# Lecture 15: Fluid Statics — Pressure and Buoyancy
## Physics I — Mechanics & Thermodynamics
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## Learning Objectives
By the end of this lecture, students will be able to:
1. Define pressure and calculate it in various contexts
2. Apply Pascal's law and the hydraulic press principle
3. Derive and use the hydrostatic pressure equation
4. Read and interpret manometers and barometers
5. State and apply Archimedes' principle to determine buoyant forces
6. Determine whether objects float or sink and calculate the fraction submerged
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## Lecture Content
### I. Fluids and Pressure
A **fluid** is a substance that flows and takes the shape of its container, encompassing both liquids and gases. **Density** (rho) is mass per unit volume, rho = m/V, measured in kg/m^3. Water has a density of 1000 kg/m^3 and air approximately 1.29 kg/m^3 at standard temperature and pressure. **Specific gravity** is the ratio of a substance's density to the density of water and is dimensionless.
**Pressure** (P) is force per unit area: P = F/A, measured in Pascals (Pa), where 1 Pa = 1 N/m^2. Pressure is a **scalar** quantity that acts equally in all directions at a point in a static fluid. Other common pressure units include 1 atm = 101,325 Pa = 760 mmHg = 760 torr = 14.7 psi. **Gauge pressure** is defined as the difference between the absolute pressure and atmospheric pressure: P_gauge = P - P_atm.
### II. Pascal's Law
**Pascal's law** states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid and to the walls of the container. This principle underlies the **hydraulic press**, in which applying a small force on a small-area piston produces a large force on a large-area piston: F_1/A_1 = F_2/A_2, so F_2 = F_1 (A_2/A_1). The mechanical advantage is the ratio A_2/A_1. The trade-off is that the small piston must move a proportionally greater distance, so the work done is the same on both sides: W = F_1 d_1 = F_2 d_2. Hydraulic brakes, car lifts, and hydraulic jacks all exploit this principle.
### III. Hydrostatic Pressure
In a static fluid, pressure increases with depth due to the weight of the fluid above: P = P_0 + rho g h, where P_0 is the pressure at the surface, h is the depth below the surface, and rho is the density of the fluid (assumed constant for incompressible fluids). This can be derived by considering a fluid column of height h and cross-sectional area A. The weight of the column is rho g h A, and the pressure at the bottom is P_0 + weight/A = P_0 + rho g h.
Several key points follow from this equation. Pressure depends only on depth, not on the shape of the container, a result sometimes called Pascal's vases. At the same depth in the same fluid, the pressure is the same regardless of the container geometry. Pressure always acts perpendicular to any surface in contact with the fluid.
Panel A: Three containers of different shapes (narrow cylinder, wide cylinder, and a funnel shape) all filled with the same fluid to the same height h. Pressure gauges at the bottom of each show the same reading P = P_0 + rho g h, illustrating that pressure depends only on depth. Panel B: A hydraulic lift showing a small piston (area A_1, force F_1) connected to a large piston (area A_2, force F_2) by an enclosed fluid. The equation F_2/F_1 = A_2/A_1 is annotated. The small piston moves distance d_1 and the large piston moves distance d_2 = d_1 A_1/A_2.
### IV. Measuring Pressure — Manometers and Barometers
A **mercury barometer** (Torricelli) consists of a tube closed at one end and inverted in a mercury dish. The mercury column rises to a height h where atmospheric pressure supports it: P_atm = rho_Hg g h. At standard atmospheric pressure, h = 760 mm = 0.760 m. An **open-tube manometer** is a U-tube open at one end and connected to a gas at the other. When the gas pressure exceeds atmospheric, the open side is higher, and P = P_atm + rho g h, where h is the height difference. A **closed-tube manometer** has one sealed, evacuated arm and measures absolute pressure. Blood pressure, measured in mmHg (for example, 120/80 mmHg), is expressed as gauge pressure.
### V. Archimedes' Principle
**Archimedes' principle** states that any object wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced: F_B = rho_fluid g V_displaced. This force arises because pressure increases with depth, so the upward pressure on the bottom of the object exceeds the downward pressure on the top. The buoyant force acts at the **center of buoyancy**, which is the centroid of the displaced fluid volume. This principle applies to all fluids and all objects.
### VI. Floating and Sinking
An object sinks if its density exceeds the fluid density (rho_object > rho_fluid), because the weight exceeds the buoyant force. An object floats if its density is less than the fluid density (rho_object < rho_fluid), settling at a depth where the buoyant force exactly equals the weight. For a floating object, rho_object g V_total = rho_fluid g V_submerged, giving the fraction submerged as V_sub/V_total = rho_object/rho_fluid. For example, ice (rho = 917 kg/m^3) floats in water with 91.7% submerged. An object with density equal to the fluid density is **neutrally buoyant** and hovers at any depth. The **apparent weight** in a fluid is W_apparent = W - F_B = mg - rho_fluid g V_object.
Three scenarios side by side. Left: An object denser than the fluid sinks — the weight arrow (down) is larger than the buoyant force arrow (up), with the object at the bottom. Center: An object with the same density as the fluid is neutrally buoyant — weight and buoyant force arrows are equal, object hovers mid-fluid. Right: An object less dense than the fluid floats — weight and buoyant force arrows are equal, but the object is partially above the surface. The fraction submerged is labeled as rho_object/rho_fluid. Each scenario includes the relevant force equations.
### VII. Applications and Examples
**Hydrometers** measure fluid density by observing how deeply they float. **Hot air balloons** rise because heated air is less dense than the surrounding cooler air, making the buoyant force exceed the weight. **Submarines** adjust their buoyancy by filling or emptying ballast tanks with seawater. **Blood pressure** is measured at the level of the heart; measurements taken at a different height require a correction of rho g h. **IV drip bags** are elevated above the patient so that hydrostatic pressure drives the fluid flow. A steel ship floats because the hull encloses a large volume of air, making the average density of the ship (steel plus enclosed air) less than that of water.
A ship floating in water. The hull is shown in cross-section with a waterline. Below the waterline, the volume of displaced water is shaded and labeled V_displaced. The weight of the ship (W = Mg, pointing down from the center of gravity) and the buoyant force (F_B = rho_water g V_displaced, pointing up from the center of buoyancy) are shown as equal arrows. An inset shows that although the steel hull is denser than water, the enclosed air space makes the overall average density less than water.