sides, but the pressure difference producing buoyancy is tied to the object’s vertical extent.